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0 | AdHoc | http://planetmath.org/AdHoc | <!DOCTYPE html>
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ad hoc
</title>
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<h1 class="ltx_title ltx_title_document">
ad hoc
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The Latin phrase
<em class="ltx_emph ltx_font_italic">
ad hoc
</em>
translates as “toward this”, and is used in mathematics to describe anything that has been made up specifically for one particular purpose.
</p>
</div>
<section class="ltx_subsubsection" id="S0.SS0.SSSx1">
<h2 class="ltx_title ltx_title_subsubsection">
Examples
</h2>
<div class="ltx_para" id="S0.SS0.SSSx1.p1">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
an ad hoc
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
is a temporary definition that might
be useful for the discussion at hand. For instance, let a
<em class="ltx_emph ltx_font_italic">
nice set
</em>
be a closed set with smooth boundary.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
If a proof makes use of an ad hoc construction, the proof
(typically) does not make use of a general theory for the problem.
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="S0.SS0.SSSx1.p2">
<p class="ltx_p">
See also
<span class="ltx_text ltx_font_typewriter">
http://en.wikipedia.org/wiki/Ad_hoc
</span>
Article on ad hoc on Wikipedia
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
ad hoc
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
AdHoc
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:45:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:45:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
9
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00-01
</td>
</tr>
</tbody>
</table>
</div>
</section>
</article>
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0 | AttachmentToTextbookProjectsOnPlanetMath | http://planetmath.org/AttachmentToTextbookProjectsOnPlanetMath | <!DOCTYPE html>
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Attachment to Textbook projects on PlanetMath
</title>
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<h1 class="ltx_title ltx_title_document">
Attachment to Textbook projects on PlanetMath
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
List of current Textbook Projects on PlanetMath:
</h2>
<div class="ltx_para" id="S1.p1">
<ol class="ltx_enumerate" id="I1">
1.
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/?op=getobj&from=books&id=289
</span>
Abelian and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
nonabelian
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/nonabelianstructures">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/nonabeliantheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/noncommutativestructure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
mathematics applications in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
quantum theories
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quantumoperatoralgebrasinquantumfieldtheories">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumfieldtheoriesqft">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and natural sciences
<math alttext="http://planetmath.org/?op=getobj&from=books&id=289" class="ltx_Math" display="inline" id="I1.m1">
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289
</mn>
</mrow>
</math>
2.
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/?op=getobj&from=books&id=288
</span>
<a class="nnexus_concept" href="http://planetmath.org/topicentryonthealgebraicfoundationsofmathematics">
Quantum Algebra
</a>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Symmetry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/symmetricrelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumsymmetry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="http://planetmath.org/?op=getobj&from=books&id=288" class="ltx_Math" display="inline" id="I1.m2">
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</mo>
<mn>
288
</mn>
</mrow>
</math>
3.
<span class="ltx_text ltx_font_typewriter">
http://planetphysics.us/?op=getobj&from=books&id=436
</span>
Higher dimensional algebroids and crossed modules
<math alttext="http://pages.bangor.ac.uk/~{}mas010/mosa-thesis.html" class="ltx_Math" display="inline" id="I1.m3">
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</mrow>
</math>
4.
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/?op=getobj&from=objects&id=12478
</span>
<a class="nnexus_concept" href="http://planetmath.org/mvlogicsinthecategoryofautomata">
Q-logics
</a>
of Quantum Automata
5a.
<span class="ltx_text ltx_font_typewriter">
http://metameso.org/ joe/docs/book.pdf
</span>
PM Book Project in 2005
5b.
<span class="ltx_text ltx_font_typewriter">
http://planetphysics.us/?op=getobj&from=books&id=435
</span>
PM Book Project in 2005, available at PlanetPhysics.org
<math alttext="http://planetphysics.us/?op=getobj&from=books&id=435" class="ltx_Math" display="inline" id="I1.m4">
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<mn>
435
</mn>
</mrow>
</math>
…
N. The result would depend on how well that tag works at PM.
</ol>
</div>
<div class="ltx_para ltx_align_right" id="S1.p2">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/attachmenttotextbookprojectsonplanetmath">
Attachment to Textbook projects on PlanetMath
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
AttachmentToTextbookProjectsOnPlanetMath
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 19:33:35
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 19:33:35
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
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14
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<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
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msc 00-01
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0 | Dimension | http://planetmath.org/Dimension | <!DOCTYPE html>
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dimension
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<h1 class="ltx_title ltx_title_document">
dimension
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The word
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
dimension
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Dimension.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dimensionofaposet">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dimension">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
in mathematics has many definitions, but
all of them are trying to quantify our intuition that, for
example, a sheet of paper has somehow one less dimension than a
stack of papers.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
One common way to define dimension is through some notion of a
number of
<a class="nnexus_concept" href="http://planetmath.org/projectivebasis">
independent
</a>
quantities needed to describe an element
of an object. For example, it is natural to say that the sheet of
paper is two-dimensional because one needs two
<a class="nnexus_concept" href="http://planetmath.org/realnumber">
real numbers
</a>
to
specify a position on the sheet, whereas the stack of papers is
three-dimension because a position in a stack is specified by a sheet
and a position on the sheet. Following this notion, in linear
algebra the
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/Dimension2
</span>
dimension of a vector space
is defined as the
<a class="nnexus_concept" href="http://planetmath.org/minimalandmaximalnumber">
minimal number
</a>
of vectors such that every other
vector in the vector space is
<a class="nnexus_concept" href="http://planetmath.org/representablefunctor">
representable
</a>
as a sum of these.
Similarly, the word
<em class="ltx_emph ltx_font_italic">
rank
</em>
denotes various dimension-like
<a class="nnexus_concept" href="http://planetmath.org/invariant">
invariants
</a>
that appear throughout the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebra
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Algebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraiccategoriesandclassesofalgebras">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
However, if we try to generalize this notion to the mathematical
objects that do not possess an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebraic structure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/algebraicsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicentryonthealgebraicfoundationsofmathematics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, then we run
into a difficulty. From the point of view of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
set theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SetTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/settheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
there are
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/
<a class="nnexus_concept" href="http://planetmath.org/cardinality">
Cardinality
</a>
</span>
as many real numbers as pairs of real
numbers since there is a
<a class="nnexus_concept" href="http://planetmath.org/bijection">
bijection
</a>
from real numbers to pairs of
real numbers. To distinguish a plane from a cube one needs to
impose
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
restrictions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/subalgebraofanalgebraicsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/subfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/restrictionofafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
on the kind of
<a class="nnexus_concept" href="http://planetmath.org/mapping">
mapping
</a>
. Surprisingly, it turns
out that the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Continuity.html">
continuity
</a>
is not enough as was pointed out by Peano.
There are
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
continuous functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ContinuousFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/classesofordinalsandenumeratingfunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
that map a square onto a cube. So,
in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
topology
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Topology.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topologicalspace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
one uses another intuitive notion that in a
high-dimensional space there are more directions than in a
low-dimensional. Hence, the (Lebesgue
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
covering
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/site">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/poset">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) dimension of a
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TopologicalSpace.html">
topological space
</a>
is defined as the smallest number
<math alttext="d" class="ltx_Math" display="inline" id="p3.m1">
<mi>
d
</mi>
</math>
such that
every covering of the space by
<a class="nnexus_concept" href="http://planetmath.org/openset">
open sets
</a>
can be refined so that no
point is
<a class="nnexus_concept" href="http://planetmath.org/superset">
contained
</a>
in more than
<math alttext="d+1" class="ltx_Math" display="inline" id="p3.m2">
<mrow>
<mi>
d
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
sets. For example, no matter
how one covers a sheet of paper by sufficiently small other sheets
of paper such that two sheets can overlap each other, but
cannot merely touch, one will always find a point that is covered
by
<math alttext="2+1=3" class="ltx_Math" display="inline" id="p3.m3">
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo>
=
</mo>
<mn>
3
</mn>
</mrow>
</math>
sheets.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Another definition of dimension rests on the idea that
higher-dimensional objects are in some sense larger than the
lower-dimensional ones. For example, to cover a cube with a side
length
<math alttext="2" class="ltx_Math" display="inline" id="p4.m1">
<mn>
2
</mn>
</math>
one needs at least
<math alttext="2^{3}=8" class="ltx_Math" display="inline" id="p4.m2">
<mrow>
<msup>
<mn>
2
</mn>
<mn>
3
</mn>
</msup>
<mo>
=
</mo>
<mn>
8
</mn>
</mrow>
</math>
cubes with a side length
<math alttext="1" class="ltx_Math" display="inline" id="p4.m3">
<mn>
1
</mn>
</math>
, but a square with a side length
<math alttext="2" class="ltx_Math" display="inline" id="p4.m4">
<mn>
2
</mn>
</math>
can be covered by only
<math alttext="2^{2}=4" class="ltx_Math" display="inline" id="p4.m5">
<mrow>
<msup>
<mn>
2
</mn>
<mn>
2
</mn>
</msup>
<mo>
=
</mo>
<mn>
4
</mn>
</mrow>
</math>
unit squares. Let
<math alttext="N(\epsilon)" class="ltx_Math" display="inline" id="p4.m6">
<mrow>
<mi>
N
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
ϵ
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
be the minimal number of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
open balls
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/metricspace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/ball">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in any covering of a bounded set
<math alttext="S" class="ltx_Math" display="inline" id="p4.m7">
<mi>
S
</mi>
</math>
by balls of radius
<math alttext="\epsilon" class="ltx_Math" display="inline" id="p4.m8">
<mi>
ϵ
</mi>
</math>
. The
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/HausdorffDimension
</span>
Besicovitch-Hausdorff dimension of
<math alttext="S" class="ltx_Math" display="inline" id="p4.m9">
<mi>
S
</mi>
</math>
is defined
as
<math alttext="-\lim_{\epsilon\to 0}\log_{\epsilon}N(\epsilon)" class="ltx_Math" display="inline" id="p4.m10">
<mrow>
<mo>
-
</mo>
<mrow>
<msub>
<mo>
lim
</mo>
<mrow>
<mi>
ϵ
</mi>
<mo>
→
</mo>
<mn>
0
</mn>
</mrow>
</msub>
<mo>
</mo>
<mrow>
<mrow>
<msub>
<mi>
log
</mi>
<mi>
ϵ
</mi>
</msub>
<mo>
</mo>
<mi>
N
</mi>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
ϵ
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</mrow>
</math>
. The
Besicovitch-Hausdorff dimension is not always defined, and when
defined it might be non-integral.
</p>
<table class="ltx_tabular ltx_align_right ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
dimension
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Canonical name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Dimension
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:02:50
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:02:50
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
bbukh (348)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
bbukh (348)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
10
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
bbukh (348)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Topic
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-01
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 15A03
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 54F45
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Dimension
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Dimension2
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
DimensionKrull
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
HausdorffDimension
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | Expression | http://planetmath.org/Expression | <!DOCTYPE html>
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<title>
expression
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
expression
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
An
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/expression">
expression
</a>
</em>
is a symbol or of symbols used to denote a quantity or value. Expressions consist of
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
constants
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/variable">
variables
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/operator">
operators
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/function">
functions
</a>
, and parentheses.
<br class="ltx_break"/>
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Note 1.
</span>
If an expression one or more operations to be performed in a certain
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/OrderOfOperations
</span>
order, the expression may be named after the last (
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/Ie
</span>
i.e. outermost) operation. For example, the expression
<math alttext="\displaystyle a^{2}-5\sqrt{a}+\frac{2}{3a}" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<mrow>
<msup>
<mi>
a
</mi>
<mn>
2
</mn>
</msup>
<mo>
-
</mo>
<mrow>
<mn>
5
</mn>
<mo>
</mo>
<msqrt>
<mi>
a
</mi>
</msqrt>
</mrow>
</mrow>
<mo>
+
</mo>
<mstyle displaystyle="true">
<mfrac>
<mn>
2
</mn>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mi>
a
</mi>
</mrow>
</mfrac>
</mstyle>
</mrow>
</math>
is a
<span class="ltx_text ltx_font_italic">
sum expression
</span>
.
<br class="ltx_break"/>
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Note 2.
</span>
An equation is a denoted equality of two expressions.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
<em class="ltx_emph ltx_font_italic">
Dictionary.com Unabridged
</em>
(version 1.1). Accessed on February 22, 2008. URL:
<span class="ltx_text ltx_font_typewriter">
http://dictionary.reference.com/browse/expression
</span>
http://dictionary.reference.com/browse/expression
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p4">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
expression
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Expression
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:50:11
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:50:11
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Wkbj79 (1863)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Wkbj79 (1863)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Wkbj79 (1863)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-01
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Equation
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
sum expression
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | FusionOfTheoreticalPhysicsWithMathematicsAtIHESOrg | http://planetmath.org/FusionOfTheoreticalPhysicsWithMathematicsAtIHESOrg | <!DOCTYPE html>
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<title>
”Fusion” of theoretical physics with mathematics at IHES org
</title>
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<h1 class="ltx_title ltx_title_document">
”Fusion” of theoretical physics with mathematics at IHES org
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_subsection" id="S0.SS1">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
0.1
</span>
A viewpoint from the IHES Organization: the ‘fusion’ of theoretical physics with mathematics
</h2>
<section class="ltx_subsubsection" id="S0.SS1.SSS1">
<h3 class="ltx_title ltx_title_subsubsection">
<span class="ltx_tag ltx_tag_subsubsection">
0.1.1
</span>
Introduction
</h3>
<div class="ltx_para" id="S0.SS1.SSS1.p1">
<p class="ltx_p">
Recent, important developments in mathematical physics that are closely related to both mathematics and quantum physics have been considered as a strong indication of the possibility of a ‘fusion between mathematics
and theoretical physics’; this viewpoint emerges from current results obtained at IHES in Paris, France,
the international institute that has formerly served well the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebraic geometry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicGeometry.html">
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</a>
</sup>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
category theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CategoryTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categorytheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bibliographyinalgebraictopologycategoriesandqat">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/graphtheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
community
during
<a class="nnexus_concept" href="http://planetmath.org/alexandergrothendieck">
Alexander Grothendieck
</a>
’s tenure at this institute. Brief excerpts of published reports by two established
mathematicians, one from France and the other from UK, are presented next together with the 2008 announcement
of the
<a class="nnexus_concept" href="http://planetmath.org/crafoordprize">
Crafoord prize
</a>
in mathematics for recent results obtained at IHES and in the US in this ‘fusion area’ between mathematics and theoretical physics (
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
quantum theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quantumoperatoralgebrasinquantumfieldtheories">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumfieldtheoriesqft">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and AQFT). Time will tell if this ‘fusion’ trend will be followed by many more mathematicians and/or theoretical physicists elsewhere, even though a precedent already exists in the application of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
non-commutative geometry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/noncommutativegeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumriemanniangeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumgeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to SUSY
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
extension
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/substructure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extension">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extensionofafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in modern physics that was initiated by Professor A. Connes.
</p>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS1.SSS2">
<h3 class="ltx_title ltx_title_subsubsection">
<span class="ltx_tag ltx_tag_subsubsection">
0.1.2
</span>
Pierre Cartier : “
<span class="ltx_text ltx_font_italic">
On the Fusion of Mathematics and Theoretical Physics at IHES
</span>
”
</h3>
<div class="ltx_para" id="S0.SS1.SSS2.p1">
<p class="ltx_p">
A verbatim quote from :
<span class="ltx_text ltx_font_typewriter">
http://www.math.jussieu.fr/ leila/grothendieckcircle/madday.pdf
</span>
“
<em class="ltx_emph ltx_font_italic">
The Evolution of
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
Concepts
</a>
of Space and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Symmetry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/symmetricrelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumsymmetry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
–
A Mad Day’s Work: From Grothendieck to Connes and Kontsevich*:
</em>
”
</p>
</div>
<div class="ltx_para" id="S0.SS1.SSS2.p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://images.
<a class="nnexus_concept" href="http://planetmath.org/planetmath">
planetmath
</a>
.org/cache/objects/11143/pdf/IHESOnTheFusionOfMathematicsAndTheoreticalPhysics2.pdf
</span>
“PM entry on IHES”
</p>
</div>
<div class="ltx_para" id="S0.SS1.SSS2.p3">
<p class="ltx_p">
“…I am in no way forgetting the facilities for work provided by the
<span class="ltx_text ltx_font_typewriter">
http://www.ihes.fr/jsp/site/Portal.jsp
</span>
Institut des Hautes Études Scientifiques (IHES) for so many years, particularly the constantly renewed opportunities for meetings and exchanges. While there have
been some difficult times, there is no point in dwelling on them.
<em class="ltx_emph ltx_font_italic">
One of the great virtues of the institute was that it erected no barriers between
mathematics and theoretical physics. There has always been a great deal of interpenetration
of these two areas of interest, which has only increased over time.
</em>
From the very beginning Louis Michel was one of the bridges due to his devotion to
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/Group
</span>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GroupTheory.html">
group theory
</a>
. At present, when the scientific outlook has changed so greatly over
the past forty years,
<em class="ltx_emph ltx_font_italic">
the fusion seems natural and no one wonders whether Connes
or Kontsevich are physicists or mathematicians. I moved between the two fields for a long time when to do so was to run counter to the current trends, and I welcome the present synthesis.
</em>
Alexander Grothendieck dominated the first ten years of the institute, and I hope no one will forget that. I knew him well during the 50s and 60s, especially through
<a class="nnexus_concept" href="http://planetmath.org/bourbakinicolas">
Bourbaki
</a>
, but we were never together at the institute, he left it in September 1970 and I arrived in July 1971. Grothendieck did not derive his inspiration from physics and its mathematical problems. Not that his mind was incapable of grasping this area—he had thought about it secretly before 1967, but the moral principles that he adhered to relegate physics to the outer darkness,
<em class="ltx_emph ltx_font_italic">
especially after Hiroshima
</em>
.
It is surprising that
<em class="ltx_emph ltx_font_italic">
some of Grothendieck’s most fertile ideas regarding the nature of space and symmetries have become naturally wed to the new directions in modern physics.
</em>
”
</p>
</div>
<div class="ltx_para" id="S0.SS1.SSS2.p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
S. Majid: On the Relationship between Mathematics and Physics:
</span>
</p>
</div>
<div class="ltx_para" id="S0.SS1.SSS2.p5">
<p class="ltx_p">
In ref.
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib7" title="">
7
</a>
]
</cite>
, S. Majid presents the following ‘thesis’ : “(roughly speaking) physics polarises down the middle into two parts, one which
<a class="nnexus_concept" href="http://planetmath.org/representablefunctor">
represents
</a>
the other, but that the latter equally represents the former, i.e. the two should be treated on an equal footing. The starting point is that Nature after all does not know or care what mathematics is already in textbooks. Therefore the quest for the ultimate theory may well entail, probably does entail, inventing entirely new mathematics in the process. In other words, at least at some intuitive level,
<span class="ltx_text ltx_font_italic">
a theoretical physicist also has to be a pure mathematician
</span>
. Then one can phrase the question ‘what is the ultimate theory of physics ?’ in the form ‘in the tableau of all mathematical concepts past present and future, is there some constrained
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Surface.html">
surface
</a>
or subset which is called physics ?’ Is there an
<a class="nnexus_concept" href="http://planetmath.org/equation">
equation
</a>
for physics itself as a subset of mathematics? I believe there is and if it were to be found it would be called the ultimate theory of physics. Moreover, I believe that it can be found and that it has a lot to do with what is different about the way a physicist looks at the world compared to a mathematician…We can then try to elevate the idea to a more general principle of representation-theoretic self-duality, that a fundamental theory of physics is incomplete unless such a role-reversal is possible. We can go further and hope to fully determine the (supposed)
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of fundamental laws of nature among all mathematical structures by this self-duality condition. Such duality considerations are certainly evident in some form in the context of quantum theory and gravity. The situation is summarised to the left in the following
<a class="nnexus_concept" href="http://planetmath.org/commutativediagram">
diagram
</a>
. For example, Lie groups provide the simplest examples of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RiemannianGeometry.html">
Riemannian geometry
</a>
, while the
<a class="nnexus_concept" href="http://planetmath.org/grouprepresentation">
representations
</a>
of
<a class="nnexus_concept" href="http://planetmath.org/equivalentmachines">
similar
</a>
Lie groups provide the quantum numbers of elementary particles in quantum theory. Thus, both quantum theory and non-Euclidean geometry are needed for a self-dual picture.
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Hopf algebras
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quantumoperatoralgebrasinquantumfieldtheories">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumgroups">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/locallycompactquantumgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
quantum groups
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quantumgroupoids">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/groupoidandgrouprepresentationsrelatedtoquantumsymmetries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/weakhopfcalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/finitequantumgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/compactquantumgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) precisely serve to unify these mutually dual structures.”
</p>
</div>
<div class="ltx_para" id="S0.SS1.SSS2.p6">
<p class="ltx_p">
The original announcement of the 2008 Crafoord award is available
<span class="ltx_text ltx_font_typewriter">
http://www.maths.qmul.ac.uk/ majid/pessay.html
</span>
on line,
and a concise, verbatim excerpt is appended here:
</p>
</div>
<div class="ltx_para" id="S0.SS1.SSS2.p7">
<p class="ltx_p">
**
<span class="ltx_text ltx_font_typewriter">
http://www.ihes.fr/jsp/site/Portal.jsp?page_id=251#
</span>
Maxim Kontsevich received the Crafoord Prize in 2008:
“Maxim Kontsevich, Daniel Iagolnitzer Prize, Prix Henri Poincaré Prize in 1997,
<a class="nnexus_concept" href="http://planetmath.org/fieldsmedal">
Fields Medal
</a>
in 1998, member of the Academy of Sciences in Paris, is a French mathematician of Russian origin and is a permament professor at IHES (since 1995). He belongs to a new generation of mathematicians who have been able to integrate in their area of work aspects of
<em class="ltx_emph ltx_font_italic">
quantum theory
</em>
, opening up radically new
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Perspective.html">
perspectives
</a>
. On the mathematical side, he drew on the systematic use of known
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebraic structure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/algebraicsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicentryonthealgebraicfoundationsofmathematics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Deformation.html">
deformations
</a>
and on the
<a class="nnexus_concept" href="http://planetmath.org/introduction">
introduction
</a>
of new ones, such as the ‘triangulated categories’ that turned out to be relevant in many other areas, with no obvious link, such as image processing.”
</p>
</div>
<div class="ltx_para" id="S0.SS1.SSS2.p8">
<p class="ltx_p">
‘The Crafoord Prize in astronomy and mathematics, biosciences, geosciences or polyarthritis research is awarded by the Royal Swedish Academy of Sciences annually according to a rotating scheme. The prize sum of USD 500,000 makes the Crafoord one of the world’ s largest scientific prizes’.
</p>
</div>
<div class="ltx_para" id="S0.SS1.SSS2.p9">
<p class="ltx_p">
“Mathematics and astrophysics were in the limelight this year, with the joint award of the Mathematics Prize to Maxim Kontsevitch, (French mathematician), and Edward Witten, (US theoretical physicist), ‘for their important contributions to mathematics inspired by modern theoretical physics’, and the award of the Astronomy Prize to Rashid Alievich Sunyaev (astrophysicist) ‘for his decisive contributions to high-energy astrophysics and cosmology’.”
</p>
</div>
</section>
</section>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
<em class="ltx_emph ltx_font_italic">
* Bulletin (New Series) of the
<a class="nnexus_concept" href="http://planetmath.org/americanmathematicalsociety">
American Mathematical Society
</a>
</em>
, Volume 38, Number 4, Pages 389–408., S 0273-0979(01)00913-2, Article published electronically on July 12, 2001, (thus anticipating the Crafoord prize award by seven years).
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
Maxim Kontsevich, Y. Chen, and A. Schwartz. Symmetries of WDVV Equations
<span class="ltx_text ltx_font_italic">
Nucl. Phys. B
</span>
, 730 (2005), 352–363.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib3">
<span class="ltx_bibtag ltx_role_refnum">
3
</span>
<span class="ltx_bibblock">
Maxim Kontsevich and A. Belov-Kanel.
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Automorphisms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/homomorphismbetweenalgebraicsystems">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structurehomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/typesofhomomorphisms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/isomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/automorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/grouphomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the Weyl Algebra
<span class="ltx_text ltx_font_italic">
Lett. Math. Phys.
</span>
74 (2005), 181–199.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib4">
<span class="ltx_bibtag ltx_role_refnum">
4
</span>
<span class="ltx_bibblock">
Maxim Kontsevich. 2008. The
<a class="nnexus_concept" href="http://planetmath.org/jacobianconjecture">
Jacobian Conjecture
</a>
is Stably
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Equivalent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Equivalent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/filterbasis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalenceofforcingnotions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalencerelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to the Dixmier Conjecture., Preprint,
<math alttext="arxiv--math/0512171" class="ltx_Math" display="inline" id="bib.bib4.m1">
<mrow>
<mi>
a
</mi>
<mi>
r
</mi>
<mi>
x
</mi>
<mi>
i
</mi>
<mi>
v
</mi>
<mo>
-
</mo>
<mo>
-
</mo>
<mi>
m
</mi>
<mi>
a
</mi>
<mi>
t
</mi>
<mi>
h
</mi>
<mo>
/
</mo>
<mn>
0512171
</mn>
</mrow>
</math>
.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib5">
<span class="ltx_bibtag ltx_role_refnum">
5
</span>
<span class="ltx_bibblock">
Maxim Kontsevich and Y. Soibelman. Integral affine structures
In:
<em class="ltx_emph ltx_font_italic">
The Unity of Mathematics in honor of the 90th birthday of I.M. Gelfand
</em>
,
<span class="ltx_text ltx_font_italic">
Progress in Mathematics 244
</span>
, Birkhaüser (2005), 321–386.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib6">
<span class="ltx_bibtag ltx_role_refnum">
6
</span>
<span class="ltx_bibblock">
Maxim Kontsevich and C. Fronsdal.
<a class="nnexus_concept" href="http://planetmath.org/quantization">
Quantization
</a>
on Curves. Preprint
<math alttext="arxivmath--ph/0507021" class="ltx_Math" display="inline" id="bib.bib6.m1">
<mrow>
<mi>
a
</mi>
<mi>
r
</mi>
<mi>
x
</mi>
<mi>
i
</mi>
<mi>
v
</mi>
<mi>
m
</mi>
<mi>
a
</mi>
<mi>
t
</mi>
<mi>
h
</mi>
<mo>
-
</mo>
<mo>
-
</mo>
<mi>
p
</mi>
<mi>
h
</mi>
<mo>
/
</mo>
<mn>
0507021
</mn>
</mrow>
</math>
.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib7">
<span class="ltx_bibtag ltx_role_refnum">
7
</span>
<span class="ltx_bibblock">
S. Majid, Principle of representation-theoretic self-duality,
<em class="ltx_emph ltx_font_italic">
Phys. Essays
</em>
. 4 (1991) 395-405.
</span>
</li>
</ul>
</section>
<div class="ltx_para" id="p2">
<p class="ltx_p">
**Source: Crafoord Prize official website.
</p>
</div>
<div class="ltx_para ltx_align_right" id="p3">
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Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
”Fusion” of theoretical physics with mathematics at IHES org
</td>
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<tr class="ltx_tr">
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</th>
<td class="ltx_td ltx_align_left ltx_border_r">
FusionOfTheoreticalPhysicsWithMathematicsAtIHESOrg
</td>
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2013-03-22 19:36:20
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7
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bci1 (20947)
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infix notation
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
infix notation
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/infixnotation">
Infix notation
</a>
is how we usually read and write
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
arithmetic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/arithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/arithmeticalhierarchy">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://planetmath.org/expression">
expressions
</a>
. In this notation, the
<a class="nnexus_concept" href="http://planetmath.org/operator">
operator
</a>
goes between the operands in the expression:
</p>
</div>
<div class="ltx_para" id="p2">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<span class="ltx_text ltx_markedasmath">
(operand1)
</span>
<span class="ltx_text ltx_markedasmath">
(operator)
</span>
<span class="ltx_text ltx_markedasmath">
(operand2)
</span>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
E.g.,
<math alttext="3+2" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mn>
3
</mn>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
</math>
, or
<math alttext="196\times 11" class="ltx_Math" display="inline" id="p3.m2">
<mrow>
<mn>
196
</mn>
<mo>
×
</mo>
<mn>
11
</mn>
</mrow>
</math>
, etc.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Infix notation suffers from some ambiguity; e.g.
</p>
</div>
<div class="ltx_para" id="p5">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="3+9\times 2" class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mn>
3
</mn>
<mo>
+
</mo>
<mrow>
<mn>
9
</mn>
<mo>
×
</mo>
<mn>
2
</mn>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
could mean
<math alttext="(3+9)\times 2" class="ltx_Math" display="inline" id="p6.m1">
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
3
</mn>
<mo>
+
</mo>
<mn>
9
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
×
</mo>
<mn>
2
</mn>
</mrow>
</math>
or
<math alttext="3+(9\times 2)" class="ltx_Math" display="inline" id="p6.m2">
<mrow>
<mn>
3
</mn>
<mo>
+
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
9
</mn>
<mo>
×
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
. Parentheses are needed to specify the
<a class="nnexus_concept" href="http://planetmath.org/orderofoperations">
order of operations
</a>
unambiguously.
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/reversepolishnotation">
Postfix notation
</a>
(or reverse-Polish notation) does not suffer this ambiguity; but it is considered harder for humans to read (hence its primary use in
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computer
</a>
applications).
</p>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
The “usual” fix for the ambiguity problem described above is to provide a convention regarding precedence of operations. This is typically done for computer parsing of mathematical expressions rather than in math done by hand, because in the former case, the computer
<em class="ltx_emph ltx_font_italic">
must
</em>
have some standard rules to proceed. For example, it is typical to make
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
multiplication
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
“higher precedence” than
<a class="nnexus_concept" href="http://planetmath.org/addition">
addition
</a>
, so in the above case,
<math alttext="9\times 2" class="ltx_Math" display="inline" id="p8.m1">
<mrow>
<mn>
9
</mn>
<mo>
×
</mo>
<mn>
2
</mn>
</mrow>
</math>
would be performed before adding the result to 3.
</p>
</div>
<div class="ltx_para" id="p9">
<p class="ltx_p">
The ambiguity problem only occurs when multiple operators are present in one expression, and thus, the associative law does not hold. E.g., there is no ambiguity in
<math alttext="1+2+3" class="ltx_Math" display="inline" id="p9.m1">
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mn>
2
</mn>
<mo>
+
</mo>
<mn>
3
</mn>
</mrow>
</math>
, because
<math alttext="(1+2)+3" class="ltx_Math" display="inline" id="p9.m2">
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
+
</mo>
<mn>
3
</mn>
</mrow>
</math>
is the same as
<math alttext="1+(2+3)" class="ltx_Math" display="inline" id="p9.m3">
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
2
</mn>
<mo>
+
</mo>
<mn>
3
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
, by the associative property of addition.
</p>
</div>
<div class="ltx_para ltx_align_right" id="p10">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
infix notation
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
InfixNotation
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:21:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:21:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
akrowne (2)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
akrowne (2)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
akrowne (2)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-01
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
infix
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
GeneralAssociativity
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
infix arithmetic
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | Rigid | http://planetmath.org/Rigid | <!DOCTYPE html>
<html>
<head>
<title>
rigid
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
rigid
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Suppose
<math alttext="C" class="ltx_Math" display="inline" id="p1.m1">
<mi>
C
</mi>
</math>
is a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
collection
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Collection.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/collection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of mathematical objects
(for
<a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">
instance
</a>
, sets, or
<a class="nnexus_concept" href="http://planetmath.org/function">
functions
</a>
).
Then we say that
<math alttext="C" class="ltx_Math" display="inline" id="p1.m2">
<mi>
C
</mi>
</math>
is
<em class="ltx_emph ltx_font_italic">
rigid
</em>
if every
<math alttext="c\in C" class="ltx_Math" display="inline" id="p1.m3">
<mrow>
<mi>
c
</mi>
<mo>
∈
</mo>
<mi>
C
</mi>
</mrow>
</math>
is uniquely determined by less information about
<math alttext="c" class="ltx_Math" display="inline" id="p1.m4">
<mi>
c
</mi>
</math>
than
one would expect.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
It should be emphasized that the above “definition” does not
define a
<em class="ltx_emph ltx_font_italic">
mathematical object
</em>
. Instead, it describes in what sense
the adjective rigid is typically used in mathematics,
by mathematicians.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Let us illustrate this by some examples:
</p>
</div>
<div class="ltx_para" id="p4">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/harmonicfunction">
Harmonic functions
</a>
on the
<a class="nnexus_concept" href="http://planetmath.org/unitdisk">
unit disk
</a>
are rigid in the sense that
they are uniquely determined by their boundary values.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
By the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental theorem of algebra
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalTheoremofAlgebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
polynomials
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Polynomial.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomialsinalgebraicsystems">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomialring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in
<math alttext="\mathbb{C}" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mi>
ℂ
</mi>
</math>
are rigid in the sense that any polynomial is completely determined by
its values on any
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
countably infinite
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CountablyInfinite.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/countablyinfinite">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
set, say
<math alttext="\mathbb{N}" class="ltx_Math" display="inline" id="I1.i2.p1.m2">
<mi>
ℕ
</mi>
</math>
, or the unit disk.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
Linear maps
<math alttext="\mathscr{L}(X,Y)" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mrow>
<mi class="ltx_font_mathscript">
ℒ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
X
</mi>
<mo>
,
</mo>
<mi>
Y
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
between
<a class="nnexus_concept" href="http://planetmath.org/vectorspace">
vector spaces
</a>
<math alttext="X,Y" class="ltx_Math" display="inline" id="I1.i3.p1.m2">
<mrow>
<mi>
X
</mi>
<mo>
,
</mo>
<mi>
Y
</mi>
</mrow>
</math>
are rigid in the sense that any
<math alttext="L\in\mathscr{L}(X,Y)" class="ltx_Math" display="inline" id="I1.i3.p1.m3">
<mrow>
<mi>
L
</mi>
<mo>
∈
</mo>
<mrow>
<mi class="ltx_font_mathscript">
ℒ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
X
</mi>
<mo>
,
</mo>
<mi>
Y
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
is completely
determined by its values on any set of basis vectors of
<math alttext="X" class="ltx_Math" display="inline" id="I1.i3.p1.m4">
<mi>
X
</mi>
</math>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
Mostow’s rigidity theorem
</p>
</div>
</li>
</ol>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
rigid
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Rigid
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:38:10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:38:10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
11
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-01
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rigidity result
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rigidity theorem
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
rigidity
</td>
</tr>
</tbody>
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</div>
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0 | Strict | http://planetmath.org/Strict | <!DOCTYPE html>
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<title>
strict
</title>
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<h1 class="ltx_title ltx_title_document">
strict
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
In mathematical writing, the adjective
<em class="ltx_emph ltx_font_italic">
strict
</em>
is used in to modify technical which have meanings. It indicates that the exclusive meaning of the term is to be understood. (More formally, one could say that this is the meaning which implies the other meanings.)
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
This term is commonly used in the context of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
inequalities
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Inequality.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inequalitiesforrealnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
— the phrases “
<a class="nnexus_concept" href="http://planetmath.org/strict">
strictly
</a>
less than” and “strictly
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Greater.html">
greater
</a>
than” mean “less than and not equal to” and “greater than and not equal to”, respectively. A related use occurs when comparing numbers to zero — “strictly
<a class="nnexus_concept" href="http://planetmath.org/positive">
positive
</a>
” and “strictly negative” mean “positive and not equal to zero” and “negative and not equal to zero”, respectively. Also, in the context of functions, the adverb “strictly ” is used to modify the terms “
<a class="nnexus_concept" href="http://planetmath.org/increasingdecreasingmonotonefunction">
monotonic
</a>
”, “increasing”, and “decreasing”.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
On the other hand, sometimes one wants to specify the inclusive meanings of terms. In the context of comparisons, one can use the phrases “non-negative”, “non-positive”, “non-increasing”, and “non-decreasing” to make it clear that the inclusive sense of the terms is intended.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Using such terminology helps avoid possible ambiguity and confusion. For instance, upon reading the phrase “x is negative”, it is not immediately clear whether x = 0 is possible, since some authors consider zero to be positive while others consider zero not to be positive. Therefore, it is prudent to write either “x is strictly negative” or “x is non-positive” unless the distinction is unimportant or the context makes
<a class="nnexus_concept" href="http://planetmath.org/obvious">
obvious
</a>
which meaning is intended or it has been explicitly stated that the term ”positive” is to be used in only one sense. (Here, in Planet Math, we have taken the third option by adopting the convention that zero is neither positive nor negative. Hence, the terms ”strictly positive” and ”positive” are synonyms here. However, the adverb ”strictly” may still be necessary in some of the other contexts described avove.)
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
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<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
strict
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Strict
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:45:23
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:45:23
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
12
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-01
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
strict
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
strictly
</td>
</tr>
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0 | TextbookProjectsOnPlanetMath | http://planetmath.org/TextbookProjectsOnPlanetMath | <!DOCTYPE html>
<html>
<head>
<title>
Textbook projects on PlanetMath
</title>
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<h1 class="ltx_title ltx_title_document">
Textbook projects on PlanetMath
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
This will be a guide to textbook projects on PM. Since I don’t know
if there currently are any textbook projects on PM, this is just a
“stub” entry for now. If you create a textbook project, please add
a link to it here. Let me know what other things you would like to
see as part of this entry. (Eventually we will probably create a
documentation item for textbook writing.) I think the 00-01 category
can and should contain articles
<em class="ltx_emph ltx_font_italic">
about
</em>
instructional exposition,
just as well as examples of instructional exposition, so perhaps such
discussions can begin here as well, and then branch off into other
entries as that becomes convenient.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
An example of the sort of textbook we might want to have would
be: a free book
<a class="nnexus_concept" href="http://planetmath.org/comparisonoffilters">
comparable
</a>
to
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/introduction">
Introduction
</a>
to
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Algorithms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Algorithm.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algorithm">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
by
Cormen
<em class="ltx_emph ltx_font_italic">
et al.
</em>
.
</p>
</div>
<div class="ltx_para" id="p3">
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/statisticsonplanetmath">
Statistics on PlanetMath
</a>
</p>
</div>
</li>
</ul>
</div>
<div class="ltx_para ltx_align_right" id="p4">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/textbookprojectsonplanetmath">
Textbook projects on PlanetMath
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
TextbookProjectsOnPlanetMath
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:46:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:46:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00-01
</td>
</tr>
</tbody>
</table>
</div>
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0 | ToyTheorem | http://planetmath.org/ToyTheorem | <!DOCTYPE html>
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toy theorem
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<h1 class="ltx_title ltx_title_document">
toy theorem
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<em class="ltx_emph ltx_font_italic">
toy theorem
</em>
is a simplified version of a more general
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
theorem
</a>
.
For
<a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">
instance
</a>
, by introducing some simplifying
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
assumptions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/deduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivationsinnaturaldeduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in a theorem,
one obtains a toy theorem.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Usually, a toy theorem is used to illustrate the claim of a theorem.
It can also be illustrative and insightful to study proofs of
a toy theorem derived from a non-trivial theorem.
Toy theorems also have a great education value.
After presenting a theorem (with, say, a highly non-trivial proof),
one can sometimes give some assurance that the theorem
really holds, by proving a toy version of the theorem.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
For instance, a toy theorem of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Brouwer fixed point theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BrouwerFixedPointTheorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/brouwerfixedpointtheorem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is obtained by restricting the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
dimension
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Dimension.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dimension">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to one.
In this case, the Brouwer fixed point theorem follows
almost immediately from the
<a class="nnexus_concept" href="http://planetmath.org/intermediatevaluetheorem">
intermediate value theorem
</a>
(see
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/BrouwerFixedPointInOneDimension
</span>
this page).
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
toy theorem
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
ToyTheorem
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:55:35
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:55:35
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00-01
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | AMSMSCClassificationOfArticlesAndConversionTables | http://planetmath.org/AMSMSCClassificationOfArticlesAndConversionTables | <!DOCTYPE html>
<html>
<head>
<title>
AMS MSC classification of articles and conversion tables
</title>
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<h1 class="ltx_title ltx_title_document">
AMS MSC classification of articles and conversion tables
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
Links to the AMS MSC 2010 Classification PDF of all MSC entries available, and the AMS MSC website
</h2>
<div class="ltx_para" id="S1.p1">
<p class="ltx_p">
Because the AMS MSC
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
classification
</a>
list or table does not seem to be available at present when creating a new entry two links are here provided to the AMS websites that list the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complete
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/ordersinanumberfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/variety">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/completemeasure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/soundcomplete">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/completebinarytree">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/completegraph">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/supplementalaxiomsforanabeliancategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/kripkesemantics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complete">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/maximallyconsistent">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
Table of AMS MSC2010 classifications:
</p>
</div>
<div class="ltx_para" id="S1.p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://www.ams.org/mathscinet/msc/pdfs/classifications2010.pdf
</span>
All MSC 2010 in one PDF :
</p>
</div>
<div class="ltx_para" id="S1.p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://www.ams.org/mathscinet/msc/msc2010.html
</span>
The AMS MSC website with its maths specialized Search Engine
</p>
</div>
<section class="ltx_subsection" id="S1.SS1">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
1.1
</span>
Conversion Tables
</h3>
<div class="ltx_para" id="S1.SS1.p1">
<p class="ltx_p">
http://www.ams.org/mathscinet/msc/pdfs/classifications2010.pdf
</p>
</div>
<div class="ltx_para" id="S1.SS1.p2">
<p class="ltx_p">
CONVERSIONS: http://www.ams.org/mathscinet/msc/conv.html?from=2000
</p>
</div>
<div class="ltx_para" id="S1.SS1.p3">
<p class="ltx_p">
<math alttext="MSC2000~{}Classification~{}Codes~{}\to~{}MSC2010~{}Classification~{}Codes~{}Update." class="ltx_Math" display="inline" id="S1.SS1.p3.m1">
<mrow>
<mrow>
<mrow>
<mi>
M
</mi>
<mo>
</mo>
<mi>
S
</mi>
<mo>
</mo>
<mi>
C
</mi>
<mo>
</mo>
<mpadded width="+3.3pt">
<mn>
2000
</mn>
</mpadded>
<mo>
</mo>
<mi>
C
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
f
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mpadded width="+3.3pt">
<mi>
n
</mi>
</mpadded>
<mo>
</mo>
<mi>
C
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mpadded width="+3.3pt">
<mi>
s
</mi>
</mpadded>
</mrow>
<mo rspace="5.8pt">
→
</mo>
<mrow>
<mi>
M
</mi>
<mo>
</mo>
<mi>
S
</mi>
<mo>
</mo>
<mi>
C
</mi>
<mo>
</mo>
<mpadded width="+3.3pt">
<mn>
2010
</mn>
</mpadded>
<mo>
</mo>
<mi>
C
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
f
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mpadded width="+3.3pt">
<mi>
n
</mi>
</mpadded>
<mo>
</mo>
<mi>
C
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mpadded width="+3.3pt">
<mi>
s
</mi>
</mpadded>
<mo>
</mo>
<mi>
U
</mi>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
e
</mi>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
Date: 14 October 2009
</p>
</div>
<div class="ltx_para" id="S1.SS1.p4">
<p class="ltx_p">
http://www.ams.org/mathscinet/msc/conv.html?from=2010
</p>
</div>
<div class="ltx_para" id="S1.SS1.p5">
<p class="ltx_p">
MSC2010 Classification Codes –¿ MSC2000 Classification Codes
</p>
</div>
</section>
<section class="ltx_subsection" id="S1.SS2">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
1.2
</span>
General Classifications
</h3>
<div class="ltx_para" id="S1.SS2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
00-01 Instructional Expositions
</span>
</p>
</div>
<div class="ltx_para" id="S1.SS2.p2">
<p class="ltx_p">
00-02 Research Expositions
</p>
</div>
<div class="ltx_para" id="S1.SS2.p3">
<p class="ltx_p">
00A05
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
General mathematics
</a>
</p>
</div>
<div class="ltx_para" id="S1.SS2.p4">
<p class="ltx_p">
00A35 Methodology of mathematics, didactics
</p>
</div>
<div class="ltx_para" id="S1.SS2.p5">
<p class="ltx_p">
00A66 Mathematics and visual arts, visualization
</p>
</div>
<div class="ltx_para" id="S1.SS2.p6">
<p class="ltx_p">
00A79 Physics
</p>
</div>
<div class="ltx_para" id="S1.SS2.p7">
<p class="ltx_p">
00A69 General applied mathematics
</p>
</div>
<div class="ltx_para" id="S1.SS2.p8">
<p class="ltx_p">
00A73 Dimensional analysis
</p>
</div>
<div class="ltx_para" id="S1.SS2.p9">
<p class="ltx_p">
00A15 Bibliographies
</p>
</div>
<div class="ltx_para" id="S1.SS2.p10">
<p class="ltx_p">
00A71 Theory of mathematical modeling
</p>
</div>
<div class="ltx_para" id="S1.SS2.p11">
<p class="ltx_p">
00A30
<a class="nnexus_concept" href="http://planetmath.org/foundationsofmathematicsoverview">
Philosophy of mathematics
</a>
and 03A05
</p>
</div>
<div class="ltx_para" id="S1.SS2.p12">
<p class="ltx_p">
00A99 Miscellaneous topics
</p>
</div>
<div class="ltx_para" id="S1.SS2.p13">
<p class="ltx_p">
00B99 None of the above, but in this
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
section
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Section.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operationsonrelations">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sectionofagroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sectionofafiberbundle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/vectorbundle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sheaf">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sectionsandretractions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/typesofmorphisms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/monic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
</section>
</section>
<section class="ltx_section" id="S2">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
2
</span>
Several Examples of AMS MSC Classifications Utilized in PM articles
</h2>
<div class="ltx_para" id="S2.p1">
<p class="ltx_p">
msc:00-01, msc:00-02
</p>
</div>
<div class="ltx_para" id="S2.p2">
<p class="ltx_p">
00A15 Bibliographies
</p>
</div>
<section class="ltx_subsection" id="S2.SS1">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.1
</span>
Algebraic Logics
</h3>
<div class="ltx_para" id="S2.SS1.p1">
<p class="ltx_p">
03G05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Boolean algebras
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BooleanAlgebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/booleanlattice">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 06Exx]
</p>
</div>
<div class="ltx_para" id="S2.SS1.p2">
<p class="ltx_p">
03G12
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
quantum logic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quantumlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categoricalquantumlmlogicalgebras">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 06C15, 81P10]
</p>
</div>
<div class="ltx_para" id="S2.SS1.p3">
<p class="ltx_p">
03G20
<math alttext="\L{}" class="ltx_Math" display="inline" id="S2.SS1.p3.m1">
<mi>
Ł
</mi>
</math>
ukasiewicz and Post algebras [See also 06D25, 06D30]
</p>
</div>
<div class="ltx_para" id="S2.SS1.p4">
<p class="ltx_p">
03G10 Lattices and related
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structures
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 06Bxx]
</p>
</div>
<div class="ltx_para" id="S2.SS1.p5">
<p class="ltx_p">
03G30 Categorical logic, topoi [See also 18B25, 18C05, 18C10]
</p>
</div>
<div class="ltx_para" id="S2.SS1.p6">
<p class="ltx_p">
03H10 Other applications of nonstandard models (economics, physics, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS1.p7">
<p class="ltx_p">
03G15 Cylindric and
<a class="nnexus_concept" href="http://planetmath.org/polyadicalgebra">
polyadic algebras
</a>
;
<a class="nnexus_concept" href="http://planetmath.org/relationalgebra">
relation algebras
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p8">
<p class="ltx_p">
03G20 Lukasiewicz and Post algebras [See also 06D25, 06D30]
</p>
</div>
<div class="ltx_para" id="S2.SS1.p9">
<p class="ltx_p">
03G25 Other
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebras
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Algebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraiccategoriesandclassesofalgebras">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
related to logic [See also 03F45, 06D20, 06E25, 06F35]
</p>
</div>
<div class="ltx_para" id="S2.SS1.p10">
<p class="ltx_p">
03G27 Abstract algebraic logic
</p>
</div>
<div class="ltx_para" id="S2.SS1.p11">
<p class="ltx_p">
03G30 Categorical logic, topoi [See also 18B25, 18C05, 18C10]
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS2">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.2
</span>
COMBINATORICS
</h3>
<div class="ltx_para" id="S2.SS2.p1">
<p class="ltx_p">
05-XX
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
COMBINATORICS
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/26.1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Combinatorics.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS2.p2">
<p class="ltx_p">
For
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
finite fields
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FiniteField.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/finitefield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, see 11Txxg
</p>
</div>
<div class="ltx_para" id="S2.SS2.p3">
<p class="ltx_p">
05-00 General reference works (handbooks, dictionaries, bibliographies,etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS2.p4">
<p class="ltx_p">
05-01 Instructional exposition (textbooks, tutorial papers, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS2.p5">
<p class="ltx_p">
05-02 Research exposition (monographs, survey articles)
</p>
</div>
<div class="ltx_para" id="S2.SS2.p6">
<p class="ltx_p">
05Axx
<a class="nnexus_concept" href="http://planetmath.org/enumerativecombinatorics">
Enumerative combinatorics
</a>
–For enumeration in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
graph theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GraphTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/graphtheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, see 05C30g , 05A05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Permutations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Permutation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/permutation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, words, matrices
</p>
</div>
<div class="ltx_para" id="S2.SS2.p7">
<p class="ltx_p">
05A10
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Factorials
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Factorial.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/factorial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
binomial coefficients
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.2#E1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://dlmf.nist.gov/26.3#SS1.p1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BinomialCoefficient.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/binomialcoefficient">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, combinatorial
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 11B65, 33Cxx]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p8">
<p class="ltx_p">
05A15 Exact
<a class="nnexus_concept" href="http://mathworld.wolfram.com/EnumerationProblem.html">
enumeration problems
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
generating functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GeneratingFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/formalpowerseries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 33Cxx,33Dxx]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p9">
<p class="ltx_p">
05A16
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Asymptotic.html">
Asymptotic
</a>
enumeration
</p>
</div>
<div class="ltx_para" id="S2.SS2.p10">
<p class="ltx_p">
05A17
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Partitions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Partition.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/partition1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integerpartition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/partition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
[See also 11P81, 11P82, 11P83]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p11">
<p class="ltx_p">
05A18 Partitions of sets
</p>
</div>
<div class="ltx_para" id="S2.SS2.p12">
<p class="ltx_p">
05A19 Combinatorial
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
identities
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/identityinaclass">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equality">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/multivaluedfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/hypergroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/category">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/group">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
bijective
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Bijective.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bijection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
combinatorics
</p>
</div>
<div class="ltx_para" id="S2.SS2.p13">
<p class="ltx_p">
05A20 Combinatorial
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
inequalities
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Inequality.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inequalitiesforrealnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS2.p14">
<p class="ltx_p">
05A30
<math alttext="q" class="ltx_Math" display="inline" id="S2.SS2.p14.m1">
<mi>
q
</mi>
</math>
-
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Calculus.html">
calculus
</a>
and related topics [See also 33Dxx]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p15">
<p class="ltx_p">
05A40
<a class="nnexus_concept" href="http://mathworld.wolfram.com/UmbralCalculus.html">
Umbral calculus
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS2.p16">
<p class="ltx_p">
05Bxx Designs and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
configurations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Configuration.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/automaton">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/unlimitedregistermachine">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
–For applications of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DesignTheory.html">
design theory
</a>
, see 94C30g
</p>
</div>
<div class="ltx_para" id="S2.SS2.p17">
<p class="ltx_p">
05B05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Block designs
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BlockDesign.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/design">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 51E05, 62K10]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p18">
<p class="ltx_p">
05B07 Triple systems
</p>
</div>
<div class="ltx_para" id="S2.SS2.p19">
<p class="ltx_p">
05B10
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Difference sets
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DifferenceSet.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/differenceset">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(number-theoretic, group-theoretic, etc.)[See also 11B13]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p20">
<p class="ltx_p">
05B15
<a class="nnexus_concept" href="http://mathworld.wolfram.com/OrthogonalArray.html">
Orthogonal arrays
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Latin squares
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LatinSquare.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/latinsquare">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RoomSquare.html">
Room squares
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS2.p21">
<p class="ltx_p">
05B20 Matrices (incidence,
<a class="nnexus_concept" href="http://planetmath.org/hadamardmatrix">
Hadamard
</a>
, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS2.p22">
<p class="ltx_p">
05B25
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FiniteGeometry.html">
Finite geometries
</a>
[See also 51D20, 51Exx]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p23">
<p class="ltx_p">
05B30 Other designs, configurations [See also 51E30]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p24">
<p class="ltx_p">
05B35
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Matroids
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Matroid.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/matroid">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, geometric lattices [See also 52B40, 90C27]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p25">
<p class="ltx_p">
05B40
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Packing.html">
Packing
</a>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
covering
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/site">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/poset">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 11H31, 52C15, 52C17]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p26">
<p class="ltx_p">
05B45
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Tessellation.html">
Tessellation
</a>
and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TilingProblem.html">
tiling problems
</a>
[See also 52C20, 52C22]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p27">
<p class="ltx_p">
05B50 Polyominoes
</p>
</div>
<div class="ltx_para" id="S2.SS2.p28">
<p class="ltx_p">
05B99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS2.p29">
<p class="ltx_p">
05Cxx Graph theory fFor applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15g
</p>
</div>
<div class="ltx_para" id="S2.SS2.p30">
<p class="ltx_p">
05C05 Trees
</p>
</div>
<div class="ltx_para" id="S2.SS2.p31">
<p class="ltx_p">
05C07
<a class="nnexus_concept" href="http://mathworld.wolfram.com/VertexDegree.html">
Vertex degrees
</a>
[See also 05E30]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p32">
<p class="ltx_p">
05C10
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Planar graphs
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PlanarGraph.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/planargraph">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
; geometric and topological aspects of graph theory [See also 57M15, 57M25]
</p>
</div>
<div class="ltx_para" id="S2.SS2.p33">
<p class="ltx_p">
05C12
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Distance
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Distance.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/distanceinagraph">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/metricspace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in graphs
</p>
</div>
<div class="ltx_para" id="S2.SS2.p34">
<p class="ltx_p">
05C15
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Coloring
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Coloring.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/coloring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of graphs and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
hypergraphs
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Hypergraph.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/hypergraph">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS3">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.3
</span>
ORDER, LATTICES, ORDERED ALGEBRAIC STRUCTURES
</h3>
<div class="ltx_para" id="S2.SS3.p1">
<p class="ltx_p">
[See also 18B35]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p2">
<p class="ltx_p">
06-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS3.p3">
<p class="ltx_p">
06-01 Instructional exposition (textbooks, tutorial papers, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS3.p4">
<p class="ltx_p">
06-02 Research exposition (monographs, survey articles)
</p>
</div>
<div class="ltx_para" id="S2.SS3.p5">
<p class="ltx_p">
06-06 Proceedings, conferences,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
collections
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Collection.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/collection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, etc.
</p>
</div>
<div class="ltx_para" id="S2.SS3.p6">
<p class="ltx_p">
06Axx
<a class="nnexus_concept" href="http://mathworld.wolfram.com/OrderedSet.html">
Ordered sets
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p7">
<p class="ltx_p">
06A05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Total order
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TotalOrder.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/totalorder">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p8">
<p class="ltx_p">
06A06
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Partial order
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PartialOrder.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/partialorder">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, general
</p>
</div>
<div class="ltx_para" id="S2.SS3.p9">
<p class="ltx_p">
06A07 Combinatorics of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PartiallyOrderedSet.html">
partially ordered sets
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p10">
<p class="ltx_p">
06A11
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Algebraic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Algebraics.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraicmap">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraic1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
aspects of posets
</p>
</div>
<div class="ltx_para" id="S2.SS3.p11">
<p class="ltx_p">
06A12 Semilattices [See also 20M10; for topological semilattices see 22A26]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p12">
<p class="ltx_p">
06A15
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Galois correspondences
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/galoisconnection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofgaloistheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
closure operators
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/consequenceoperator">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/closuremap">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/closureaxioms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p13">
<p class="ltx_p">
06A75
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Generalizations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/hilbertsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiomsystemforfirstorderlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of ordered sets
</p>
</div>
<div class="ltx_para" id="S2.SS3.p14">
<p class="ltx_p">
06A99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS3.p15">
<p class="ltx_p">
06Bxx Lattices [See also 03G10]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p16">
<p class="ltx_p">
06B05 Structure theory
</p>
</div>
<div class="ltx_para" id="S2.SS3.p17">
<p class="ltx_p">
06B10 Ideals,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
congruence relations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/congruenceaxioms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruencerelationonanalgebraicsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruenceonapartialalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p18">
<p class="ltx_p">
06B15
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RepresentationTheory.html">
Representation theory
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p19">
<p class="ltx_p">
06B20
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Varieties
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Variety.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equationalclass">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/varietyofgroups">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of lattices
</p>
</div>
<div class="ltx_para" id="S2.SS3.p20">
<p class="ltx_p">
06B23 Complete lattices,
<a class="nnexus_concept" href="http://planetmath.org/completion">
completions
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p21">
<p class="ltx_p">
06B25 Free lattices, projective lattices,
<a class="nnexus_concept" href="http://planetmath.org/wordproblem">
word problems
</a>
[See also 03D40,08A50, 20F10]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p22">
<p class="ltx_p">
06B30 Topological lattices,
<a class="nnexus_concept" href="http://planetmath.org/ordertopology">
order topologies
</a>
[See also 06F30, 22A26, 54F05, 54H12]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p23">
<p class="ltx_p">
06B35
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Continuous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Continuous.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/classesofordinalsandenumeratingfunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
lattices and posets, applications [See also 06B30, 06D10,06F30, 18B35, 22A26, 68Q55]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p24">
<p class="ltx_p">
06B75 Generalizations of lattices
</p>
</div>
<div class="ltx_para" id="S2.SS3.p25">
<p class="ltx_p">
06B99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS3.p26">
<p class="ltx_p">
06Cxx
<a class="nnexus_concept" href="http://planetmath.org/modularlattice">
Modular
</a>
lattices,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complemented
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/complementedlattice">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/wellpointedtopos">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
lattices
</p>
</div>
<div class="ltx_para" id="S2.SS3.p27">
<p class="ltx_p">
06C05 Modular lattices,
<a class="nnexus_concept" href="http://planetmath.org/desarguestheorem">
Desarguesian
</a>
lattices
</p>
</div>
<div class="ltx_para" id="S2.SS3.p28">
<p class="ltx_p">
06C10 Semimodular lattices, geometric lattices
</p>
</div>
<div class="ltx_para" id="S2.SS3.p29">
<p class="ltx_p">
06C15 Complemented lattices,
<a class="nnexus_concept" href="http://planetmath.org/orthocomplementedlattice">
orthocomplemented
</a>
lattices and posets [See also 03G12, 81P10]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p30">
<p class="ltx_p">
06C20 Complemented modular lattices,
<a class="nnexus_concept" href="http://planetmath.org/continuousgeometry">
continuous geometries
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p31">
<p class="ltx_p">
06C99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS3.p32">
<p class="ltx_p">
06Dxx
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Distributive
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/distributivity">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/ternaryring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
lattices
</p>
</div>
<div class="ltx_para" id="S2.SS3.p33">
<p class="ltx_p">
06D05 Structure and representation theory
</p>
</div>
<div class="ltx_para" id="S2.SS3.p34">
<p class="ltx_p">
06D10
<a class="nnexus_concept" href="http://planetmath.org/completedistributivity">
Complete distributivity
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p35">
<p class="ltx_p">
06D15 Pseudocomplemented lattices
</p>
</div>
<div class="ltx_para" id="S2.SS3.p36">
<p class="ltx_p">
06D20
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Heyting algebras
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/HeytingAlgebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/heytingalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 03G25]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p37">
<p class="ltx_p">
06D22 Frames, locales For topological questions see 54XXg
</p>
</div>
<div class="ltx_para" id="S2.SS3.p38">
<p class="ltx_p">
06D25 Post algebras [See also 03G20]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p39">
<p class="ltx_p">
06D30
<a class="nnexus_concept" href="http://planetmath.org/demorganalgebra">
De Morgan algebras
</a>
, Lukasiewicz algebras [See also 03G20]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p40">
<p class="ltx_p">
06D35 MV–algebras
</p>
</div>
<div class="ltx_para" id="S2.SS3.p41">
<p class="ltx_p">
06D50 Lattices and
<a class="nnexus_concept" href="http://planetmath.org/polarity">
duality
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p42">
<p class="ltx_p">
06D72 Fuzzy lattices (soft algebras) and related topics
</p>
</div>
<div class="ltx_para" id="S2.SS3.p43">
<p class="ltx_p">
06D75 Other generalizations of distributive lattices
</p>
</div>
<div class="ltx_para" id="S2.SS3.p44">
<p class="ltx_p">
06D99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS3.p45">
<p class="ltx_p">
06Exx Boolean algebras (
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Boolean rings
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BooleanRing.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/booleanring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) [See also 03G05]
</p>
</div>
<div class="ltx_para" id="S2.SS3.p46">
<p class="ltx_p">
06E05 Structure theory
</p>
</div>
<div class="ltx_para" id="S2.SS3.p47">
<p class="ltx_p">
06E10
<a class="nnexus_concept" href="http://planetmath.org/chaincondition">
Chain conditions
</a>
, complete algebras
</p>
</div>
<div class="ltx_para" id="S2.SS3.p48">
<p class="ltx_p">
06E15
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Stone spaces
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/StoneSpace.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/stonespace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(Boolean spaces) and related structures
</p>
</div>
<div class="ltx_para" id="S2.SS3.p49">
<p class="ltx_p">
06E20 Ring–theoretic
<a class="nnexus_concept" href="http://planetmath.org/property">
properties
</a>
[See also 16E50, 16G30]
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS4">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.4
</span>
General Algebraic Systems
</h3>
<div class="ltx_para" id="S2.SS4.p1">
<p class="ltx_p">
08-XX GENERAL
<a class="nnexus_concept" href="http://planetmath.org/algebraicsystem">
ALGEBRAIC SYSTEMS
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS4.p2">
<p class="ltx_p">
08-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS4.p3">
<p class="ltx_p">
08-02 Research exposition (monographs, survey articles)
</p>
</div>
<div class="ltx_para" id="S2.SS4.p4">
<p class="ltx_p">
08-06 Proceedings, conferences, collections, etc.
</p>
</div>
<div class="ltx_para" id="S2.SS4.p5">
<p class="ltx_p">
08Axx
<a class="nnexus_concept" href="http://planetmath.org/topicentryonthealgebraicfoundationsofmathematics">
algebraic structures
</a>
[See also 03C05]
</p>
</div>
<div class="ltx_para" id="S2.SS4.p6">
<p class="ltx_p">
08A02
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Relational systems
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RelationalSystem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relationalsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, laws of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
composition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Composition.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/countingcompositionsofaninteger">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
08A05 Structure theory
</p>
</div>
<div class="ltx_para" id="S2.SS4.p7">
<p class="ltx_p">
08A30
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Subalgebras
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Subalgebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/subalgebraofanalgebraicsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/subalgebraofapartialalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, congruence relations
</p>
</div>
<div class="ltx_para" id="S2.SS4.p8">
<p class="ltx_p">
08A35
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Automorphisms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Automorphism.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homomorphismbetweenalgebraicsystems">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structurehomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/isomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/typesofhomomorphisms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/automorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/grouphomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Endomorphism.html">
endomorphisms
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS4.p9">
<p class="ltx_p">
08A70 Applications of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/UniversalAlgebra.html">
universal algebra
</a>
in computer science
</p>
</div>
<div class="ltx_para" id="S2.SS4.p10">
<p class="ltx_p">
08A72 Fuzzy algebraic structures
</p>
</div>
<div class="ltx_para" id="S2.SS4.p11">
<p class="ltx_p">
08Cxx Other
<a class="nnexus_concept" href="http://planetmath.org/conceptsinabstractalgebra">
classes of algebra
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS4.p12">
<p class="ltx_p">
08C05
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Category.html">
Categories
</a>
of algebras [See also 18C05]
</p>
</div>
<div class="ltx_para" id="S2.SS4.p13">
<p class="ltx_p">
08C10 Axiomatic model classes [See also 03Cxx, in particular 03C60]
</p>
</div>
<div class="ltx_para" id="S2.SS4.p14">
<p class="ltx_p">
08C15
<a class="nnexus_concept" href="http://planetmath.org/implicationalclass">
Quasivarieties
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS4.p15">
<p class="ltx_p">
08C20 Natural dualities for classes of algebras [See also 06E15, 18A40, 22A30]
</p>
</div>
<div class="ltx_para" id="S2.SS4.p16">
<p class="ltx_p">
08C99 None of the above, but in this section
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS5">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.5
</span>
Algebraic number theory, Galois theory, cohomology and polynomials
</h3>
<div class="ltx_para" id="S2.SS5.p1">
<p class="ltx_p">
11Sxx
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Algebraic number theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicNumberTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraicnumbertheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
: local and
<math alttext="p" class="ltx_Math" display="inline" id="S2.SS5.p1.m1">
<mi>
p
</mi>
</math>
-adic fields
</p>
</div>
<div class="ltx_para" id="S2.SS5.p2">
<p class="ltx_p">
11S05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Polynomials
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Polynomial.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomialsinalgebraicsystems">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomialring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS5.p3">
<p class="ltx_p">
11S15
<a class="nnexus_concept" href="http://planetmath.org/ramificationindex">
Ramification
</a>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
extension
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/substructure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extension">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extensionofafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extensionfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
theory
</p>
</div>
<div class="ltx_para" id="S2.SS5.p4">
<p class="ltx_p">
11S20
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GaloisTheory.html">
Galois theory
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS5.p5">
<p class="ltx_p">
11S23
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Integral
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.4#iv">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lebesgueintegral">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integral">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://planetmath.org/grouprepresentation">
representations
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS5.p6">
<p class="ltx_p">
11S25
<a class="nnexus_concept" href="http://planetmath.org/galoisrepresentation">
Galois cohomology
</a>
[See also 12Gxx, 16H05]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p7">
<p class="ltx_p">
11S31
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ClassFieldTheory.html">
Class field theory
</a>
;
<math alttext="p" class="ltx_Math" display="inline" id="S2.SS5.p7.m1">
<mi>
p
</mi>
</math>
-adic formal groups [See also 14L05]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p8">
<p class="ltx_p">
11S37 Langlands–
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Weil conjectures
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/WeilConjectures.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/weilconjectures">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
nonabelian
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/nonabelianstructures">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/nonabeliantheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/noncommutativestructure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
class field theory [See also 11Fxx, 22E50]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p9">
<p class="ltx_p">
11S40
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Zeta functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ZetaFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/weierstrasssigmafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and
<math alttext="L" class="ltx_Math" display="inline" id="S2.SS5.p9.m1">
<mi>
L
</mi>
</math>
–functions [See also 11M41, 19F27]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p10">
<p class="ltx_p">
11S45 Algebras and orders, and their zeta functions [See also 11R52, 11R54, 16Hxx, 16Kxx]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p11">
<p class="ltx_p">
11S70
<math alttext="K" class="ltx_Math" display="inline" id="S2.SS5.p11.m1">
<mi>
K
</mi>
</math>
–theory of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
local fields
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LocalField.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/localfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 19Fxx]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p12">
<p class="ltx_p">
11S80 Other
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
analytic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/logicism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/analytic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
theory (analogues of beta and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
gamma functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/5#PT2">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://dlmf.nist.gov/5.2#E1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GammaFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/gammafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<math alttext="p" class="ltx_Math" display="inline" id="S2.SS5.p12.m1">
<mi>
p
</mi>
</math>
-adic integration, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS5.p13">
<p class="ltx_p">
11S82 Non–
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Archimedean
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/partiallyorderedgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/infinitesimal">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/archimedeansemigroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/valuation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://planetmath.org/groupoidcdynamicalsystem">
dynamical systems
</a>
[See mainly 37Pxx]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p14">
<p class="ltx_p">
11S85 Other nonanalytic theory
</p>
</div>
<div class="ltx_para" id="S2.SS5.p15">
<p class="ltx_p">
11S90 Prehomogeneous vector spaces
</p>
</div>
<div class="ltx_para" id="S2.SS5.p16">
<p class="ltx_p">
11S99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS5.p17">
<p class="ltx_p">
11Txx Finite fields and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CommutativeRing.html">
commutative rings
</a>
(number–theoretic aspects)
</p>
</div>
<div class="ltx_para" id="S2.SS5.p18">
<p class="ltx_p">
11T06 Polynomials
</p>
</div>
<div class="ltx_para" id="S2.SS5.p19">
<p class="ltx_p">
11T22 Cyclotomy
</p>
</div>
<div class="ltx_para" id="S2.SS5.p20">
<p class="ltx_p">
11T23 Exponential sums
</p>
</div>
<div class="ltx_para" id="S2.SS5.p21">
<p class="ltx_p">
11T24 Other character sums and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Gauss sums
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/27.10#E9">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/gausssum">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS5.p22">
<p class="ltx_p">
11T30 Structure theory
</p>
</div>
<div class="ltx_para" id="S2.SS5.p23">
<p class="ltx_p">
11T55
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Arithmetic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/arithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/arithmeticalhierarchy">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
theory of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PolynomialRing.html">
polynomial rings
</a>
over finite fields
</p>
</div>
<div class="ltx_para" id="S2.SS5.p24">
<p class="ltx_p">
11T60 Finite upper half–planes
</p>
</div>
<div class="ltx_para" id="S2.SS5.p25">
<p class="ltx_p">
11T71 Algebraic coding theory; cryptography
</p>
</div>
<div class="ltx_para" id="S2.SS5.p26">
<p class="ltx_p">
11T99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS5.p27">
<p class="ltx_p">
11Uxx
<a class="nnexus_concept" href="http://planetmath.org/doublegroupoidwithconnection">
Connections
</a>
with logic
</p>
</div>
<div class="ltx_para" id="S2.SS5.p28">
<p class="ltx_p">
11U05 Decidability [See also 03B25]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p29">
<p class="ltx_p">
11U07
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Ultraproducts
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Ultraproduct.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reduceddirectproduct">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 03C20]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p30">
<p class="ltx_p">
11U09
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Model theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ModelTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/modeltheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 03Cxx]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p31">
<p class="ltx_p">
11U10 Nonstandard arithmetic [See also 03H15]
</p>
</div>
<div class="ltx_para" id="S2.SS5.p32">
<p class="ltx_p">
11U99 None of the above, but in this section
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS6">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.6
</span>
POLYNOMIALS and Field Theory
</h3>
<div class="ltx_para" id="S2.SS6.p1">
<p class="ltx_p">
12-XX FIELD THEORY AND POLYNOMIALS
</p>
</div>
<div class="ltx_para" id="S2.SS6.p2">
<p class="ltx_p">
12-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS6.p3">
<p class="ltx_p">
12-01 Instructional exposition (textbooks, tutorial papers, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS6.p4">
<p class="ltx_p">
12-02 Research exposition (monographs, survey articles)
</p>
</div>
<div class="ltx_para" id="S2.SS6.p5">
<p class="ltx_p">
12-06 Proceedings, conferences, collections, etc.
</p>
</div>
<div class="ltx_para" id="S2.SS6.p6">
<p class="ltx_p">
12Dxx Real and
<a class="nnexus_concept" href="http://planetmath.org/complex">
complex
</a>
fields
</p>
</div>
<div class="ltx_para" id="S2.SS6.p7">
<p class="ltx_p">
12D05 Polynomials:
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Factorization.html">
factorization
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS6.p8">
<p class="ltx_p">
12D10 Polynomials: location of zeros (algebraic
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
theorems
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) -For the analytic
theory, see 26C10, 30C15g
</p>
</div>
<div class="ltx_para" id="S2.SS6.p9">
<p class="ltx_p">
12D15 Fields related with sums of squares (
<a class="nnexus_concept" href="http://planetmath.org/formallyrealfield">
formally real fields
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Pythagorean fields
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PythagoreanField.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/pythagoreanfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, etc.) [See also 11Exx]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p10">
<p class="ltx_p">
12D99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS6.p11">
<p class="ltx_p">
12Exx General field theory
</p>
</div>
<div class="ltx_para" id="S2.SS6.p12">
<p class="ltx_p">
12E05 Polynomials (irreducibility, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS6.p13">
<p class="ltx_p">
12E10 Special polynomials
</p>
</div>
<div class="ltx_para" id="S2.SS6.p14">
<p class="ltx_p">
12E12
<a class="nnexus_concept" href="http://planetmath.org/equation">
Equations
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS6.p15">
<p class="ltx_p">
12E15
<a class="nnexus_concept" href="http://planetmath.org/divisionring">
Skew fields
</a>
, division rings [See also 11R52, 11R54, 11S45, 16Kxx]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p16">
<p class="ltx_p">
12E20 Finite fields (field–theoretic aspects)
</p>
</div>
<div class="ltx_para" id="S2.SS6.p17">
<p class="ltx_p">
12E25
<a class="nnexus_concept" href="http://planetmath.org/hilbertsirreducibilitytheorem">
Hilbertian fields
</a>
; Hilbert’ s irreducibility theorem
</p>
</div>
<div class="ltx_para" id="S2.SS6.p18">
<p class="ltx_p">
12E30 Field arithmetic
</p>
</div>
<div class="ltx_para" id="S2.SS6.p19">
<p class="ltx_p">
12E99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS6.p20">
<p class="ltx_p">
12Fxx Field extensions
</p>
</div>
<div class="ltx_para" id="S2.SS6.p21">
<p class="ltx_p">
12F05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Algebraic extensions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicExtension.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraicextension">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS6.p22">
<p class="ltx_p">
12F10
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Separable extensions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SeparableExtension.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/separable">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, Galois theory
</p>
</div>
<div class="ltx_para" id="S2.SS6.p23">
<p class="ltx_p">
12F12
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Inverse
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Inverse.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inversenumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/regularsemigroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inversestatement">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inversefunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
Galois theory
</p>
</div>
<div class="ltx_para" id="S2.SS6.p24">
<p class="ltx_p">
12F15 Inseparable extensions
</p>
</div>
<div class="ltx_para" id="S2.SS6.p25">
<p class="ltx_p">
12F20
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TranscendentalExtension.html">
Transcendental extensions
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS6.p26">
<p class="ltx_p">
12F99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS6.p27">
<p class="ltx_p">
12Gxx Homological methods (field theory)
</p>
</div>
<div class="ltx_para" id="S2.SS6.p28">
<p class="ltx_p">
12G05 Galois cohomology [See also 14F22, 16Hxx, 16K50]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p29">
<p class="ltx_p">
12G10 Cohomological dimension
</p>
</div>
<div class="ltx_para" id="S2.SS6.p30">
<p class="ltx_p">
12G99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS6.p31">
<p class="ltx_p">
12Hxx
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Differential
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Differential.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/totaldifferential">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and difference algebra
</p>
</div>
<div class="ltx_para" id="S2.SS6.p32">
<p class="ltx_p">
12H05 Differential algebra [See also 13Nxx]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p33">
<p class="ltx_p">
12H10 Difference algebra [See also 39Axx]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p34">
<p class="ltx_p">
12H20 Abstract differential equations [See also 34Mxx]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p35">
<p class="ltx_p">
12H25
<math alttext="p" class="ltx_Math" display="inline" id="S2.SS6.p35.m1">
<mi>
p
</mi>
</math>
-adic differential equations [See also 11S80, 14G20]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p36">
<p class="ltx_p">
12H99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS6.p37">
<p class="ltx_p">
12Jxx
<a class="nnexus_concept" href="http://planetmath.org/topologicalring">
Topological fields
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS6.p38">
<p class="ltx_p">
12J05
<a class="nnexus_concept" href="http://planetmath.org/gelfandtornheimtheorem">
Normed fields
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS6.p39">
<p class="ltx_p">
12J10 Valued fields
</p>
</div>
<div class="ltx_para" id="S2.SS6.p40">
<p class="ltx_p">
12J12 Formally
<math alttext="p" class="ltx_Math" display="inline" id="S2.SS6.p40.m1">
<mi>
p
</mi>
</math>
-adic fields
</p>
</div>
<div class="ltx_para" id="S2.SS6.p41">
<p class="ltx_p">
12J15
<a class="nnexus_concept" href="http://planetmath.org/orderedring">
Ordered fields
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS6.p42">
<p class="ltx_p">
12J17 Topological semi fields
</p>
</div>
<div class="ltx_para" id="S2.SS6.p43">
<p class="ltx_p">
12J20 General
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ValuationTheory.html">
valuation theory
</a>
[See also 13A18]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p44">
<p class="ltx_p">
12J25 Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p45">
<p class="ltx_p">
12J27 Krasner–Tate algebras [See mainly 32P05; see also 46S10, 47S10]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p46">
<p class="ltx_p">
12J99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS6.p47">
<p class="ltx_p">
12Kxx Generalizations of fields
</p>
</div>
<div class="ltx_para" id="S2.SS6.p48">
<p class="ltx_p">
12K05 Near–fields [See also 16Y30]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p49">
<p class="ltx_p">
12K10 Semi fields [See also 16Y60]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p50">
<p class="ltx_p">
12K99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS6.p51">
<p class="ltx_p">
12Lxx Connections with logic
</p>
</div>
<div class="ltx_para" id="S2.SS6.p52">
<p class="ltx_p">
12L05 Decidability [See also 03B25]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p53">
<p class="ltx_p">
12L10 Ultraproducts [See also 03C20]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p54">
<p class="ltx_p">
12L12 Model theory [See also 03C60]
</p>
</div>
<div class="ltx_para" id="S2.SS6.p55">
<p class="ltx_p">
12L15 Nonstandard arithmetic [See also 03H15]
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS7">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.7
</span>
COMMUTATIVE ALGEBRA
</h3>
<div class="ltx_para" id="S2.SS7.p1">
<p class="ltx_p">
13-XX
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CommutativeAlgebra.html">
COMMUTATIVE ALGEBRA
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS7.p2">
<p class="ltx_p">
13-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS7.p3">
<p class="ltx_p">
13-01 Instructional exposition (textbooks, tutorial papers, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS7.p4">
<p class="ltx_p">
13-02 Research exposition (monographs, survey articles)
</p>
</div>
<div class="ltx_para" id="S2.SS7.p5">
<p class="ltx_p">
13D05
<a class="nnexus_concept" href="http://planetmath.org/globaldimension">
Homological dimension
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS7.p6">
<p class="ltx_p">
13D07 Homological functors on modules (Tor, Ext, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS7.p7">
<p class="ltx_p">
13D09
<a class="nnexus_concept" href="http://planetmath.org/derivedcategory">
Derived categories
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS7.p8">
<p class="ltx_p">
13Axx General commutative ring theory
</p>
</div>
<div class="ltx_para" id="S2.SS7.p9">
<p class="ltx_p">
13A02
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Graded rings
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GradedRing.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/gradedring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 16W50]
</p>
</div>
<div class="ltx_para" id="S2.SS7.p10">
<p class="ltx_p">
13A05
<a class="nnexus_concept" href="http://planetmath.org/divisibility">
Divisibility
</a>
; factorizations [See also 13F15]
</p>
</div>
<div class="ltx_para" id="S2.SS7.p11">
<p class="ltx_p">
13A15 Ideals;
<a class="nnexus_concept" href="http://planetmath.org/multiplicativefunction">
multiplicative
</a>
ideal theory
</p>
</div>
<div class="ltx_para" id="S2.SS7.p12">
<p class="ltx_p">
13A18
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Valuations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Valuation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/truthvaluesemanticsforclassicalpropositionallogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and their generalizations [See also 12J20]
</p>
</div>
<div class="ltx_para" id="S2.SS7.p13">
<p class="ltx_p">
13A30 Associated graded rings of ideals (
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ReesRing.html">
Rees ring
</a>
, form ring), analytic spread and related topics
</p>
</div>
<div class="ltx_para" id="S2.SS7.p14">
<p class="ltx_p">
13A35
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Characteristic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/briggsianlogarithms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/characteristicsubgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/characteristic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
–methods (
<a class="nnexus_concept" href="http://planetmath.org/frobeniushomomorphism">
Frobenius endomorphism
</a>
) and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
reduction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/diamondlemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reducedword">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/division">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to characteristic ;
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TightClosure.html">
tight closure
</a>
[See also 13B22]
</p>
</div>
<div class="ltx_para" id="S2.SS7.p15">
<p class="ltx_p">
13A50 Actions of groups on commutative rings; invariant theory [See also 14L24]
</p>
</div>
<div class="ltx_para" id="S2.SS7.p16">
<p class="ltx_p">
13A99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS7.p17">
<p class="ltx_p">
13Bxx Ring extensions and related topics
</p>
</div>
<div class="ltx_para" id="S2.SS7.p18">
<p class="ltx_p">
13B02 Extension theory
</p>
</div>
<div class="ltx_para" id="S2.SS7.p19">
<p class="ltx_p">
13B05 Galois theory
</p>
</div>
<div class="ltx_para" id="S2.SS7.p20">
<p class="ltx_p">
13-03 Historical (must also be assigned at least one classification number from Section 01)
</p>
</div>
<div class="ltx_para" id="S2.SS7.p21">
<p class="ltx_p">
13-04 Explicit
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
machine
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Machine.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categoryofautomata">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
computation and programs (not the theory of computation or programming)
</p>
</div>
<div class="ltx_para" id="S2.SS7.p22">
<p class="ltx_p">
13-06 Proceedings, conferences, collections, etc.
</p>
</div>
<div class="ltx_para" id="S2.SS7.p23">
<p class="ltx_p">
13B21 Integral dependence; going up, going down
</p>
</div>
<div class="ltx_para" id="S2.SS7.p24">
<p class="ltx_p">
13B22
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Integral closure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/IntegralClosure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integralclosure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of rings and ideals [See also 13A35];
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
integrally closed
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/IntegrallyClosed.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integrallyclosed">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
rings, related rings (Japanese, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS7.p25">
<p class="ltx_p">
13B25 Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
</p>
</div>
<div class="ltx_para" id="S2.SS7.p26">
<p class="ltx_p">
13B30
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Rings of fractions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RingofFractions.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/localization">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extensionbylocalization">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Localization.html">
localization
</a>
[See also 16S85] 13B35 Completion [See also 13J10]
</p>
</div>
<div class="ltx_para" id="S2.SS7.p27">
<p class="ltx_p">
13B40 Etale and at extensions; Henselization; Artin approximation [See also 13J15, 14B12, 14B25]
</p>
</div>
<div class="ltx_para" id="S2.SS7.p28">
<p class="ltx_p">
13B99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS7.p29">
<p class="ltx_p">
13Cxx Theory of modules and ideals
</p>
</div>
<div class="ltx_para" id="S2.SS7.p30">
<p class="ltx_p">
13B02 Extension theory
</p>
</div>
<div class="ltx_para" id="S2.SS7.p31">
<p class="ltx_p">
13B05 Galois theory
</p>
</div>
<div class="ltx_para" id="S2.SS7.p32">
<p class="ltx_p">
13B10
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Morphism.html">
Morphisms
</a>
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS8">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.8
</span>
ALGEBRAIC GEOMETRY
</h3>
<div class="ltx_para" id="S2.SS8.p1">
<p class="ltx_p">
14-XX
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
ALGEBRAIC GEOMETRY
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicGeometry.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraicgeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicsinalgebraictopology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS8.p2">
<p class="ltx_p">
14-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS8.p3">
<p class="ltx_p">
14-01 Instructional exposition (textbooks, tutorial papers, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS8.p4">
<p class="ltx_p">
14-02 Research exposition (monographs, survey articles)
</p>
</div>
<div class="ltx_para" id="S2.SS8.p5">
<p class="ltx_p">
14-06 Proceedings, conferences, collections, etc.
</p>
</div>
<div class="ltx_para" id="S2.SS8.p6">
<p class="ltx_p">
14Axx
<a class="nnexus_concept" href="http://planetmath.org/axiomoffoundation">
Foundations
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS8.p7">
<p class="ltx_p">
14A05 Relevant commutative algebra [See also 13XX] 14A10 Varieties and morphisms
</p>
</div>
<div class="ltx_para" id="S2.SS8.p8">
<p class="ltx_p">
14A15 Schemes and morphisms
</p>
</div>
<div class="ltx_para" id="S2.SS8.p9">
<p class="ltx_p">
14A20 Generalizations (algebraic spaces, stacks)
</p>
</div>
<div class="ltx_para" id="S2.SS8.p10">
<p class="ltx_p">
14A22 Noncommutative algebraic geometry [See also 16S38] 14A25
<a class="nnexus_concept" href="http://planetmath.org/elementaryrecursivefunction">
Elementary
</a>
questions
</p>
</div>
<div class="ltx_para" id="S2.SS8.p11">
<p class="ltx_p">
14A99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS8.p12">
<p class="ltx_p">
14Bxx Local theory
</p>
</div>
<div class="ltx_para" id="S2.SS8.p13">
<p class="ltx_p">
14B05 Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p14">
<p class="ltx_p">
14B07
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Deformation.html">
deformations
</a>
of singularities [See also 14D15, 32S30]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p15">
<p class="ltx_p">
14B10 Infnitesimal methods [See also 13D10]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p16">
<p class="ltx_p">
14B12 Local
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DeformationTheory.html">
deformation theory
</a>
, Artin approximation, etc. [See also 13B40, 13D10]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p17">
<p class="ltx_p">
14B15 Local cohomology [See also 13D45, 32C36]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p18">
<p class="ltx_p">
14B20 Formal neighborhoods
</p>
</div>
<div class="ltx_para" id="S2.SS8.p19">
<p class="ltx_p">
14B25 Local structure of morphisms: etale, at, etc. [See also 13B40]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p20">
<p class="ltx_p">
14B25 Local structure of morphisms: etale, at, etc. [See also 13B40],
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinitesimal
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinitesimal.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/hyperreal">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
methods [See also 14B10, 14B12, 14D15, 32Gxx]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p21">
<p class="ltx_p">
13D15
<a class="nnexus_concept" href="http://planetmath.org/grothendieckgroup">
Grothendieck groups
</a>
,
<math alttext="K" class="ltx_Math" display="inline" id="S2.SS8.p21.m1">
<mi>
K
</mi>
</math>
–theory [See also 14C35, 18F30, 19Axx, 19D50]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p22">
<p class="ltx_p">
13D22 Homological conjectures (intersection theorems)
</p>
</div>
<div class="ltx_para" id="S2.SS8.p23">
<p class="ltx_p">
13D30 Torsion theory [See also 13C12, 18E40]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p24">
<p class="ltx_p">
13D40 Hilbert–Samuel and Hilbert–Kunz functions; Poincare series
</p>
</div>
<div class="ltx_para" id="S2.SS8.p25">
<p class="ltx_p">
13D45 Local cohomology [See also 14B15]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p26">
<p class="ltx_p">
13D99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS8.p27">
<p class="ltx_p">
13Exx Chain conditions, finiteness conditions
</p>
</div>
<div class="ltx_para" id="S2.SS8.p28">
<p class="ltx_p">
13E05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Noetherian rings
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NoetherianRing.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/noetherianring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and modules
</p>
</div>
<div class="ltx_para" id="S2.SS8.p29">
<p class="ltx_p">
13E10
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ArtinianRing.html">
Artinian rings
</a>
and modules, finite–dimensional algebras
</p>
</div>
<div class="ltx_para" id="S2.SS8.p30">
<p class="ltx_p">
13E15 Rings and modules of finite generation or
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
presentation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Presentation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/presentationofinversemonoidsandinversesemigroups">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/presentationsofalgebraicobjects">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/presentationofagroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
; number of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
generators
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/submodule">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/grothendieckcategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generatorofacategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS8.p31">
<p class="ltx_p">
13E99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS8.p32">
<p class="ltx_p">
13Fxx Arithmetic rings and other special rings
</p>
</div>
<div class="ltx_para" id="S2.SS8.p33">
<p class="ltx_p">
13F05 Dedekind, Prufer, Krull and Mori rings and their extensions
</p>
</div>
<div class="ltx_para" id="S2.SS8.p34">
<p class="ltx_p">
14Hxx
<span class="ltx_text ltx_font_bold">
Curves
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS8.p35">
<p class="ltx_p">
14H05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Algebraic functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraicfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
;
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
function fields
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FunctionField.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functionfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 11R58]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p36">
<p class="ltx_p">
14H10 Families, moduli (algebraic)
</p>
</div>
<div class="ltx_para" id="S2.SS8.p37">
<p class="ltx_p">
14H15 Families, moduli (analytic) [See also 30F10, 32G15]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p38">
<p class="ltx_p">
14H20 Singularities,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
local rings
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LocalRing.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/localring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 13Hxx, 14B05]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p39">
<p class="ltx_p">
14H25 Arithmetic
<a class="nnexus_concept" href="http://planetmath.org/groundfieldsandrings">
ground fields
</a>
[See also 11Dxx, 11G05, 14Gxx]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p40">
<p class="ltx_p">
14H30 coverings,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental group
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalGroup.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentalgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generalizedvankampensieferttheoremfordoublegroupoids">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 14E20, 14F35]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p41">
<p class="ltx_p">
14H37 Automorphisms
</p>
</div>
<div class="ltx_para" id="S2.SS8.p42">
<p class="ltx_p">
14H40
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Jacobians
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.5#E38">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/jacobianmatrix">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, Prym varieties [See also 32G20]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p43">
<p class="ltx_p">
14H42
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Theta functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/20.2#i">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ThetaFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
</sup>
; Schottky problem [See also 14K25, 32G20]
</p>
</div>
<div class="ltx_para" id="S2.SS8.p44">
<p class="ltx_p">
14H45 Special curves and curves of low genus
</p>
</div>
<div class="ltx_para" id="S2.SS8.p45">
<p class="ltx_p">
14H50 Plane and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SpaceCurve.html">
space curves
</a>
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS9">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.9
</span>
Category Theory
</h3>
<div class="ltx_para" id="S2.SS9.p1">
<p class="ltx_p">
18Axx general theory of categories and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functors
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Functor.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functor">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS9.p2">
<p class="ltx_p">
18A05
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definitions
</a>
, generalizations
</p>
</div>
<div class="ltx_para" id="S2.SS9.p3">
<p class="ltx_p">
18A10 graphs,
<a class="nnexus_concept" href="http://planetmath.org/precategory">
diagram schemes
</a>
, precategories [See especially 20L05]
</p>
</div>
<div class="ltx_para" id="S2.SS9.p4">
<p class="ltx_p">
18A15 Foundations,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
relations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relationonobjects">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to logic and
<a class="nnexus_concept" href="http://planetmath.org/deductivesystem">
deductive systems
</a>
[See also 03-XX]
</p>
</div>
<div class="ltx_para" id="S2.SS9.p5">
<p class="ltx_p">
18A20
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
epimorphisms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Epimorphism.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/epi">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/abeliancategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/ringhomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/typesofhomomorphisms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Monomorphism.html">
monomorphisms
</a>
, special classes of morphisms,
<a class="nnexus_concept" href="http://planetmath.org/kernelofamorphism">
null Morphisms
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS9.p6">
<p class="ltx_p">
18A22 Special properties of functors (
<a class="nnexus_concept" href="http://planetmath.org/groupaction">
faithful
</a>
, full, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS9.p7">
<p class="ltx_p">
18A23 Natural morphisms, dinatural morphisms
</p>
</div>
<div class="ltx_para" id="S2.SS9.p8">
<p class="ltx_p">
18A25
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functor categories
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/functorcategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functorcategory1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/commacategory">
comma categories
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS9.p9">
<p class="ltx_p">
18A30 limits and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
colimits
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Colimit.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/limitofafunctor">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
products
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Product.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/product">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/producttopology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categoricaldirectproduct">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, sums, directed limits,
<a class="nnexus_concept" href="http://planetmath.org/categoricalpullback">
pushouts
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/fibreproduct">
fiber products
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
equalizers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Equalizer.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equalizer">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, kernels, ends and coends, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS9.p10">
<p class="ltx_p">
18A32 Factorization of morphisms, substructures,
<a class="nnexus_concept" href="http://planetmath.org/quotientstructure">
quotient structures
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
congruences
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Congruence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence12">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/freeproductwithamalgamatedsubgroup">
amalgams
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS9.p11">
<p class="ltx_p">
18A35 Categories admitting limits (
<a class="nnexus_concept" href="http://planetmath.org/completecategory">
complete categories
</a>
), functors preserving limits, completions
</p>
</div>
<div class="ltx_para" id="S2.SS9.p12">
<p class="ltx_p">
18A40
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
adjoint functors
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AdjointFunctor.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/similarityandanalogoussystemsdynamicadjointnessandtopologicalequivalence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/adjointfunctor">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraiccategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(universal constructions, reective
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
subcategories
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Subcategory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/subcategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, Kan extensions, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS9.p13">
<p class="ltx_p">
18A99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS9.p14">
<p class="ltx_p">
18Bxx Special categories
</p>
</div>
<div class="ltx_para" id="S2.SS9.p15">
<p class="ltx_p">
18B05
<a class="nnexus_concept" href="http://planetmath.org/categoryofsets">
Category of sets
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
characterizations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Characterization.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/characterisation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 03XX]
</p>
</div>
<div class="ltx_para" id="S2.SS9.p16">
<p class="ltx_p">
18D05 (
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Category theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CategoryTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categorytheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bibliographyinalgebraictopologycategoriesandqat">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
;
<a class="nnexus_concept" href="http://mathworld.wolfram.com/HomologicalAlgebra.html">
homological algebra
</a>
, Categories with structure:
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Double categories
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/fundamentalgroupoidfunctor">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homotopydoublegroupoidofahausdorffspace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/doublecategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<math alttext="2" class="ltx_Math" display="inline" id="S2.SS9.p16.m1">
<mn>
2
</mn>
</math>
-categories, bicategories and generalizations)
</p>
</div>
<div class="ltx_para" id="S2.SS9.p17">
<p class="ltx_p">
18-00 (Category theory; homological algebra: General reference works (handbooks, dictionaries, bibliographies, etc.))
</p>
</div>
<div class="ltx_para" id="S2.SS9.p18">
<p class="ltx_p">
18E05 (Category theory; homological algebra:
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Abelian categories
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AbelianCategory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/standardabeliancategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/grothendieckcategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, Preadditive,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
additive categories
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AdditiveCategory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/additivecategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
)
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS10">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.10
</span>
Group Theory
</h3>
<div class="ltx_para" id="S2.SS10.p1">
<p class="ltx_p">
20C30 Representations of finite
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
symmetric groups
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SymmetricGroup.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/symmetricgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/symmetricgroup1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS10.p2">
<p class="ltx_p">
20C32 Representations of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinite
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinite.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extendedrealnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/infinite">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
symmetric groups
</p>
</div>
<div class="ltx_para" id="S2.SS10.p3">
<p class="ltx_p">
20F05 Generators, relations, and presentations
</p>
</div>
<div class="ltx_para" id="S2.SS10.p4">
<p class="ltx_p">
20F06 Cancellation theory; application of van Kampen diagrams [See also 57M05]
</p>
</div>
<div class="ltx_para" id="S2.SS10.p5">
<p class="ltx_p">
20F11 Groups of finite Morley rank [See also 03C45, 03C60]
</p>
</div>
<div class="ltx_para" id="S2.SS10.p6">
<p class="ltx_p">
20F12
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Commutator
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Commutator.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/commutatorbracket">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivedsubgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
calculus
</p>
</div>
<div class="ltx_para" id="S2.SS10.p7">
<p class="ltx_p">
20F14 Derived series, central series, and generalizations
</p>
</div>
<div class="ltx_para" id="S2.SS10.p8">
<p class="ltx_p">
20F16
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Solvable groups
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SolvableGroup.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/solvablegroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/supersolvablegroup">
supersolvable groups
</a>
[See also 20D10]
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS11">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.11
</span>
REAL FUNCTIONS
</h3>
<div class="ltx_para" id="S2.SS11.p1">
<p class="ltx_p">
26-XX
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RealFunction.html">
REAL FUNCTIONS
</a>
[See also 54C30]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p2">
<p class="ltx_p">
26-00 General reference works (handbooks, dictionaries, bibliographies,etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS11.p3">
<p class="ltx_p">
26-01 Instructional exposition (textbooks, tutorial papers, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS11.p4">
<p class="ltx_p">
26-02 Research exposition (monographs, survey articles)
</p>
</div>
<div class="ltx_para" id="S2.SS11.p5">
<p class="ltx_p">
26Axx Functions of one
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
variable
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Variable.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/variable">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS11.p6">
<p class="ltx_p">
26A03 Foundations: limits and generalizations, elementary
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
topology
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Topology.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topologicalspace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the line
</p>
</div>
<div class="ltx_para" id="S2.SS11.p7">
<p class="ltx_p">
26A06 One–variable calculus
</p>
</div>
<div class="ltx_para" id="S2.SS11.p8">
<p class="ltx_p">
26A09
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Elementary functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ElementaryFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/elementaryfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS11.p9">
<p class="ltx_p">
26A12 Rate of growth of functions,
<a class="nnexus_concept" href="http://planetmath.org/formaldefinitionoflandaunotation">
orders of infinity
</a>
, slowly varying functions [See also 26A48]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p10">
<p class="ltx_p">
26A15
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Continuity.html">
Continuity
</a>
and related questions (modulus of continuity, semicontinuity,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Discontinuity.html">
discontinuities
</a>
, etc.)
-For properties determined by
<a class="nnexus_concept" href="http://planetmath.org/fouriercoefficients">
Fourier coefficients
</a>
, see 42A16; for those determined by
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ApproximationProperty.html">
approximation properties
</a>
, see 41A25, 41A27g
</p>
</div>
<div class="ltx_para" id="S2.SS11.p11">
<p class="ltx_p">
26A16
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Lipschitz
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/lipschitzcondition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lipschitzfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(Holder) classes
</p>
</div>
<div class="ltx_para" id="S2.SS11.p12">
<p class="ltx_p">
26A18
<a class="nnexus_concept" href="http://planetmath.org/iteration">
Iteration
</a>
[See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p13">
<p class="ltx_p">
26A21 Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p14">
<p class="ltx_p">
26A24
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Differentiation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Differentiation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/higherorderderivatives">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(functions of one variable): general theory, generalized derivatives, mean–value theorems [See also 28A15]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p15">
<p class="ltx_p">
26A27 Non-differentiability (nondifferentiable functions, points of non-differentiability),
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
discontinuous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Discontinuous.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/discontinuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
derivatives
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/fixedpointsofnormalfunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivative">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS11.p16">
<p class="ltx_p">
26A30
<a class="nnexus_concept" href="http://planetmath.org/singularfunction">
Singular functions
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Cantor functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CantorFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cantorfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, functions with other special properties
</p>
</div>
<div class="ltx_para" id="S2.SS11.p17">
<p class="ltx_p">
26A33
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Fractional derivatives
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.15#E51">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FractionalDerivative.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fractionaldifferentiation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and integrals
</p>
</div>
<div class="ltx_para" id="S2.SS11.p18">
<p class="ltx_p">
26A36 Anti-differentiation
</p>
</div>
<div class="ltx_para" id="S2.SS11.p19">
<p class="ltx_p">
26A39 Denjoy and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PerronIntegral.html">
Perron integrals
</a>
, other special integrals
</p>
</div>
<div class="ltx_para" id="S2.SS11.p20">
<p class="ltx_p">
26A42 Integrals of Riemann, Stieltjes and Lebesgue type [See also 28XX]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p21">
<p class="ltx_p">
26A45
<a class="nnexus_concept" href="http://planetmath.org/bvfunction">
Functions of bounded variation
</a>
, generalizations
</p>
</div>
<div class="ltx_para" id="S2.SS11.p22">
<p class="ltx_p">
26A46
<a class="nnexus_concept" href="http://planetmath.org/absolutelycontinuousfunction">
Absolutely continuous functions
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS11.p23">
<p class="ltx_p">
26A48
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Monotonic functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MonotonicFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/orderpreservingmap">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, generalizations
</p>
</div>
<div class="ltx_para" id="S2.SS11.p24">
<p class="ltx_p">
26A51 Convexity, generalizations
</p>
</div>
<div class="ltx_para" id="S2.SS11.p25">
<p class="ltx_p">
26A99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS11.p26">
<p class="ltx_p">
26Bxx Functions of several variables
</p>
</div>
<div class="ltx_para" id="S2.SS11.p27">
<p class="ltx_p">
26B05 Continuity and differentiation questions
</p>
</div>
<div class="ltx_para" id="S2.SS11.p28">
<p class="ltx_p">
26B10
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Implicit function theorems
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ImplicitFunctionTheorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/implicitfunctiontheorem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, Jacobians,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
transformations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Transformation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functorialmorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/transformation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
with several variables
</p>
</div>
<div class="ltx_para" id="S2.SS11.p29">
<p class="ltx_p">
26B12 Calculus of vector functions
</p>
</div>
<div class="ltx_para" id="S2.SS11.p30">
<p class="ltx_p">
26B15 Integration: length, area, volume [See also 28A75, 51M25]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p31">
<p class="ltx_p">
26B20 Integral
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
formulas
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Formula.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/firstorderlanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(Stokes, Gauss, Green, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS11.p32">
<p class="ltx_p">
26B25 Convexity, generalizations
</p>
</div>
<div class="ltx_para" id="S2.SS11.p33">
<p class="ltx_p">
26B30 Absolutely continuous functions, functions of bounded variation
</p>
</div>
<div class="ltx_para" id="S2.SS11.p34">
<p class="ltx_p">
26B35 Special
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
properties of functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/propertiesofafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/propertiesoffunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of several variables,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/HoelderCondition.html">
Holder conditions
</a>
, etc.
</p>
</div>
<div class="ltx_para" id="S2.SS11.p35">
<p class="ltx_p">
26B40 Representation and superposition of functions
</p>
</div>
<div class="ltx_para" id="S2.SS11.p36">
<p class="ltx_p">
26B99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS11.p37">
<p class="ltx_p">
26Cxx Polynomials,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
rational functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RationalFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/rationalfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS11.p38">
<p class="ltx_p">
26C05 Polynomials: analytic properties, etc. [See also 12Dxx, 12Exx]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p39">
<p class="ltx_p">
26C10 Polynomials: location of zeros [See also 12D10, 30C15, 65H05]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p40">
<p class="ltx_p">
26C15 Rational functions [See also 14Pxx]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p41">
<p class="ltx_p">
26C99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS11.p42">
<p class="ltx_p">
26Dxx Inequalities
-For maximal function inequalities, see 42B25; for
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functional
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/interpretationofintuitionisticlogicbymeansoffunctionals">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/intuitionisticlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functional">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
inequalities, see 39B72; for probabilistic inequalities, see 60E15g
</p>
</div>
<div class="ltx_para" id="S2.SS11.p43">
<p class="ltx_p">
26D05 Inequalities for
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
trigonometric functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/4#PT3">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TrigonometricFunctions.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/definitionsintrigonometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and polynomials
</p>
</div>
<div class="ltx_para" id="S2.SS11.p44">
<p class="ltx_p">
26D07 Inequalities involving other types of functions
</p>
</div>
<div class="ltx_para" id="S2.SS11.p45">
<p class="ltx_p">
26D10 Inequalities involving derivatives and differential and integral operators
</p>
</div>
<div class="ltx_para" id="S2.SS11.p46">
<p class="ltx_p">
26D15 Inequalities for sums, series and integrals
</p>
</div>
<div class="ltx_para" id="S2.SS11.p47">
<p class="ltx_p">
26D20 Other analytical inequalities
</p>
</div>
<div class="ltx_para" id="S2.SS11.p48">
<p class="ltx_p">
26D99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS11.p49">
<p class="ltx_p">
26Exx Miscellaneous topics [See also 58Cxx]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p50">
<p class="ltx_p">
26E05 Real–
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AnalyticFunction.html">
analytic functions
</a>
[See also 32B05, 32C05]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p51">
<p class="ltx_p">
26E10 C1–functions, quasi–analytic functions [See also 58C25]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p52">
<p class="ltx_p">
26E15 Calculus of functions on infinite–dimensional spaces [See also 46G05, 58Cxx]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p53">
<p class="ltx_p">
26E20 Calculus of functions taking values in infinite–dimensional spaces [See also 46E40, 46G10, 58Cxx]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p54">
<p class="ltx_p">
26E25 Set-valued functions [See also 28B20, 49J53, 54C60] –For nonsmooth
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
analysis
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Analysis.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicentryonanalysis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, see 49J52, 58Cxx, 90Cxxg
</p>
</div>
<div class="ltx_para" id="S2.SS11.p55">
<p class="ltx_p">
26E30 Non–Archimedean analysis [See also 12J25]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p56">
<p class="ltx_p">
26E35
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Nonstandard analysis
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NonstandardAnalysis.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/nonstandardanalysis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 03H05, 28E05, 54J05]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p57">
<p class="ltx_p">
26E40 Constructive
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RealAnalysis.html">
real analysis
</a>
[See also 03F60]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p58">
<p class="ltx_p">
26E50 Fuzzy real analysis [See also 03E72, 28E10]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p59">
<p class="ltx_p">
26E60 Means [See also 47A64]
</p>
</div>
<div class="ltx_para" id="S2.SS11.p60">
<p class="ltx_p">
26E70 Real analysis on time scales or
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
measure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Measure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/measureonabooleanalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/measure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
chains –For dynamic equations on time scales or measure chains see 34N05g
</p>
</div>
<div class="ltx_para" id="S2.SS11.p61">
<p class="ltx_p">
26E99 None of the above, but in this section
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS12">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.12
</span>
MEASURE AND INTEGRATION
</h3>
<div class="ltx_para" id="S2.SS12.p1">
<p class="ltx_p">
28-XX MEASURE AND INTEGRATION
</p>
</div>
<div class="ltx_para" id="S2.SS12.p2">
<p class="ltx_p">
For analysis on manifolds, see 58-XXg
</p>
</div>
<div class="ltx_para" id="S2.SS12.p3">
<p class="ltx_p">
28-00 General reference works (handbooks, dictionaries, bibliographies,etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS12.p4">
<p class="ltx_p">
28-01 Instructional exposition (textbooks, tutorial papers, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS12.p5">
<p class="ltx_p">
28-02 Research exposition (monographs, survey articles)
</p>
</div>
<div class="ltx_para" id="S2.SS12.p6">
<p class="ltx_p">
28-06 Proceedings, conferences, collections, etc.
</p>
</div>
<div class="ltx_para" id="S2.SS12.p7">
<p class="ltx_p">
28Axx Classical
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MeasureTheory.html">
measure theory
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS12.p8">
<p class="ltx_p">
28A05 Classes of sets (
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BorelField.html">
Borel fields
</a>
, B–rings, etc.),
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
measurable sets
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MeasurableSet.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/measurablespace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/analyticset1">
Suslin sets
</a>
,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AnalyticSet.html">
analytic sets
</a>
[See also 03E15, 26A21, 54H05]
</p>
</div>
<div class="ltx_para" id="S2.SS12.p9">
<p class="ltx_p">
28A10 Real- or complex- valued
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
set functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SetFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/minimalandmaximalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS12.p10">
<p class="ltx_p">
28A12
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Content.html">
Contents
</a>
, measures,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
outer measures
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/OuterMeasure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/outermeasure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lebesgueoutermeasure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/choquetcapacity">
capacities
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS12.p11">
<p class="ltx_p">
28A15 Abstract differentiation theory, differentiation of set functions [See also 26A24]
</p>
</div>
<div class="ltx_para" id="S2.SS12.p12">
<p class="ltx_p">
28A20
<a class="nnexus_concept" href="http://planetmath.org/riemannmultipleintegral">
Measurable
</a>
and nonmeasurable functions,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
sequences
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Sequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categoricalsequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
measurable functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MeasurableFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/measurablefunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, modes of convergence
</p>
</div>
<div class="ltx_para" id="S2.SS12.p13">
<p class="ltx_p">
28A25 Integration with respect to measures and other set functions
</p>
</div>
<div class="ltx_para" id="S2.SS12.p14">
<p class="ltx_p">
28A33 Spaces of measures, convergence of measures [See also 46E27, 60Bxx]
</p>
</div>
<div class="ltx_para" id="S2.SS12.p15">
<p class="ltx_p">
28A35 Measures and integrals in
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ProductSpace.html">
product spaces
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS12.p16">
<p class="ltx_p">
28A50 Integration and disintegration of measures
</p>
</div>
<div class="ltx_para" id="S2.SS12.p17">
<p class="ltx_p">
28A51 Lifting theory [See also 46G15]
</p>
</div>
<div class="ltx_para" id="S2.SS12.p18">
<p class="ltx_p">
28A60 Measures on Boolean rings,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MeasureAlgebra.html">
measure algebras
</a>
[See also 54H10]
</p>
</div>
<div class="ltx_para" id="S2.SS12.p19">
<p class="ltx_p">
28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
</p>
</div>
<div class="ltx_para" id="S2.SS12.p20">
<p class="ltx_p">
28A78
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Hausdorff
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/separationaxioms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/hausdorffspace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and packing measures
</p>
</div>
<div class="ltx_para" id="S2.SS12.p21">
<p class="ltx_p">
28A80
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Fractals
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Fractal.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fractal">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 37Fxx]
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS13">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.13
</span>
FUNCTIONS OF A COMPLEX VARIABLE
</h3>
<div class="ltx_para" id="S2.SS13.p1">
<p class="ltx_p">
30-XX FUNCTIONS OF A
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexVariable.html">
COMPLEX VARIABLE
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS13.p2">
<p class="ltx_p">
For analysis on manifolds, see 58-XXg
</p>
</div>
<div class="ltx_para" id="S2.SS13.p3">
<p class="ltx_p">
30-00 General reference works (handbooks, dictionaries, bibliographies,etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS13.p4">
<p class="ltx_p">
30-01 Instructional exposition (textbooks, tutorial papers, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS13.p5">
<p class="ltx_p">
30-02 Research exposition (monographs, survey articles)
</p>
</div>
<div class="ltx_para" id="S2.SS13.p6">
<p class="ltx_p">
30-06 Proceedings, conferences, collections, etc.
</p>
</div>
<div class="ltx_para" id="S2.SS13.p7">
<p class="ltx_p">
30Axx General properties
</p>
</div>
<div class="ltx_para" id="S2.SS13.p8">
<p class="ltx_p">
30A05 Monogenic properties of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexFunction.html">
complex functions
</a>
(including polygenic and areolar
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MonogenicFunction.html">
monogenic functions
</a>
)
</p>
</div>
<div class="ltx_para" id="S2.SS13.p9">
<p class="ltx_p">
30A10 Inequalities in the complex domain
</p>
</div>
<div class="ltx_para" id="S2.SS13.p10">
<p class="ltx_p">
30A99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p11">
<p class="ltx_p">
30Bxx
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SeriesExpansion.html">
Series expansions
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS13.p12">
<p class="ltx_p">
30B10
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Power series
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PowerSeries.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/powerseries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(including lacunary series)
</p>
</div>
<div class="ltx_para" id="S2.SS13.p13">
<p class="ltx_p">
30B20 Random power series
</p>
</div>
<div class="ltx_para" id="S2.SS13.p14">
<p class="ltx_p">
30B30
<a class="nnexus_concept" href="http://planetmath.org/boundaryfrontier">
Boundary
</a>
<a class="nnexus_concept" href="http://planetmath.org/behavior">
behavior
</a>
of power series, over-convergence
</p>
</div>
<div class="ltx_para" id="S2.SS13.p15">
<p class="ltx_p">
30B40
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AnalyticContinuation.html">
Analytic continuation
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS13.p16">
<p class="ltx_p">
30B50
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Dirichlet series
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DirichletSeries.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dirichletseries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and other series expansions,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
exponential
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Exponential.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/exponential">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
series[See also 11M41, 42XX]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p17">
<p class="ltx_p">
30B60 Completeness problems,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
closure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Closure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/closure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of a system of functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p18">
<p class="ltx_p">
30B70
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Continued fractions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.12#i">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ContinuedFraction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuedfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 11A55, 40A15]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p19">
<p class="ltx_p">
30B99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p20">
<p class="ltx_p">
30Cxx Geometric function theory
</p>
</div>
<div class="ltx_para" id="S2.SS13.p21">
<p class="ltx_p">
30C10 Polynomials
</p>
</div>
<div class="ltx_para" id="S2.SS13.p22">
<p class="ltx_p">
30C15 Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
bounded
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/topologyofthecomplexplane">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bounded">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bounded1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/upperbound">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DirichletIntegrals.html">
Dirichlet integral
</a>
) For algebraic theory, see 12D10; for real methods, see 26C10g
</p>
</div>
<div class="ltx_para" id="S2.SS13.p23">
<p class="ltx_p">
30C20
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Conformal mappings
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ConformalMapping.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/riemannianmanifoldscategoryrm">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/conformalmapping">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of special domains
</p>
</div>
<div class="ltx_para" id="S2.SS13.p24">
<p class="ltx_p">
30C25 Covering theorems in conformal mapping theory
</p>
</div>
<div class="ltx_para" id="S2.SS13.p25">
<p class="ltx_p">
30C30 Numerical methods in conformal mapping theory [See also 65E05]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p26">
<p class="ltx_p">
30C35 General theory of conformal mappings
</p>
</div>
<div class="ltx_para" id="S2.SS13.p27">
<p class="ltx_p">
30C40 Kernel functions and applications
</p>
</div>
<div class="ltx_para" id="S2.SS13.p28">
<p class="ltx_p">
30C45 Special classes of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
univalent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Univalent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/univalentanalyticfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and multivalent functions (starlike, convex, bounded
<a class="nnexus_concept" href="http://planetmath.org/euclideantransformation">
rotation
</a>
, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS13.p29">
<p class="ltx_p">
30C50 Coe_cient problems for univalent and multivalent functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p30">
<p class="ltx_p">
30C55 General theory of univalent and multivalent functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p31">
<p class="ltx_p">
30C62
<a class="nnexus_concept" href="http://planetmath.org/quasiconformalmapping">
Quasiconformal mappings
</a>
in the plane
</p>
</div>
<div class="ltx_para" id="S2.SS13.p32">
<p class="ltx_p">
30C65 Quasiconformal mappings in
<math alttext="R^{n}" class="ltx_Math" display="inline" id="S2.SS13.p32.m1">
<msup>
<mi>
R
</mi>
<mi>
n
</mi>
</msup>
</math>
, other generalizations
</p>
</div>
<div class="ltx_para" id="S2.SS13.p33">
<p class="ltx_p">
30C70 Extremal problems for
<a class="nnexus_concept" href="http://planetmath.org/conformalmapping">
conformal
</a>
and quasiconformal mappings, variational methods
</p>
</div>
<div class="ltx_para" id="S2.SS13.p34">
<p class="ltx_p">
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
</p>
</div>
<div class="ltx_para" id="S2.SS13.p35">
<p class="ltx_p">
30C80
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Maximum principle
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/hausdorffsmaximumprinciple">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/maximumprinciple">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
; Schwarz’s lemma, Lindel method of principle, analogues and generalizations; subordination
</p>
</div>
<div class="ltx_para" id="S2.SS13.p36">
<p class="ltx_p">
30C85 Capacity and harmonic measure in the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexPlane.html">
complex plane
</a>
[See also 31A15]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p37">
<p class="ltx_p">
30C99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p38">
<p class="ltx_p">
30Dxx Entire and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MeromorphicFunction.html">
meromorphic functions
</a>
, and related topics
</p>
</div>
<div class="ltx_para" id="S2.SS13.p39">
<p class="ltx_p">
30D05 Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39XX]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p40">
<p class="ltx_p">
30D10 Representations of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
entire functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/EntireFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/entirefunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
by series and integrals
</p>
</div>
<div class="ltx_para" id="S2.SS13.p41">
<p class="ltx_p">
30D15 Special classes of entire functions and growth estimates
</p>
</div>
<div class="ltx_para" id="S2.SS13.p42">
<p class="ltx_p">
30D20 Entire functions, general theory
</p>
</div>
<div class="ltx_para" id="S2.SS13.p43">
<p class="ltx_p">
30D30 Meromorphic functions, general theory
</p>
</div>
<div class="ltx_para" id="S2.SS13.p44">
<p class="ltx_p">
30D35
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.16#SS1.p5">
Distribution
</a>
of values,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Nevanlinna theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NevanlinnaTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/nevanlinnatheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS13.p45">
<p class="ltx_p">
30D40 Cluster sets, prime ends, boundary behavior
</p>
</div>
<div class="ltx_para" id="S2.SS13.p46">
<p class="ltx_p">
30D45 Bloch functions,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NormalFunction.html">
normal functions
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/normalfamily">
normal families
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS13.p47">
<p class="ltx_p">
30D60 Quasi-analytic and other classes of functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p48">
<p class="ltx_p">
30D99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p49">
<p class="ltx_p">
30Exx Miscellaneous topics of analysis in the complex domain
</p>
</div>
<div class="ltx_para" id="S2.SS13.p50">
<p class="ltx_p">
30E05 Moment problems, interpolation problems
</p>
</div>
<div class="ltx_para" id="S2.SS13.p51">
<p class="ltx_p">
30E10 Approximation in the complex domain
</p>
</div>
<div class="ltx_para" id="S2.SS13.p52">
<p class="ltx_p">
30E15 Asymptotic representations in the complex domain
</p>
</div>
<div class="ltx_para" id="S2.SS13.p53">
<p class="ltx_p">
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p54">
<p class="ltx_p">
30E25 Boundary value problems [See also 45Exx]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p55">
<p class="ltx_p">
30E99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p56">
<p class="ltx_p">
30Fxx
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Riemann surfaces
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/21.7">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RiemannSurface.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/riemannsurface">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS13.p57">
<p class="ltx_p">
30F10 Compact Riemann surfaces and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Uniformization.html">
uniformization
</a>
[See also 14H15, 32G15]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p58">
<p class="ltx_p">
30F15
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Harmonic functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/harmonicfunction1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/harmonicfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
on Riemann surfaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p59">
<p class="ltx_p">
30F20 Classification theory of Riemann surfaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p60">
<p class="ltx_p">
30F25 Ideal boundary theory
</p>
</div>
<div class="ltx_para" id="S2.SS13.p61">
<p class="ltx_p">
30F30 Differentials on Riemann surfaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p62">
<p class="ltx_p">
30F35 Fuchsian groups and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AutomorphicFunction.html">
automorphic functions
</a>
[See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p63">
<p class="ltx_p">
30F40
<a class="nnexus_concept" href="http://mathworld.wolfram.com/KleinianGroup.html">
Kleinian groups
</a>
[See also 20H10]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p64">
<p class="ltx_p">
30F45 Conformal metrics (hyperbolic, Poincar
<math alttext="\'{e}" class="ltx_Math" display="inline" id="S2.SS13.p64.m1">
<mi mathvariant="normal">
é
</mi>
</math>
, distance functions)
</p>
</div>
<div class="ltx_para" id="S2.SS13.p65">
<p class="ltx_p">
30F50 Klein surfaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p66">
<p class="ltx_p">
30F60 Teichmuller theory [See also 32G15]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p67">
<p class="ltx_p">
30F99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p68">
<p class="ltx_p">
30Gxx Generalized function theory
</p>
</div>
<div class="ltx_para" id="S2.SS13.p69">
<p class="ltx_p">
30G06 Non-Archimedean function theory [See also 12J25]; nonstandard function theory [See also 03H05]
</p>
</div>
<div class="ltx_para" id="S2.SS13.p70">
<p class="ltx_p">
30G12 Finely
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
holomorphic functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/HolomorphicFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/holomorphic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and topological function theory
</p>
</div>
<div class="ltx_para" id="S2.SS13.p71">
<p class="ltx_p">
30G20 Generalizations of Bers or Vekua type (pseudoanalytic,
<math alttext="p" class="ltx_Math" display="inline" id="S2.SS13.p71.m1">
<mi>
p
</mi>
</math>
–analytic, etc.)
</p>
</div>
<div class="ltx_para" id="S2.SS13.p72">
<p class="ltx_p">
30G25
<a class="nnexus_concept" href="http://planetmath.org/discrete">
Discrete
</a>
analytic functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p73">
<p class="ltx_p">
30G30 Other generalizations of analytic functions (including abstract–valued functions)
</p>
</div>
<div class="ltx_para" id="S2.SS13.p74">
<p class="ltx_p">
30G35 Functions of hypercomplex variables and generalized variables 30G99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p75">
<p class="ltx_p">
30Hxx Spaces and algebras of analytic functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p76">
<p class="ltx_p">
30H05 Bounded analytic functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p77">
<p class="ltx_p">
30H10 Hardy spaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p78">
<p class="ltx_p">
30H15 Nevanlinna class and Smirnov class
</p>
</div>
<div class="ltx_para" id="S2.SS13.p79">
<p class="ltx_p">
30H20 Bergman spaces, Fock spaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p80">
<p class="ltx_p">
30H25 Besov spaces and
<math alttext="Q_{p}" class="ltx_Math" display="inline" id="S2.SS13.p80.m1">
<msub>
<mi>
Q
</mi>
<mi>
p
</mi>
</msub>
</math>
-spaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p81">
<p class="ltx_p">
30H30 Bloch spaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p82">
<p class="ltx_p">
30H35 BMO–spaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p83">
<p class="ltx_p">
30H50 Algebras of analytic functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p84">
<p class="ltx_p">
30H80 Corona theorems
</p>
</div>
<div class="ltx_para" id="S2.SS13.p85">
<p class="ltx_p">
30H99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p86">
<p class="ltx_p">
30Jxx Function theory on the disc
</p>
</div>
<div class="ltx_para" id="S2.SS13.p87">
<p class="ltx_p">
30J05
<a class="nnexus_concept" href="http://planetmath.org/innerfunction">
Inner functions
</a>
</p>
</div>
<div class="ltx_para" id="S2.SS13.p88">
<p class="ltx_p">
30J10
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Blaschke products
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BlaschkeProduct.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/blaschkeproduct">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS13.p89">
<p class="ltx_p">
30J15 Singular inner functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p90">
<p class="ltx_p">
30J99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p91">
<p class="ltx_p">
30Kxx
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Universal
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/universalmappingproperty">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universalstructure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universalrelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
holomorphic functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p92">
<p class="ltx_p">
30K05 Universal
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Taylor series
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TaylorSeries.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/taylorseries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
<div class="ltx_para" id="S2.SS13.p93">
<p class="ltx_p">
30K10 Universal Dirichlet series
</p>
</div>
<div class="ltx_para" id="S2.SS13.p94">
<p class="ltx_p">
30K15 Bounded universal functions
</p>
</div>
<div class="ltx_para" id="S2.SS13.p95">
<p class="ltx_p">
30K20 Compositional universality
</p>
</div>
<div class="ltx_para" id="S2.SS13.p96">
<p class="ltx_p">
30K99 None of the above, but in this section
</p>
</div>
<div class="ltx_para" id="S2.SS13.p97">
<p class="ltx_p">
30Lxx Analysis on metric spaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p98">
<p class="ltx_p">
30L05 Geometric embeddings of metric spaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p99">
<p class="ltx_p">
30L10 Quasiconformal mappings in metric spaces
</p>
</div>
<div class="ltx_para" id="S2.SS13.p100">
<p class="ltx_p">
35Q40 Partial differential equations
</p>
</div>
<div class="ltx_para" id="S2.SS13.p101">
<p class="ltx_p">
81Q05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Quantum theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quantumoperatoralgebrasinquantumfieldtheories">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumfieldtheoriesqft">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
: General mathematical topics and methods in quantum theory
</p>
</div>
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<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
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Title
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<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/amsmscclassificationofarticlesandconversiontables">
AMS MSC classification of articles and conversion tables
</a>
</td>
</tr>
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<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
AMSMSCClassificationOfArticlesAndConversionTables
</td>
</tr>
<tr class="ltx_tr">
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Date of creation
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<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 19:22:28
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
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<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 19:22:28
</td>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
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<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
26
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
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Topic
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Classification
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<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-02
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-01
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
MSC
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
philosophy of mathematics classification
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
of articles and conversion tablesmiscellaneous mathematics classifications
</td>
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0 | FailureFunctionTests | http://planetmath.org/FailureFunctionTests | <!DOCTYPE html>
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failure function tests
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<h1 class="ltx_title ltx_title_document">
failure function tests
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
FAILURE FUNCTION
</h2>
<div class="ltx_para" id="S1.p1">
<p class="ltx_p">
An abstract
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
is:
</p>
</div>
<div class="ltx_para" id="S1.p2">
<p class="ltx_p">
Let
<math alttext="\phi(x)" class="ltx_Math" display="inline" id="S1.p2.m1">
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
be a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<math alttext="x" class="ltx_Math" display="inline" id="S1.p2.m2">
<mi>
x
</mi>
</math>
. Then,
<math alttext="x=\psi(x_{0})" class="ltx_Math" display="inline" id="S1.p2.m3">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mi>
ψ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
is a failure function if the values of
<math alttext="x" class="ltx_Math" display="inline" id="S1.p2.m4">
<mi>
x
</mi>
</math>
generated by
<math alttext="\psi(x_{0})" class="ltx_Math" display="inline" id="S1.p2.m5">
<mrow>
<mi>
ψ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
, when substituted in
<math alttext="\phi(x)" class="ltx_Math" display="inline" id="S1.p2.m6">
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
, generate only failures in accordance with our definition of a failure. Here
<math alttext="x_{0}" class="ltx_Math" display="inline" id="S1.p2.m7">
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</math>
is a specific value of
<math alttext="x" class="ltx_Math" display="inline" id="S1.p2.m8">
<mi>
x
</mi>
</math>
.
</p>
</div>
<section class="ltx_subsection" id="S1.SS1">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
1.1
</span>
Examples
</h3>
<div class="ltx_para" id="S1.SS1.p1">
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
(i) Let the mother function be a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
polynomial
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/polynomial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomialring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in x (coeffficients belong to
<math alttext="\mathcal{Z}" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mi class="ltx_font_mathcaligraphic">
𝒵
</mi>
</math>
), say
<math alttext="\phi(x)" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
.
Let our definition of a failure be a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
composite number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CompositeNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/compositenumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. Then,
<math alttext="x=\psi(x_{0})=x_{0}+k(\phi(x_{0}))" class="ltx_Math" display="inline" id="I1.i1.p1.m3">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mi>
ψ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo>
+
</mo>
<mrow>
<mi>
k
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</mrow>
</math>
is a failure function because the values of x generated by
<math alttext="\phi(x_{0})" class="ltx_Math" display="inline" id="I1.i1.p1.m4">
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
, when substituted in
<math alttext="\phi(x)" class="ltx_Math" display="inline" id="I1.i1.p1.m5">
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
, generate only failures.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
(ii) Let the mother function be an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
exponential function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/4.2#iii">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://dlmf.nist.gov/4.2#E19">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ExponentialFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
</sup>
, say
<math alttext="\phi(x)=a^{x}+c" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mrow>
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
a
</mi>
<mi>
x
</mi>
</msup>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mrow>
</math>
.
Then
<math alttext="x=\psi(x_{0})=x_{0}+k.Eulerphi(\phi(x_{0}))" class="ltx_Math" display="inline" id="I1.i2.p1.m2">
<mrow>
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mi>
ψ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo>
+
</mo>
<mi>
k
</mi>
</mrow>
</mrow>
<mo>
.
</mo>
<mrow>
<mi>
E
</mi>
<mo>
</mo>
<mi>
u
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
is a failure function since the values of x generated by
<math alttext="\psi(x_{0})" class="ltx_Math" display="inline" id="I1.i2.p1.m3">
<mrow>
<mi>
ψ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
, when substituted in the mother function, generate only failures.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
(iii) Let our definition of a failure be a non-Carmichael number.
Let the mother function
<math alttext="\phi(x)" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
be
<math alttext="2^{n}+49" class="ltx_Math" display="inline" id="I1.i3.p1.m2">
<mrow>
<msup>
<mn>
2
</mn>
<mi>
n
</mi>
</msup>
<mo>
+
</mo>
<mn>
49
</mn>
</mrow>
</math>
. Then,
<math alttext="n=5+6k" class="ltx_Math" display="inline" id="I1.i3.p1.m3">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mrow>
<mn>
5
</mn>
<mo>
+
</mo>
<mrow>
<mn>
6
</mn>
<mo>
</mo>
<mi>
k
</mi>
</mrow>
</mrow>
</mrow>
</math>
is its failure function
<math alttext="\psi(x)" class="ltx_Math" display="inline" id="I1.i3.p1.m4">
<mrow>
<mi>
ψ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
</li>
</ul>
</div>
</section>
<section class="ltx_subsection" id="S1.SS2">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
1.2
</span>
Note
</h3>
<div class="ltx_para" id="S1.SS2.p1">
<p class="ltx_p">
Here too our definition of a failure is a composite number and k belongs to N.
</p>
</div>
<div class="ltx_para ltx_align_right" id="S1.SS2.p2">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/failurefunctiontests">
failure function tests
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
FailureFunctionTests
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 19:33:30
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 19:33:30
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
14
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-02
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00-01
</td>
</tr>
</tbody>
</table>
</div>
</section>
</section>
</article>
</div>
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0 | GeorgePolya | http://planetmath.org/GeorgePolya | <!DOCTYPE html>
<html>
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<title>
George Pólya
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
George Pólya
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
George Pólya
</h2>
<div class="ltx_para" id="S1.p1">
<p class="ltx_p">
American mathematician,
Born: György Pólya in Budapest, Hungary in 1887, (
<span class="ltx_text ltx_font_italic">
d.
</span>
1985 in Palo Alto, USA)
</p>
</div>
<div class="ltx_para" id="S1.p2">
<p class="ltx_p">
An excellent problem solver. He designed a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complete
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/soundcomplete">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/completebinarytree">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/completegraph">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/kripkesemantics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/maximallyconsistent">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
strategy for problem solving
that can help both the beginner and the advanced mathematician to solve both mathematical
and physical problems.
</p>
</div>
<div class="ltx_para" id="S1.p3">
<p class="ltx_p">
“
<span class="ltx_text ltx_font_italic">
His first job was to tutor the young son, Gregor, of a Hungarian baron. Gregor struggled due to his lack of problem solving skills.
</span>
” Thus, according to Long (
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
]
</cite>
), Polya insisted that the skill of “
<span class="ltx_text ltx_font_italic">
solving problems was not an inborn quality but, something that could be taught
</span>
”.
</p>
</div>
<div class="ltx_para" id="S1.p4">
<p class="ltx_p">
In 1940,
<a class="nnexus_concept" href="http://planetmath.org/georgepolya">
George Polya
</a>
and his wife, Stella, (the only daughter of Swiss Dr. Weber, in Zurich) moved to the
<a class="nnexus_concept" href="http://planetmath.org/historyofmathematicsintheunitedstatesofamerica">
United States
</a>
because of their justified fear of Nazism in Germany (
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
]
</cite>
).
</p>
</div>
<div class="ltx_para" id="S1.p5">
<p class="ltx_p">
He taught at first, at Brown University, and then he moved permanently with his wife
to Stanford University. Became Professor Emeritus at Stanford in 1953. He also taught many classes to
<a class="nnexus_concept" href="http://planetmath.org/elementaryrecursivefunction">
elementary
</a>
and secondary classroom teachers, inspiring them how to motivate and teach their students how to solve problems.
His research was in several mathematical areas:
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
functional analysis
</a>
, probability, number theory, algebra,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Combinatorics.html">
combinatorics
</a>
and
<a class="nnexus_concept" href="http://planetmath.org/moscowmathematicalpapyrus">
geometry
</a>
.
Recieved The
<a class="nnexus_concept" href="http://planetmath.org/mathematicalassociationofamerica">
Mathematical Association of America
</a>
Award ”for articles of expository excellence published in the College Mathematics Journal”.
He published in 1945 the book “
<span class="ltx_text ltx_font_italic">
How to Solve It
</span>
” that sold in more than one million copies in 18
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
languages
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/language">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/signature">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. Although an appropriate strategy can be learned by solving many problems, it is learned much faster if several,
<a class="nnexus_concept" href="http://planetmath.org/equivalentmachines">
similar
</a>
examples are worked out first with a teacher on an one–on–one basis. Here are some of the highlights of his simple strategies for problem solving:
</p>
</div>
<div class="ltx_para" id="S1.p6">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Understand the Problem
</span>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
Devise a Plan on how to approach the Problem; such a plan may include one or several of the following:
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
Make a first guess to begin with, and then verify the answer
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
Solve a simpler problem
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
Consider special cases that are much easier to solve
</p>
</div>
</li>
<li class="ltx_item" id="I1.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I1.i6.p1">
<p class="ltx_p">
Look for a pattern
</p>
</div>
</li>
<li class="ltx_item" id="I1.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I1.i7.p1">
<p class="ltx_p">
Draw a picture
</p>
</div>
</li>
<li class="ltx_item" id="I1.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I1.i8.p1">
<p class="ltx_p">
Use a model
</p>
</div>
</li>
<li class="ltx_item" id="I1.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I1.i9.p1">
<p class="ltx_p">
Use direct reasoning but double-check your results
</p>
</div>
</li>
<li class="ltx_item" id="I1.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
10.
</span>
<div class="ltx_para" id="I1.i10.p1">
<p class="ltx_p">
Use a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
formula
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Formula.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/firstorderlanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
that you fully understand and have used before
</p>
</div>
</li>
<li class="ltx_item" id="I1.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
11.
</span>
<div class="ltx_para" id="I1.i11.p1">
<p class="ltx_p">
Eliminate possibilities
</p>
</div>
</li>
<li class="ltx_item" id="I1.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
12.
</span>
<div class="ltx_para" id="I1.i12.p1">
<p class="ltx_p">
Carry out the Plan, as modified by partial solutions
</p>
</div>
</li>
<li class="ltx_item" id="I1.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
13.
</span>
<div class="ltx_para" id="I1.i13.p1">
<p class="ltx_p">
If plan doesn’t work, make an improved plan but do not give up
</p>
</div>
</li>
<li class="ltx_item" id="I1.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
14.
</span>
<div class="ltx_para" id="I1.i14.p1">
<p class="ltx_p">
Last-but-not-least, look back and examine critically your solution(s):
</p>
</div>
</li>
<li class="ltx_item" id="I1.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
15.
</span>
<div class="ltx_para" id="I1.i15.p1">
<p class="ltx_p">
Does the solution make sense? Does it check out in particular cases?
</p>
</div>
</li>
<li class="ltx_item" id="I1.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
16.
</span>
<div class="ltx_para" id="I1.i16.p1">
<p class="ltx_p">
Make sure there are no gaps and no steps missing.
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="S1.p7">
<p class="ltx_p">
He published also a two-volume book, “
<span class="ltx_text ltx_font_italic">
Mathematics and Plausible Reasoning
</span>
” in 1954, and
<span class="ltx_text ltx_font_italic">
Mathematical Discovery
</span>
in 1962.
</p>
</div>
</section>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Long, C. T., & DeTemple, D. W.,
<span class="ltx_text ltx_font_italic">
Mathematical reasoning for elementary teachers
</span>
. (1996). Reading MA: Addison-Wesley
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
Reimer, L., & Reimer, W.
<span class="ltx_text ltx_font_italic">
Mathematicians are people too
</span>
. (Volume 2). (1995) Dale Seymour Publications
</span>
</li>
<li class="ltx_bibitem" id="bib.bib3">
<span class="ltx_bibtag ltx_role_refnum">
3
</span>
<span class="ltx_bibblock">
Polya, G.
<span class="ltx_text ltx_font_italic">
How to solve it
</span>
. (1957) Garden City, NY: Doubleday and Co., Inc.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib4">
<span class="ltx_bibtag ltx_role_refnum">
4
</span>
<span class="ltx_bibblock">
A. Motter,,
<span class="ltx_text ltx_font_typewriter">
http://www.math.wichita.edu/history/men/polya.html
</span>
“A Biography of George Polya”
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p2">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
George Pólya
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
GeorgePolya
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:26:56
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:26:56
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
12
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Biography
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-02
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 01A70
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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MSC classification of objects: articles search
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
MSC classification of objects: articles search
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
Links to the AMS MSC 2010 Classification PDF of all MSC entries available, and the AMS MSC website
</h2>
<div class="ltx_para" id="S1.p1">
<p class="ltx_p">
Because the AMS MSC does not seem to be available at present as a link with NS 1.5, here are the links to
the AMS that provide the MSC classifications:
</p>
</div>
<div class="ltx_para" id="S1.p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://www.ams.org/mathscinet/msc/pdfs/classifications2010.pdf
</span>
All MSC 2010 in one PDF :
</p>
</div>
<div class="ltx_para" id="S1.p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://www.ams.org/mathscinet/msc/msc2010.html
</span>
The AMS MSC website with its maths specialized Search Engine
</p>
</div>
<section class="ltx_subsection" id="S1.SS1">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
1.1
</span>
Conversion Tables
</h3>
<div class="ltx_para" id="S1.SS1.p1">
<p class="ltx_p">
http://www.ams.org/mathscinet/msc/pdfs/classifications2010.pdf
</p>
</div>
<div class="ltx_para" id="S1.SS1.p2">
<p class="ltx_p">
CONVERSIONS: http://www.ams.org/mathscinet/msc/conv.html?from=2000
</p>
</div>
<div class="ltx_para" id="S1.SS1.p3">
<p class="ltx_p">
<math alttext="MSC2000~{}Classification~{}Codes~{}\to~{}MSC2010~{}Classification~{}Codes~{}Update." class="ltx_Math" display="inline" id="S1.SS1.p3.m1">
<mrow>
<mrow>
<mrow>
<mi>
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</mi>
<mo>
</mo>
<mi>
S
</mi>
<mo>
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</mi>
<mo>
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<mpadded width="+3.3pt">
<mn>
2000
</mn>
</mpadded>
<mo>
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</mi>
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</mpadded>
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</mrow>
<mo rspace="5.8pt">
→
</mo>
<mrow>
<mi>
M
</mi>
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S
</mi>
<mo>
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C
</mi>
<mo>
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<mpadded width="+3.3pt">
<mn>
2010
</mn>
</mpadded>
<mo>
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C
</mi>
<mo>
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l
</mi>
<mo>
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</mi>
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<mi>
c
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
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</mi>
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</mi>
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n
</mi>
</mpadded>
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<mpadded width="+3.3pt">
<mi>
s
</mi>
</mpadded>
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U
</mi>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
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</mi>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
Date: 14 October 2009
</p>
</div>
<div class="ltx_para" id="S1.SS1.p4">
<p class="ltx_p">
http://www.ams.org/mathscinet/msc/conv.html?from=2010
</p>
</div>
<div class="ltx_para" id="S1.SS1.p5">
<p class="ltx_p">
MSC2010 Classification Codes –¿ MSC2000 Classification Codes
</p>
</div>
</section>
<section class="ltx_subsection" id="S1.SS2">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
1.2
</span>
General Classifications
</h3>
<div class="ltx_para" id="S1.SS2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
00-01 Instructional Expositions
</span>
</p>
</div>
<div class="ltx_para" id="S1.SS2.p2">
<p class="ltx_p">
00-02 Research Expositions
</p>
</div>
<div class="ltx_para" id="S1.SS2.p3">
<p class="ltx_p">
00A05
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
General mathematics
</a>
</p>
</div>
<div class="ltx_para" id="S1.SS2.p4">
<p class="ltx_p">
00A35 Methodology of mathematics, didactics
</p>
</div>
<div class="ltx_para" id="S1.SS2.p5">
<p class="ltx_p">
00A66 Mathematics and visual arts, visualization
</p>
</div>
<div class="ltx_para" id="S1.SS2.p6">
<p class="ltx_p">
00A79 Physics
</p>
</div>
<div class="ltx_para" id="S1.SS2.p7">
<p class="ltx_p">
00A69 General applied mathematics
</p>
</div>
<div class="ltx_para" id="S1.SS2.p8">
<p class="ltx_p">
00A73 Dimensional analysis
</p>
</div>
<div class="ltx_para" id="S1.SS2.p9">
<p class="ltx_p">
00A15 Bibliographies
</p>
</div>
<div class="ltx_para" id="S1.SS2.p10">
<p class="ltx_p">
00A71 Theory of mathematical modeling
</p>
</div>
<div class="ltx_para" id="S1.SS2.p11">
<p class="ltx_p">
00A30
<a class="nnexus_concept" href="http://planetmath.org/foundationsofmathematicsoverview">
Philosophy of mathematics
</a>
and 03A05
</p>
</div>
<div class="ltx_para" id="S1.SS2.p12">
<p class="ltx_p">
00A99 Miscellaneous topics
</p>
</div>
<div class="ltx_para" id="S1.SS2.p13">
<p class="ltx_p">
00B99 None of the above, but in this
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
section
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/operationsonrelations">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sectionofafiberbundle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/vectorbundle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sheaf">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sectionsandretractions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/typesofmorphisms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/monic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
</section>
<section class="ltx_subsection" id="S1.SS3">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
1.3
</span>
Examples of Objects at PM with General Classifications
</h3>
<div class="ltx_para" id="S1.SS3.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
00-01 - General: Instructional exposition (textbooks, tutorial papers, etc.–here actually shown are only Encyclopedia articles):
</span>
</p>
</div>
<div class="ltx_para" id="S1.SS3.p2">
<p class="ltx_p">
<math alttext="(1+\alpha/n)^{n}" class="ltx_Math" display="inline" id="S1.SS3.p2.m1">
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mrow>
<mi>
α
</mi>
<mo>
/
</mo>
<mi>
n
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mi>
n
</mi>
</msup>
</math>
is
<a class="nnexus_concept" href="http://planetmath.org/poset">
monotone
</a>
for large
<math alttext="n" class="ltx_Math" display="inline" id="S1.SS3.p2.m2">
<mi>
n
</mi>
</math>
owned by Uri Weiss
</p>
</div>
<div class="ltx_para" id="S1.SS3.p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
ad hoc
</span>
, owned by Cam McLeman
</p>
</div>
<div class="ltx_para" id="S1.SS3.p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concept" href="http://planetmath.org/amsmscclassificationofarticlesandconversiontables">
AMS MSC classification of articles and conversion tables
</a>
</span>
, owned by bci1
</p>
</div>
<div class="ltx_para" id="S1.SS3.p5">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
dimension
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/dimensionofaposet">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dimension">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
, owned by Boris Bukh
</p>
</div>
<div class="ltx_para" id="S1.SS3.p6">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concept" href="http://planetmath.org/expression">
expression
</a>
</span>
, owned by Warren Buck
</p>
</div>
<div class="ltx_para" id="S1.SS3.p7">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
infix
</span>
, notation owned by Aaron Krowne
</p>
</div>
<div class="ltx_para" id="S1.SS3.p8">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
rigid
</span>
, owned by matte
</p>
</div>
<div class="ltx_para" id="S1.SS3.p9">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
strict
</span>
, owned by Raymond Puzio
</p>
</div>
<div class="ltx_para" id="S1.SS3.p10">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concept" href="http://planetmath.org/textbookprojectsonplanetmath">
Textbook projects on PlanetMath
</a>
</span>
, owned by John Smith
</p>
</div>
<div class="ltx_para" id="S1.SS3.p11">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concept" href="http://planetmath.org/toytheorem">
toy theorem
</a>
, owned by matte
</span>
</p>
</div>
<div class="ltx_para" id="S1.SS3.p12">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold ltx_font_italic">
One can see many such examples by clicking on the AMS MSC classification specified at the bottom of any published article at PM.
<span class="ltx_text ltx_font_upright">
</span>
</span>
</p>
</div>
</section>
</section>
<section class="ltx_section" id="S2">
<h2 class="ltx_title ltx_font_bold ltx_title_section">
<span class="ltx_tag ltx_tag_section">
2
</span>
Algebraic Logic Examples of AMS MSC Classifications Utilized in PM articles
</h2>
<div class="ltx_para" id="S2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
msc:00-01, msc:00-02
</span>
</p>
</div>
<div class="ltx_para" id="S2.p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
00A15 Bibliographies
</span>
</p>
</div>
<section class="ltx_subsection" id="S2.SS1">
<h3 class="ltx_title ltx_font_bold ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.1
</span>
Algebraic Logics
</h3>
<div class="ltx_para" id="S2.SS1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G05
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Boolean algebras
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BooleanAlgebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/booleanlattice">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 06Exx]
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G12
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
quantum logic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quantumlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categoricalquantumlmlogicalgebras">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 06C15, 81P10]
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G20
</span>
<math alttext="\L{}" class="ltx_Math" display="inline" id="S2.SS1.p3.m1">
<mi>
Ł
</mi>
</math>
<span class="ltx_text ltx_font_bold">
ukasiewicz and Post algebras [See also 06D25, 06D30]
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G10 Lattices and related
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structures
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
[See also 06Bxx]
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p5">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G30 Categorical logic, topoi [See also 18B25, 18C05, 18C10]
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p6">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03H10 Other applications of nonstandard models (economics, physics, etc.)
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p7">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G15 Cylindric and
<a class="nnexus_concept" href="http://planetmath.org/polyadicalgebra">
polyadic algebras
</a>
;
<a class="nnexus_concept" href="http://planetmath.org/relationalgebra">
relation algebras
</a>
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p8">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G20 Lukasiewicz and Post algebras [See also 06D25, 06D30]
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p9">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G25 Other
<a class="nnexus_concept" href="http://planetmath.org/algebraiccategoriesandclassesofalgebras">
algebras
</a>
related to logic [See also 03F45, 06D20, 06E25, 06F35]
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p10">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G27 Abstract algebraic logic
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS1.p11">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
03G30 Categorical logic, topoi [See also 18B25, 18C05, 18C10]
</span>
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS2">
<h3 class="ltx_title ltx_font_bold ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.2
</span>
Topology-General
</h3>
<div class="ltx_para" id="S2.SS2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
54-00 (General topology : General reference works (handbooks, dictionaries, bibliographies, etc.))
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS2.p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
54D30 (General topology : Fairly general
<a class="nnexus_concept" href="http://planetmath.org/property">
properties
</a>
:
<a class="nnexus_concept" href="http://planetmath.org/compactness">
Compactness
</a>
)
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS2.p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
54-01 (General topology : Instructional exposition (textbooks, tutorial papers, etc.))
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS2.p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
54B10 (General topology : Basic constructions : Product spaces)
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS2.p5">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
54D05 (General topology : Fairly general properties :
<a class="nnexus_concept" href="http://planetmath.org/connectedposet">
Connected
</a>
and locally connected spaces (general aspects))
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS2.p6">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
54D45 (General topology : Fairly general properties : Local compactness,
</span>
<math alttext="\sigma" class="ltx_Math" display="inline" id="S2.SS2.p6.m1">
<mi>
σ
</mi>
</math>
<span class="ltx_text ltx_font_bold">
-compactness)
</span>
</p>
</div>
</section>
<section class="ltx_subsection" id="S2.SS3">
<h3 class="ltx_title ltx_font_bold ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
2.3
</span>
Quantum Algebra
</h3>
<div class="ltx_para" id="S2.SS3.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://aux.planetmath.org/files/books/288/QuantumAlgebraSymmetry.pdf
</span>
<span class="ltx_text ltx_font_bold">
Book:
<a class="nnexus_concept" href="http://planetmath.org/topicentryonthealgebraicfoundationsofmathematics">
Quantum Algebra
</a>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Symmetries
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/symmetricrelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumsymmetry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, Second Edition, 780 pages, 27 Mb pdf, (printed book 1,120 pages, in three volumes with an extensive Index of Contents and comprehensive References list)
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
AMS MSC: 18-00 (
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Category theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CategoryTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categorytheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bibliographyinalgebraictopologycategoriesandqat">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/graphtheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicsinalgebraictopology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
; homological algebra :: General reference works (handbooks, dictionaries, bibliographies, etc.))
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
55R40 (
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicTopology.html">
Algebraic topology
</a>
::
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FiberSpace.html">
Fiber spaces
</a>
and bundles ::
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Homology
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Homology.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homologyofachaincomplex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
classifying spaces
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/groupcohomologytopologicaldefinition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universalbundle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CharacteristicClass.html">
characteristic classes
</a>
)
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
81T05 (
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Quantum theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quantumoperatoralgebrasinquantumfieldtheories">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumfieldtheoriesqft">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
::
<a class="nnexus_concept" href="http://planetmath.org/mathematicalfoundationsofquantumfieldtheories">
Quantum field theory
</a>
; related classical field theories :: Axiomatic quantum field theory; operator algebras)
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p5">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
81V05 (Quantum theory :: Applications to specific physical systems :: Strong interaction, including quantum chromodynamics)
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p6">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
81Q60 (Quantum theory :: General mathematical topics and methods in quantum theory :: Supersymmetric quantum mechanics)
</span>
</p>
</div>
<div class="ltx_para" id="S2.SS3.p7">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
(Cited from the Books section at PM).
</span>
</p>
</div>
<div class="ltx_para ltx_align_right" id="S2.SS3.p8">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
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<span class="ltx_text ltx_font_bold">
Title
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_bold">
MSC classification of objects: articles search
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
MSCClassificationOfObjectsArticlesSearch
</span>
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2013-03-22 19:22:32
</span>
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<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Last modified on
</span>
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<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
2013-03-22 19:22:32
</span>
</td>
</tr>
<tr class="ltx_tr">
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<span class="ltx_text ltx_font_bold">
Owner
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
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bci1 (20947)
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Last modified by
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
bci1 (20947)
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Numerical id
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
24
</span>
</td>
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<span class="ltx_text ltx_font_bold">
Author
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
bci1 (20947)
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Entry type
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
Bibliography
</span>
</td>
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<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Classification
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
msc 00-02
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Classification
</span>
</td>
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<span class="ltx_text ltx_font_bold">
msc 00-01
</span>
</td>
</tr>
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<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
AMS classification for: general applied mathematics
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebraic geometry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicGeometry.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraicgeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
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<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
number theory
</span>
</td>
</tr>
<tr class="ltx_tr">
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<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
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<span class="ltx_text ltx_font_bold">
functional analysis
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
harmonic analysis
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
algebraic logics
</span>
</td>
</tr>
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<span class="ltx_text ltx_font_bold">
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</span>
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<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
physics
</span>
</td>
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<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
general mathematics
</span>
</td>
</tr>
<tr class="ltx_tr">
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<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
mathematical physics
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
dimensional analysis
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
geometry
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
algebra
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concept" href="http://planetmath.org/topologicalspace">
topology
</a>
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
<span class="ltx_text ltx_font_bold">
Defines
</span>
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
<span class="ltx_text ltx_font_bold">
algebr
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0 | PMPlanetaryUpdateOnDevelopmentalWebServersAndSitesincludingPlanetPhysicsorgWebServers | http://planetmath.org/PMPlanetaryUpdateOnDevelopmentalWebServersAndSitesincludingPlanetPhysicsorgWebServers | <!DOCTYPE html>
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PM Planetary Update on Developmental web Servers and Sites–including PlanetPhysics.org web servers
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<h1 class="ltx_title ltx_title_document">
PM Planetary Update on Developmental web Servers and Sites–including PlanetPhysics.org web servers
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
Web servers Updates:
</h2>
<div class="ltx_para" id="S1.p1">
<p class="ltx_p">
(In reverse chronological order, with the first website/server listed being the latest one)
Currently all servers are running at full speed…
Books can be edited and saved in PDF format at the Media Wiki version 1.17
<span class="ltx_text ltx_font_typewriter">
http://wiki.planetphysics.info:8888
</span>
PlanetPhysics.org website:
</p>
</div>
<div class="ltx_para" id="S1.p2">
<p class="ltx_p">
(Yes, you need to add :8888 after the URL for protection against spamming!)
</p>
</div>
<div class="ltx_para" id="S1.p3">
<p class="ltx_p">
“
<span class="ltx_text ltx_font_italic">
Planetary.PlanetPhysics.info
</span>
” is a new PM and PlanetPhysics.org developmental server/website for PlanetMath.org, running Drupal under Ubuntu 12.04.01 OS with
<span class="ltx_text ltx_font_italic">
Planetary.PlanetMath.Beta3B
</span>
; it is currently dedicated as a Developmental PM new server for high storage backup and website
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Development.html">
development
</a>
; it is also planned in the future to add to it a Planetary.PlanetPhysics website, running Drupal under Ubuntu 12.04.1 OS. This is currently the only PM developmental website that allows direct upload of Full (not filtered) HTML source code;
</p>
</div>
<div class="ltx_para" id="S1.p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Beta2
</span>
<span class="ltx_text ltx_font_italic">
”PlanetPhysics.org developmental website”
</span>
is actually an earlier PM developmental planetary website running on a different server
</p>
</div>
<div class="ltx_para" id="S1.p5">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Beta.PlanetMath.org
</span>
is running Beta3 Planetary for the PM developmental website with Drupal under Ubuntu 12.04.1 OS
</p>
</div>
<div class="ltx_para" id="S1.p6">
<p class="ltx_p">
PlanetPhysics.org R- wiki is an Organizational Blog and Discussion R-wiki for backing up contents
</p>
</div>
<div class="ltx_para" id="S1.p7">
<p class="ltx_p">
PlanetPhysics.org R-wiki on Noosphere 1.0 main server running under Ubuntu 10.04
</p>
</div>
<div class="ltx_para" id="S1.p8">
<p class="ltx_p">
PlanetPhysics.org’s MediaWiki v.1.17 server runs under Scientific Linux version 6.1 OS
</p>
</div>
<div class="ltx_para" id="S1.p9">
<p class="ltx_p">
PlanetPhysics.us- is the Noosphere 1.0 website of PlanetPhysics.org
</p>
</div>
<div class="ltx_para" id="S1.p10">
<p class="ltx_p">
PlanetPhysics.org ’s
<span class="ltx_text ltx_font_typewriter">
http://pp-dev.org:8080/
</span>
Developmental-redux website runs on internet port 8080 Noosphere v. 1.5.
</p>
</div>
<div class="ltx_para ltx_align_right" id="S1.p11">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
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Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
PM Planetary Update on Developmental web Servers and Sites–including PlanetPhysics.org web servers
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PMPlanetaryUpdateOnDevelopmentalWebServersAndSitesincludingPlanetPhysicsorgWebServers
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 19:36:31
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 19:36:31
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
31
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00-02
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00-01
</td>
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0 | AnExampleOfMathematicalInduction | http://planetmath.org/AnExampleOfMathematicalInduction | <!DOCTYPE html>
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an example of mathematical induction
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
an example of mathematical induction
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Below is a step-by-step demonstration of how mathematical induction (the
<a class="nnexus_concept" href="http://planetmath.org/principleoffiniteinduction">
Principle of Finite Induction
</a>
) works.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Proposition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/booleanvaluedfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/propositionallogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</span>
For any positive integer
<math alttext="n" class="ltx_Math" display="inline" id="p2.m1">
<mi>
n
</mi>
</math>
,
<math alttext="2^{n-1}\leq n!" class="ltx_Math" display="inline" id="p2.m2">
<mrow>
<msup>
<mn>
2
</mn>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
≤
</mo>
<mrow>
<mi>
n
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_proof">
<h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">
Proof.
</h6>
<div class="ltx_para" id="p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Setup.
</span>
Let
<math alttext="S" class="ltx_Math" display="inline" id="p3.m1">
<mi>
S
</mi>
</math>
be the set of positive integers
<math alttext="n" class="ltx_Math" display="inline" id="p3.m2">
<mi>
n
</mi>
</math>
satisfying the rule:
<math alttext="2^{n-1}\leq n!" class="ltx_Math" display="inline" id="p3.m3">
<mrow>
<msup>
<mn>
2
</mn>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
≤
</mo>
<mrow>
<mi>
n
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mrow>
</math>
. We want to show that
<math alttext="S" class="ltx_Math" display="inline" id="p3.m4">
<mi>
S
</mi>
</math>
is the set of
<em class="ltx_emph ltx_font_italic">
all
</em>
positive integers, which would prove our proposition.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Initial Step.
</span>
For
<math alttext="n=1" class="ltx_Math" display="inline" id="p4.m1">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
,
<math alttext="2^{n-1}=2^{1-1}=2^{0}=1" class="ltx_Math" display="inline" id="p4.m2">
<mrow>
<msup>
<mn>
2
</mn>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
=
</mo>
<msup>
<mn>
2
</mn>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
=
</mo>
<msup>
<mn>
2
</mn>
<mn>
0
</mn>
</msup>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
, while
<math alttext="n!=1!=1" class="ltx_Math" display="inline" id="p4.m3">
<mrow>
<mrow>
<mi>
n
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
1
</mn>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
. So
<math alttext="2^{n-1}=n!" class="ltx_Math" display="inline" id="p4.m4">
<mrow>
<msup>
<mn>
2
</mn>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
=
</mo>
<mrow>
<mi>
n
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mrow>
</math>
for
<math alttext="n=1" class="ltx_Math" display="inline" id="p4.m5">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
and thus
<math alttext="2^{n-1}\leq n!" class="ltx_Math" display="inline" id="p4.m6">
<mrow>
<msup>
<mn>
2
</mn>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
≤
</mo>
<mrow>
<mi>
n
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mrow>
</math>
all the more so. This shows that
<math alttext="1\in S" class="ltx_Math" display="inline" id="p4.m7">
<mrow>
<mn>
1
</mn>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Induction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Induction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/induction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
Step 1.
</span>
Assume that for
<math alttext="n=k" class="ltx_Math" display="inline" id="p5.m1">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mi>
k
</mi>
</mrow>
</math>
,
<math alttext="k" class="ltx_Math" display="inline" id="p5.m2">
<mi>
k
</mi>
</math>
a positive integer,
<math alttext="2^{n-1}\leq n!" class="ltx_Math" display="inline" id="p5.m3">
<mrow>
<msup>
<mn>
2
</mn>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
≤
</mo>
<mrow>
<mi>
n
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mrow>
</math>
. In other words, we assume that
<math alttext="k\in S" class="ltx_Math" display="inline" id="p5.m4">
<mrow>
<mi>
k
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</math>
, or that
<math alttext="2^{k-1}\leq k!" class="ltx_Math" display="inline" id="p5.m5">
<mrow>
<msup>
<mn>
2
</mn>
<mrow>
<mi>
k
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
≤
</mo>
<mrow>
<mi>
k
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Induction Step 2.
</span>
Next, we want to show that
<math alttext="k+1\in S" class="ltx_Math" display="inline" id="p6.m1">
<mrow>
<mrow>
<mi>
k
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</math>
. If we let
<math alttext="n=k+1" class="ltx_Math" display="inline" id="p6.m2">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mrow>
<mi>
k
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
</math>
, then by the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
assumption
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/deduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivationsinnaturaldeduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the proposition, showing
<math alttext="n=k+1\in S" class="ltx_Math" display="inline" id="p6.m3">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mrow>
<mi>
k
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</math>
is the same as showing
<math alttext="2^{n-1}\leq n!" class="ltx_Math" display="inline" id="p6.m4">
<mrow>
<msup>
<mn>
2
</mn>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
≤
</mo>
<mrow>
<mi>
n
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mrow>
</math>
, or
<math alttext="2^{k}\leq(k+1)!" class="ltx_Math" display="inline" id="p6.m5">
<mrow>
<msup>
<mn>
2
</mn>
<mi>
k
</mi>
</msup>
<mo>
≤
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
k
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mrow>
</math>
. This can be done by the following calculation:
</p>
<table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S0.EGx1">
<tbody id="S0.E1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 2^{k}" class="ltx_Math" display="inline" id="S0.E1.m1">
<msup>
<mn>
2
</mn>
<mi>
k
</mi>
</msup>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S0.E1.m2">
<mo>
=
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle 2^{k-1}2" class="ltx_Math" display="inline" id="S0.E1.m3">
<mrow>
<msup>
<mn>
2
</mn>
<mrow>
<mi>
k
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
</mo>
<mn>
2
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(1)
</span>
</td>
</tr>
</tbody>
<tbody id="S0.E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S0.E2.m2">
<mo>
≤
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle k!2" class="ltx_Math" display="inline" id="S0.E2.m3">
<mrow>
<mrow>
<mi>
k
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
<mo>
</mo>
<mn>
2
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(2)
</span>
</td>
</tr>
</tbody>
<tbody id="S0.E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S0.E3.m2">
<mo>
≤
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle k!(k+1)" class="ltx_Math" display="inline" id="S0.E3.m3">
<mrow>
<mrow>
<mi>
k
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
k
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(3)
</span>
</td>
</tr>
</tbody>
<tbody id="S0.E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S0.E4.m2">
<mo>
=
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle(k+1)!," class="ltx_Math" display="inline" id="S0.E4.m3">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
k
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(4)
</span>
</td>
</tr>
</tbody>
</table>
<p class="ltx_p">
where Equations (1) and (4) are just
<a class="nnexus_concept" href="http://planetmath.org/definition">
definitions
</a>
of the power and the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
factorial
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Factorial.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/factorial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of a number, respectively. Step (3) is the fact that
<math alttext="2\leq k+1" class="ltx_Math" display="inline" id="p6.m6">
<mrow>
<mn>
2
</mn>
<mo>
≤
</mo>
<mrow>
<mi>
k
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
</math>
for any positive integer
<math alttext="k" class="ltx_Math" display="inline" id="p6.m7">
<mi>
k
</mi>
</math>
(which, incidentally, can be proved by mathematical induction as well). Step (2) follows from the
<em class="ltx_emph ltx_font_italic">
induction step
</em>
, the assumption that we made in the
<span class="ltx_text ltx_font_bold">
Induction Step 1.
</span>
in the previous paragraph. Because
<math alttext="2^{k}\leq(k+1)!" class="ltx_Math" display="inline" id="p6.m8">
<mrow>
<msup>
<mn>
2
</mn>
<mi>
k
</mi>
</msup>
<mo>
≤
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
k
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mrow>
</math>
, we have thus shown that
<math alttext="n=k+1\in S" class="ltx_Math" display="inline" id="p6.m9">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mrow>
<mi>
k
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</math>
, proving the proposition.
∎
</p>
</div>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Remark.
</span>
All these steps are essential in any proof by (mathematical) induction. However, in more advanced math articles and books, some or all of these steps are omitted with the assumption that the reader is already familiar with the
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
concepts
</a>
and the steps involved.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
an example of mathematical induction
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
AnExampleOfMathematicalInduction
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2013-03-22 15:39:46
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2013-03-22 15:39:46
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Owner
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<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
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Last modified by
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CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
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Author
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CWoo (3771)
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Entry type
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Example
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
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Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A35
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
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<td class="ltx_td ltx_align_left ltx_border_r">
msc 03E25
</td>
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Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
PrincipleOfFiniteInduction
</td>
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0 | ArgMinAndArgMax | http://planetmath.org/ArgMinAndArgMax | <!DOCTYPE html>
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<title>
arg min and arg max
</title>
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<h1 class="ltx_title ltx_title_document">
arg min and arg max
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
For a real-valued function
<math alttext="f" class="ltx_Math" display="inline" id="p1.m1">
<mi>
f
</mi>
</math>
with domain
<math alttext="S" class="ltx_Math" display="inline" id="p1.m2">
<mi>
S
</mi>
</math>
,
<math alttext="\arg\min_{x\in S}f(x)" class="ltx_Math" display="inline" id="p1.m3">
<mrow>
<mrow>
<mi>
arg
</mi>
<mo>
</mo>
<mrow>
<msub>
<mi>
min
</mi>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</msub>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is the set of elements in
<math alttext="S" class="ltx_Math" display="inline" id="p1.m4">
<mi>
S
</mi>
</math>
that achieve the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
global minimum
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GlobalMinimum.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extremum">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in
<math alttext="S" class="ltx_Math" display="inline" id="p1.m5">
<mi>
S
</mi>
</math>
,
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="{\arg\min}_{x\in S}f(x)=\{x\in S:\,f(x)=\min_{y\in S}f(y)\}." class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mrow>
<mrow>
<mi>
arg
</mi>
<mo>
</mo>
<mrow>
<munder>
<mi>
min
</mi>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</munder>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
<mo rspace="4.2pt">
:
</mo>
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<munder>
<mi>
min
</mi>
<mrow>
<mi>
y
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</munder>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
y
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
<math alttext="\arg\max_{x\in S}f(x)" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<mrow>
<mi>
arg
</mi>
<mo>
</mo>
<mrow>
<msub>
<mi>
max
</mi>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</msub>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is the set of elements in
<math alttext="S" class="ltx_Math" display="inline" id="p2.m2">
<mi>
S
</mi>
</math>
that achieve the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GlobalMaximum.html">
global maximum
</a>
in
<math alttext="S" class="ltx_Math" display="inline" id="p2.m3">
<mi>
S
</mi>
</math>
,
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="{\arg\max}_{x\in S}f(x)=\{x\in S:\,f(x)=\max_{y\in S}f(y)\}." class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mrow>
<mrow>
<mrow>
<mi>
arg
</mi>
<mo>
</mo>
<mrow>
<munder>
<mi>
max
</mi>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</munder>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
<mo rspace="4.2pt">
:
</mo>
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<munder>
<mi>
max
</mi>
<mrow>
<mi>
y
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</munder>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
y
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
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<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/argminandargmax">
arg min and arg max
</a>
</td>
</tr>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
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Canonical
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name
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ArgMinAndArgMax
</td>
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Date of creation
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<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:27:55
</td>
</tr>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
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<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:27:55
</td>
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Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
kshum (5987)
</td>
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Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
kshum (5987)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
11
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
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<td class="ltx_td ltx_align_left ltx_border_r">
kshum (5987)
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Entry type
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<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
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Classification
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msc 00A05
</td>
</tr>
<tr class="ltx_tr">
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Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
argmin argmax
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0 | Arithmetic | http://planetmath.org/Arithmetic | <!DOCTYPE html>
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arithmetic
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<h1 class="ltx_title ltx_title_document">
arithmetic
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Arithmetic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/arithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/arithmeticalhierarchy">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
is ” the science of numbers and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
on sets of numbers. Arithmetic is understood to include problems on the origin and development of the
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
concept
</a>
of a number, methods and means of calculation, the study of operations on numbers of different kinds, as well as
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Analysis.html">
analysis
</a>
of the axiomatic
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of number sets and the
<a class="nnexus_concept" href="http://planetmath.org/property">
properties
</a>
of numbers.” The four basic operations are
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/subtraction">
subtraction
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
multiplication
</a>
and
<a class="nnexus_concept" href="http://planetmath.org/division">
division
</a>
; in the absence of parentheses these are performed according to the rules of
<a class="nnexus_concept" href="http://planetmath.org/orderofoperations">
operator precedence
</a>
. From multiplication follows
<a class="nnexus_concept" href="http://planetmath.org/exponentiation">
exponentiation
</a>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Most civilizations usually develop arithmetic before any other branches of mathematics, and in modern times
<a class="nnexus_concept" href="http://planetmath.org/treesettheoretic">
children
</a>
are usually taught arithmetic first (though in some curricula they might be taught
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
set theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SetTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/settheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
first).
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/numerationsystem">
Numeral systems
</a>
of ancient civilizations often took letter symbols for numbers. Boyer called these letter symbols
<a class="nnexus_concept" href="http://planetmath.org/injectivefunction">
one-to-one
</a>
ciphered numerals. The ancient Greeks, for example, used letters in Ionian and Dorian
<a class="nnexus_concept" href="http://planetmath.org/alphabet">
alphabets
</a>
to represent numbers, as did the Romans in one alphabet. These systems were
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
additive
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/additivefunction1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/additive">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in one sense (that is, the value of the overall “word” was merely the sum of the values of the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Individual.html">
individual
</a>
symbols). To convert
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
rational numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/egyptianfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/liberabaci">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/rationalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
into a ciphered numeration system,
<a class="nnexus_concept" href="http://planetmath.org/leonardodapisa">
Fibonacci
</a>
implemented seven rules, one being a LCM
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
multiple
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Multiple.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generalassociativity">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divisibility">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. For example, four of Fibonacci’s rules were used by earlier cultures. One was an Egyptian scribe. In 1650 BCE Ahmes converted
<math alttext="\frac{n}{p}" class="ltx_Math" display="inline" id="p3.m1">
<mfrac>
<mi>
n
</mi>
<mi>
p
</mi>
</mfrac>
</math>
by using large multiples. About 200 years earlier, a student converted
<math alttext="\frac{n}{pq}" class="ltx_Math" display="inline" id="p3.m2">
<mfrac>
<mi>
n
</mi>
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mi>
q
</mi>
</mrow>
</mfrac>
</math>
by using small LCM multiples.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
It was not just the invention of zero as a symbol for that
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
between 1 and -1 allowed Hindu-Arabic numerals to significant expand man’s ability to perform arithmetic operations. Significantly larger numbers came into use after zero became a placeholder in the new additive-exponential system. An algorithm written in Hindu-Arabic numbers before 1600 AD was also added. Modern decimals apply few lessons learned during the 3,200 year life of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Egyptian fractions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/rhindmathematicalpapyrus">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/egyptianmathematicalleatherroll">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/egyptianfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/hultschbruinsmethod">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/remainderarithmeticvsegyptianfractions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/remainderarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/unitfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. One exception is
<a class="nnexus_concept" href="http://planetmath.org/properdivisor">
aliquot parts
</a>
, used in the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental theorem of arithmetic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/HigherArithmetic.html">
higher arithmetic
</a>
.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
The next great expansion of arithmetic power occurred with the invention of
<a class="nnexus_concept" href="http://planetmath.org/calculator">
calculators
</a>
and
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computers
</a>
in the 20th Century, with both kinds of devices performing arithmetic in a binary algorithm. The algorithm, stated in a cursive form, had been used prior to 2,000 BCE.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
As
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
number theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NumberTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/numbertheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
generalizes the principles of arithmetic to all numbers or all numbers of a given kind, it is sometimes called the “higher arithmetic.”
</p>
</div>
<section class="ltx_subsection" id="S0.SS1">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
0.1
</span>
External links
</h2>
<div class="ltx_para" id="S0.SS1.p1">
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://liberabaci.blogspot.com
</span>
Fibonacci
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://rmprectotable.blogspot.com
</span>
n/p
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://emlr.blogspot.com
</span>
n/pq
</p>
</div>
</li>
</ul>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
arithmetic
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Arithmetic
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:16:18
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:16:18
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
29
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
arithmetics
</td>
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</div>
</section>
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0 | ArithmeticProgression | http://planetmath.org/ArithmeticProgression | <!DOCTYPE html>
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<title>
arithmetic progression
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
arithmetic progression
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Arithmetic progression
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ArithmeticProgression.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/remainderarithmeticvsegyptianfractions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/arithmeticprogression">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of length
<math alttext="n" class="ltx_Math" display="inline" id="p1.m1">
<mi>
n
</mi>
</math>
, initial term
<math alttext="a_{1}" class="ltx_Math" display="inline" id="p1.m2">
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
</math>
and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CommonDifference.html">
common difference
</a>
<math alttext="d" class="ltx_Math" display="inline" id="p1.m3">
<mi>
d
</mi>
</math>
is the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
sequence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Sequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="a_{1},a_{1}+d,a_{1}+2d,\ldots,a_{1}+(n-1)d" class="ltx_Math" display="inline" id="p1.m4">
<mrow>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
<mo>
,
</mo>
<mrow>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
<mo>
+
</mo>
<mi>
d
</mi>
</mrow>
<mo>
,
</mo>
<mrow>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
<mo>
+
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
d
</mi>
</mrow>
</mrow>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
,
</mo>
<mrow>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
<mo>
+
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mi>
d
</mi>
</mrow>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The sum of terms of an arithmetic progression can be computed using Gauss’s trick:
</p>
</div>
<div class="ltx_para" id="p3">
<span class="ltx_inline-block ltx_parbox ltx_align_middle" style="width:433.6pt;">
<span class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx1">
<span class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex1">
<span class="ltx_eqn_cell ltx_eqn_center_padleft">
</span>
<span class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle S" class="ltx_Math" display="inline" id="S0.Ex1.m1">
<mi>
S
</mi>
</math>
</span>
<span class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle=\makebox[70.0pt]{$(a_{1}+0)$}+\makebox[70.0pt]{$(a_{1}+d)$}+%
\cdots+\makebox[70.0pt]{$(a_{1}+(n-2)d)$}+\makebox[70.0pt]{$(a_{1}+(n-1)d)$}" class="ltx_Math" display="inline" id="S0.Ex1.m2">
<mrow>
<mi>
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<mo>
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<mo stretchy="false">
)
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</mrow>
</mrow>
</mrow>
</math>
</span>
<span class="ltx_eqn_cell ltx_eqn_center_padright">
</span>
</span>
<span class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex2">
<span class="ltx_eqn_cell ltx_eqn_center_padleft">
</span>
<span class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle+\underline{S\vphantom{\makebox[70.0pt]{$(a_{1}+(n-1)d)$}}}" class="ltx_Math" display="inline" id="S0.Ex2.m1">
<mrow>
<mo>
+
</mo>
<munder accentunder="true">
<mi>
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<mo>
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</mo>
</munder>
</mrow>
</math>
</span>
<span class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle\underline{{}=\makebox[70.0pt]{$(a_{1}+(n-1)d)$}+\makebox[70.0pt]%
{$(a_{1}+(n-2)d)$}+\cdots+\makebox[70.0pt]{$(a_{1}+d)$}+\makebox[70.0pt]{$(a_{%
1}+0)$}}" class="ltx_Math" display="inline" id="S0.Ex2.m2">
<munder accentunder="true">
<mrow>
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<mo stretchy="false">
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</mrow>
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</mrow>
<mo>
¯
</mo>
</munder>
</math>
</span>
<span class="ltx_eqn_cell ltx_eqn_center_padright">
</span>
</span>
<span class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex3">
<span class="ltx_eqn_cell ltx_eqn_center_padleft">
</span>
<span class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 2S" class="ltx_Math" display="inline" id="S0.Ex3.m1">
<mrow>
<mn>
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<mo>
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<mi>
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</mi>
</mrow>
</math>
</span>
<span class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle=\makebox[70.0pt]{$(2a_{1}+(n-1)d)$}+\makebox[70.0pt]{$(2a_{1}+(n%
-1)d)$}+\cdots+\makebox[70.0pt]{$(2a_{1}+(n-1)d)$}+\makebox[70.0pt]{$(2a_{1}+(%
n-1)d)$}." class="ltx_Math" display="inline" id="S0.Ex3.m2">
<mrow>
<mrow>
<mi>
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</mn>
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</mrow>
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</mo>
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</mrow>
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<mo stretchy="false">
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</mo>
</mrow>
</mrow>
</mrow>
<mo>
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</mo>
</mrow>
</math>
</span>
<span class="ltx_eqn_cell ltx_eqn_center_padright">
</span>
</span>
</span>
</span>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
We just add the sum with itself written backwards, and the sum of each of the columns equals to
<math alttext="(2a_{1}+(n-1)d)" class="ltx_Math" display="inline" id="p4.m1">
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mo>
+
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mi>
d
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
. The sum is then
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="S=\frac{(2a_{1}+(n-1)d)n}{2}." class="ltx_Math" display="block" id="S0.Ex4.m1">
<mrow>
<mrow>
<mi>
S
</mi>
<mo>
=
</mo>
<mfrac>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mo>
+
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mi>
d
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
<mn>
2
</mn>
</mfrac>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
arithmetic progression
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
ArithmeticProgression
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:39:00
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:39:00
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bbukh (348)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bbukh (348)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
bbukh (348)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 11B25
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
MulidimensionalArithmeticProgression
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
SumOfKthPowersOfTheFirstNPositiveIntegers
</td>
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</article>
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0 | ArithmeticgeometricharmonicMeansInequality | http://planetmath.org/ArithmeticgeometricharmonicMeansInequality | <!DOCTYPE html>
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<title>
arithmetic-geometric-harmonic means inequality
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<h1 class="ltx_title ltx_title_document">
arithmetic-geometric-harmonic means inequality
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Let
<math alttext="x_{1},x_{2},\ldots,x_{n}" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<msub>
<mi>
x
</mi>
<mn>
1
</mn>
</msub>
<mo>
,
</mo>
<msub>
<mi>
x
</mi>
<mn>
2
</mn>
</msub>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
,
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
</mrow>
</math>
be positive numbers.
Then
</p>
<table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S0.EGx1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex4">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle\max\{x_{1},x_{2},\ldots,x_{n}\}" class="ltx_Math" display="inline" id="S0.Ex1.m1">
<mrow>
<mi>
max
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<msub>
<mi>
x
</mi>
<mn>
1
</mn>
</msub>
<mo>
,
</mo>
<msub>
<mi>
x
</mi>
<mn>
2
</mn>
</msub>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
,
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S0.Ex1.m2">
<mo>
≥
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}" class="ltx_Math" display="inline" id="S0.Ex1.m3">
<mstyle displaystyle="true">
<mfrac>
<mrow>
<msub>
<mi>
x
</mi>
<mn>
1
</mn>
</msub>
<mo>
+
</mo>
<msub>
<mi>
x
</mi>
<mn>
2
</mn>
</msub>
<mo>
+
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
+
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mi>
n
</mi>
</mfrac>
</mstyle>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S0.Ex2.m2">
<mo>
≥
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle\sqrt[n]{x_{1}x_{2}\cdots x_{n}}" class="ltx_Math" display="inline" id="S0.Ex2.m3">
<mroot>
<mrow>
<msub>
<mi>
x
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<msub>
<mi>
x
</mi>
<mn>
2
</mn>
</msub>
<mo>
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mi>
n
</mi>
</mroot>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S0.Ex3.m2">
<mo>
≥
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}}" class="ltx_Math" display="inline" id="S0.Ex3.m3">
<mstyle displaystyle="true">
<mfrac>
<mi>
n
</mi>
<mrow>
<mfrac>
<mn>
1
</mn>
<msub>
<mi>
x
</mi>
<mn>
1
</mn>
</msub>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<msub>
<mi>
x
</mi>
<mn>
2
</mn>
</msub>
</mfrac>
<mo>
+
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
</mfrac>
</mrow>
</mfrac>
</mstyle>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S0.Ex4.m2">
<mo>
≥
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle\min\{x_{1},x_{2},\ldots,x_{n}\}" class="ltx_Math" display="inline" id="S0.Ex4.m3">
<mrow>
<mi>
min
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<msub>
<mi>
x
</mi>
<mn>
1
</mn>
</msub>
<mo>
,
</mo>
<msub>
<mi>
x
</mi>
<mn>
2
</mn>
</msub>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
,
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The equality is obtained if and only if
<math alttext="x_{1}=x_{2}=\cdots=x_{n}" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<msub>
<mi>
x
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<msub>
<mi>
x
</mi>
<mn>
2
</mn>
</msub>
<mo>
=
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
=
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
There are several
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
generalizations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/hilbertsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiomsystemforfirstorderlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to this
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
inequality
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Inequality.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inequalitiesforrealnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
using
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
power means
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PowerMean.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/powermean">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and
<a class="nnexus_concept" href="http://planetmath.org/weightedpowermean">
weighted power means
</a>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/arithmeticgeometricharmonicmeansinequality">
arithmetic-geometric-harmonic means inequality
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
ArithmeticgeometricharmonicMeansInequality
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 11:42:32
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 11:42:32
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
drini (3)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
drini (3)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
22
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
drini (3)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 20-XX
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 26D15
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
harmonic-geometric-arithmetic means inequality
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
arithmetic-geometric means inequality
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
AGM inequality
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
AGMH inequality
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
ArithmeticMean
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
GeometricMean
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
HarmonicMean
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
GeneralMeansInequality
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
WeightedPowerMean
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
PowerMean
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
RootMeanSquare3
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
ProofOfGeneralMeansInequality
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
JensensInequality
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
DerivationOfHarmonicMeanAsTheLimitOfThePowerMean
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
MinimalAndMaximalNumber
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
ProofOfArithm
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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|
0 | Arity | http://planetmath.org/Arity | <!DOCTYPE html>
<html>
<head>
<title>
arity
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
arity
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The
<em class="ltx_emph ltx_font_italic">
arity
</em>
of something is the number of arguments it takes. This is usually applied to functions: an
<math alttext="n" class="ltx_Math" display="inline" id="p1.m1">
<mi>
n
</mi>
</math>
-ary function is one that takes
<math alttext="n" class="ltx_Math" display="inline" id="p1.m2">
<mi>
n
</mi>
</math>
arguments.
<em class="ltx_emph ltx_font_italic">
Unary
</em>
is a synonym for
<math alttext="1" class="ltx_Math" display="inline" id="p1.m3">
<mn>
1
</mn>
</math>
-ary, and binary for
<math alttext="2" class="ltx_Math" display="inline" id="p1.m4">
<mn>
2
</mn>
</math>
-ary.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
arity
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Arity
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:00:01
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:00:01
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
4
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
ary
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
-arity
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
BinaryOperation
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
unary
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
binary
</td>
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0 | BabylonianMethodOfComputingSquareRoots | http://planetmath.org/BabylonianMethodOfComputingSquareRoots | <!DOCTYPE html>
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Babylonian method of computing square roots
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
Babylonian method of computing square roots
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
Description
</h2>
<div class="ltx_para" id="S1.p1">
<p class="ltx_p">
To compute the
<a class="nnexus_concept" href="http://planetmath.org/squareroot">
square root
</a>
of a number which lies between 0 and 2, one may use a method of successive approximations which involves only the operations of squaring and averaging. The basis of method is the binomial identity:
</p>
<table class="ltx_equation ltx_eqn_table" id="S1.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="(x+y)^{2}=x^{2}+2xy+y^{2}" class="ltx_Math" display="block" id="S1.Ex1.m1">
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<mi>
y
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
x
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="S1.p2">
<p class="ltx_p">
Write the number whose square root is to be computed as
<math alttext="1-x" class="ltx_Math" display="inline" id="S1.p2.m1">
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mi>
x
</mi>
</mrow>
</math>
. Write the square root of the number as
<math alttext="1-y" class="ltx_Math" display="inline" id="S1.p2.m2">
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mi>
y
</mi>
</mrow>
</math>
. By
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
, one has
</p>
<table class="ltx_equation ltx_eqn_table" id="S1.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="(1-y)^{2}=1-x" class="ltx_Math" display="block" id="S1.Ex2.m1">
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mi>
y
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
<mo>
=
</mo>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Using the binomial identity,
</p>
<table class="ltx_equation ltx_eqn_table" id="S1.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="1-2y+y^{2}=1-x." class="ltx_Math" display="block" id="S1.Ex3.m1">
<mrow>
<mrow>
<mrow>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Cancelling,
</p>
<table class="ltx_equation ltx_eqn_table" id="S1.Ex4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="-2y+y^{2}=-x." class="ltx_Math" display="block" id="S1.Ex4.m1">
<mrow>
<mrow>
<mrow>
<mrow>
<mo>
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Moving terms from one side to the other,
</p>
<table class="ltx_equation ltx_eqn_table" id="S1.Ex5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="2y=x+y^{2}" class="ltx_Math" display="block" id="S1.Ex5.m1">
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Dividing by 2,
</p>
<table class="ltx_equation ltx_eqn_table" id="S1.Ex6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y=\frac{1}{2}(x+y^{2})." class="ltx_Math" display="block" id="S1.Ex6.m1">
<mrow>
<mrow>
<mi>
y
</mi>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="S1.p3">
<p class="ltx_p">
Even though
<math alttext="y" class="ltx_Math" display="inline" id="S1.p3.m1">
<mi>
y
</mi>
</math>
appears alone on one side of the equation, we cannot use it directly to obtain
<math alttext="y" class="ltx_Math" display="inline" id="S1.p3.m2">
<mi>
y
</mi>
</math>
because
<math alttext="y" class="ltx_Math" display="inline" id="S1.p3.m3">
<mi>
y
</mi>
</math>
also appears on the other side. However, if
<math alttext="y" class="ltx_Math" display="inline" id="S1.p3.m4">
<mi>
y
</mi>
</math>
lies between
<math alttext="-1" class="ltx_Math" display="inline" id="S1.p3.m5">
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</math>
and
<math alttext="+1" class="ltx_Math" display="inline" id="S1.p3.m6">
<mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
, then
<math alttext="y^{2}" class="ltx_Math" display="inline" id="S1.p3.m7">
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
</math>
is smaller than
<math alttext="y" class="ltx_Math" display="inline" id="S1.p3.m8">
<mi>
y
</mi>
</math>
and
<math alttext="y^{2}/2" class="ltx_Math" display="inline" id="S1.p3.m9">
<mrow>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</math>
is even smaller. So, as a first approximation, the occurrence of
<math alttext="y" class="ltx_Math" display="inline" id="S1.p3.m10">
<mi>
y
</mi>
</math>
on the right hand side can be ignored. One then obtains a second approximation to
<math alttext="y" class="ltx_Math" display="inline" id="S1.p3.m11">
<mi>
y
</mi>
</math>
as
<math alttext="x/2" class="ltx_Math" display="inline" id="S1.p3.m12">
<mrow>
<mi>
x
</mi>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</math>
. To obtain a third approximation to
<math alttext="y" class="ltx_Math" display="inline" id="S1.p3.m13">
<mi>
y
</mi>
</math>
, one can substitute the second approximation into the right hand side. One can continue this process of substituting the preceding approximation into the right hand side and computing the next approximation forever. In symbols, this procedure may be described as follows:
</p>
<table class="ltx_equation ltx_eqn_table" id="S1.Ex7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{1}=0" class="ltx_Math" display="block" id="S1.Ex7.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S1.Ex8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{n+1}=\frac{1}{2}(x+y_{n}^{2})" class="ltx_Math" display="block" id="S1.Ex8.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
As we shall see, this procedure
<a class="nnexus_concept" href="http://planetmath.org/convergentsequence">
converges
</a>
to the correct value of
<math alttext="y" class="ltx_Math" display="inline" id="S1.p3.m14">
<mi>
y
</mi>
</math>
in the limit as
<math alttext="n\to\infty" class="ltx_Math" display="inline" id="S1.p3.m15">
<mrow>
<mi>
n
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</math>
.
</p>
</div>
</section>
<section class="ltx_section" id="S2">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
2
</span>
Examples
</h2>
<div class="ltx_para" id="S2.p1">
<p class="ltx_p">
To illustrate this, consider
<math alttext="\sqrt{1/2}" class="ltx_Math" display="inline" id="S2.p1.m1">
<msqrt>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</msqrt>
</math>
. Since
<math alttext="1/2=1-1/2" class="ltx_Math" display="inline" id="S2.p1.m2">
<mrow>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</mrow>
</mrow>
</math>
, it is the case that
<math alttext="x=1/2" class="ltx_Math" display="inline" id="S2.p1.m3">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</mrow>
</math>
so the recursion is as follows:
</p>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{n+1}=\frac{1}{2}(\frac{1}{2}+y_{n}^{2})=\frac{1}{4}+\frac{1}{2}y_{n}^{2}" class="ltx_Math" display="block" id="S2.Ex9.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
+
</mo>
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
4
</mn>
</mfrac>
<mo>
+
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
The first few approximations look as follows:
</p>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{1}=0" class="ltx_Math" display="block" id="S2.Ex10.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{1}=0.25" class="ltx_Math" display="block" id="S2.Ex11.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<mn>
0.25
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{2}=0.28125" class="ltx_Math" display="block" id="S2.Ex12.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
2
</mn>
</msub>
<mo>
=
</mo>
<mn>
0.28125
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{3}=0.28955" class="ltx_Math" display="block" id="S2.Ex13.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
3
</mn>
</msub>
<mo>
=
</mo>
<mn>
0.28955
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{4}=0.29192" class="ltx_Math" display="block" id="S2.Ex14.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
4
</mn>
</msub>
<mo>
=
</mo>
<mn>
0.29192
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{5}=0.29261" class="ltx_Math" display="block" id="S2.Ex15.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
5
</mn>
</msub>
<mo>
=
</mo>
<mn>
0.29261
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{6}=0.29281" class="ltx_Math" display="block" id="S2.Ex16.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
6
</mn>
</msub>
<mo>
=
</mo>
<mn>
0.29281
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex17">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{7}=0.29287" class="ltx_Math" display="block" id="S2.Ex17.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
7
</mn>
</msub>
<mo>
=
</mo>
<mn>
0.29287
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex18">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{8}=0.29289" class="ltx_Math" display="block" id="S2.Ex18.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
8
</mn>
</msub>
<mo>
=
</mo>
<mn>
0.29289
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Hence, the eighth approximation for
<math alttext="\sqrt{1/2}" class="ltx_Math" display="inline" id="S2.p1.m4">
<msqrt>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</msqrt>
</math>
is
<math alttext="0.70711" class="ltx_Math" display="inline" id="S2.p1.m5">
<mn>
0.70711
</mn>
</math>
. Squaring this, we see that it agrees with
<math alttext="0.5" class="ltx_Math" display="inline" id="S2.p1.m6">
<mn>
0.5
</mn>
</math>
to five
<a class="nnexus_concept" href="http://planetmath.org/decimalplace">
decimal places
</a>
. To be sure, the Babylonians would have written their numbers is base 60 rather than base 10 as we do, yet the calculation of
<math alttext="\sqrt{1/2}" class="ltx_Math" display="inline" id="S2.p1.m7">
<msqrt>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</msqrt>
</math>
given above is substantially the same as was carried out centuries ago by ancient scholars on clay tablets.
</p>
</div>
<div class="ltx_para" id="S2.p2">
<p class="ltx_p">
As it shall be seen, this method requires that the number whose square root is to be computed lie between 0 and 2 in order to converge. However, with a few simple hacks, it is possible to use it to compute the square roots of numbers which lie outside of this range. One possibility is to compute the square root of the
<a class="nnexus_concept" href="http://planetmath.org/inversenumber">
reciprocal
</a>
of the number. For example, one can take the reciprocal of the result obtained above to obtain 1.41421 as a value for
<math alttext="\sqrt{2}" class="ltx_Math" display="inline" id="S2.p2.m1">
<msqrt>
<mn>
2
</mn>
</msqrt>
</math>
.
</p>
</div>
<div class="ltx_para" id="S2.p3">
<p class="ltx_p">
However, this is not so good a strategy for computing the square roots of large numbers — the reciprocal of a large number is a small number, and will turn out that the method takes a long time to converge. A better strategy is to divide the number in question by a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
perfect square
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PerfectSquare.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/square1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to obtain a number in the suitable range.
</p>
</div>
<div class="ltx_para" id="S2.p4">
<p class="ltx_p">
How this works can be illustrated by computing
<math alttext="\sqrt{10}" class="ltx_Math" display="inline" id="S2.p4.m1">
<msqrt>
<mn>
10
</mn>
</msqrt>
</math>
. The
<a class="nnexus_concept" href="http://planetmath.org/perfectfield">
perfect
</a>
sqare
<math alttext="9=3^{2}" class="ltx_Math" display="inline" id="S2.p4.m2">
<mrow>
<mn>
9
</mn>
<mo>
=
</mo>
<msup>
<mn>
3
</mn>
<mn>
2
</mn>
</msup>
</mrow>
</math>
is close to
<math alttext="10" class="ltx_Math" display="inline" id="S2.p4.m3">
<mn>
10
</mn>
</math>
. So, instead of computing
<math alttext="\sqrt{10}" class="ltx_Math" display="inline" id="S2.p4.m4">
<msqrt>
<mn>
10
</mn>
</msqrt>
</math>
, consider
<math alttext="\sqrt{10/9}" class="ltx_Math" display="inline" id="S2.p4.m5">
<msqrt>
<mrow>
<mn>
10
</mn>
<mo>
/
</mo>
<mn>
9
</mn>
</mrow>
</msqrt>
</math>
. Once this has been computed, one can multiply by
<math alttext="3" class="ltx_Math" display="inline" id="S2.p4.m6">
<mn>
3
</mn>
</math>
to obtain the answer because
<math alttext="3\sqrt{10/9}=\sqrt{9}\sqrt{10/9}=\sqrt{10}" class="ltx_Math" display="inline" id="S2.p4.m7">
<mrow>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<msqrt>
<mrow>
<mn>
10
</mn>
<mo>
/
</mo>
<mn>
9
</mn>
</mrow>
</msqrt>
</mrow>
<mo>
=
</mo>
<mrow>
<msqrt>
<mn>
9
</mn>
</msqrt>
<mo>
</mo>
<msqrt>
<mrow>
<mn>
10
</mn>
<mo>
/
</mo>
<mn>
9
</mn>
</mrow>
</msqrt>
</mrow>
<mo>
=
</mo>
<msqrt>
<mn>
10
</mn>
</msqrt>
</mrow>
</math>
. In this case,
<math alttext="x=-1/9" class="ltx_Math" display="inline" id="S2.p4.m8">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
9
</mn>
</mrow>
</mrow>
</mrow>
</math>
, the recursion looks like
</p>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex19">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{n+1}=-\frac{1}{18}+\frac{1}{2}y_{n}^{2}" class="ltx_Math" display="block" id="S2.Ex19.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
=
</mo>
<mrow>
<mrow>
<mo>
-
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
18
</mn>
</mfrac>
</mrow>
<mo>
+
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
and the approximations go as follows:
</p>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex20">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{1}=0" class="ltx_Math" display="block" id="S2.Ex20.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{2}=-0.05556" class="ltx_Math" display="block" id="S2.Ex21.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
2
</mn>
</msub>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mn>
0.05556
</mn>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex22">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{3}=-0.05401" class="ltx_Math" display="block" id="S2.Ex22.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
3
</mn>
</msub>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mn>
0.05401
</mn>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex23">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{4}=-0.05410" class="ltx_Math" display="block" id="S2.Ex23.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
4
</mn>
</msub>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mn>
0.05410
</mn>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S2.Ex24">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{5}=-0.05409" class="ltx_Math" display="block" id="S2.Ex24.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
5
</mn>
</msub>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mn>
0.05409
</mn>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Subtracting from 1 and multiplying by 3, one obtains 3.16227 as a value for
<math alttext="\sqrt{10}" class="ltx_Math" display="inline" id="S2.p4.m9">
<msqrt>
<mn>
10
</mn>
</msqrt>
</math>
, which is good to 5 decimal places.
</p>
</div>
</section>
<section class="ltx_section" id="S3">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
3
</span>
Convergence
</h2>
<div class="ltx_para" id="S3.p1">
<p class="ltx_p">
Having shown that how this method works, it is now time to make sure that it really does work. As a first step, it will be shown that, if
<math alttext="x" class="ltx_Math" display="inline" id="S3.p1.m1">
<mi>
x
</mi>
</math>
lies between
<math alttext="-2" class="ltx_Math" display="inline" id="S3.p1.m2">
<mrow>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
</math>
and
<math alttext="+1" class="ltx_Math" display="inline" id="S3.p1.m3">
<mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
, then all approximations
<math alttext="y_{n}" class="ltx_Math" display="inline" id="S3.p1.m4">
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</math>
lie between
<math alttext="-1" class="ltx_Math" display="inline" id="S3.p1.m5">
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</math>
and
<math alttext="+1" class="ltx_Math" display="inline" id="S3.p1.m6">
<mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
as well. By definition
<math alttext="y_{0}" class="ltx_Math" display="inline" id="S3.p1.m7">
<msub>
<mi>
y
</mi>
<mn>
0
</mn>
</msub>
</math>
lies in the required range, since it is defined to be 0. Suppose that
<math alttext="y_{n}" class="ltx_Math" display="inline" id="S3.p1.m8">
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</math>
lies in the range. Then, by definition,
</p>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex25">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{n+1}=\frac{1}{2}(x+y_{n})." class="ltx_Math" display="block" id="S3.Ex25.m1">
<mrow>
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Since both
<math alttext="x" class="ltx_Math" display="inline" id="S3.p1.m9">
<mi>
x
</mi>
</math>
and
<math alttext="y_{n}" class="ltx_Math" display="inline" id="S3.p1.m10">
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</math>
lie between
<math alttext="-1" class="ltx_Math" display="inline" id="S3.p1.m11">
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</math>
and
<math alttext="+1" class="ltx_Math" display="inline" id="S3.p1.m12">
<mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
, it follows that their
<a class="nnexus_concept" href="http://planetmath.org/arithmeticmean">
average
</a>
must lie in the same range.
</p>
</div>
<div class="ltx_para" id="S3.p2">
<p class="ltx_p">
To show
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Convergence.html">
convergence
</a>
, the recursion will subtracted from a shifted version of itself like so:
</p>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex26">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{n+2}=\frac{1}{2}(x+y_{n+1}^{2})" class="ltx_Math" display="block" id="S3.Ex26.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
</msub>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msubsup>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mn>
2
</mn>
</msubsup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex27">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{n+1}=\frac{1}{2}(x+y_{n}^{2})" class="ltx_Math" display="block" id="S3.Ex27.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex28">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{n+2}-y_{n+1}=\frac{1}{2}(y_{n+1}^{2}-y_{n}^{2})" class="ltx_Math" display="block" id="S3.Ex28.m1">
<mrow>
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
</msub>
<mo>
-
</mo>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
</mrow>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msubsup>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mn>
2
</mn>
</msubsup>
<mo>
-
</mo>
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
The right hand side, being a
<a class="nnexus_concept" href="http://planetmath.org/differenceofsquares">
difference of squares
</a>
, may be factored.
</p>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex29">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{n+2}-y_{n+1}=\frac{1}{2}(y_{n+1}+y_{n})(y_{n+1}-y_{n})" class="ltx_Math" display="block" id="S3.Ex29.m1">
<mrow>
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
</msub>
<mo>
-
</mo>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
</mrow>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
+
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
-
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
For ease of explanation, define the quantity
<math alttext="z_{n}=y_{n+1}-y_{n}" class="ltx_Math" display="inline" id="S3.p2.m1">
<mrow>
<msub>
<mi>
z
</mi>
<mi>
n
</mi>
</msub>
<mo>
=
</mo>
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
-
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
</mrow>
</math>
. In terms of this entity, the above equation may be written as
</p>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex30">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="z_{n+1}=\frac{1}{2}(y_{n+1}+y_{n})z_{n}" class="ltx_Math" display="block" id="S3.Ex30.m1">
<mrow>
<msub>
<mi>
z
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
+
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<msub>
<mi>
z
</mi>
<mi>
n
</mi>
</msub>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="S3.p3">
<p class="ltx_p">
The proof will now proceed differently for the cases of
<math alttext="x" class="ltx_Math" display="inline" id="S3.p3.m1">
<mi>
x
</mi>
</math>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
positive
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/positive">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/positiveelement">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and
<math alttext="x" class="ltx_Math" display="inline" id="S3.p3.m2">
<mi>
x
</mi>
</math>
negative. (The in-between case
<math alttext="x=0" class="ltx_Math" display="inline" id="S3.p3.m3">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
is trivial.) If
<math alttext="x" class="ltx_Math" display="inline" id="S3.p3.m4">
<mi>
x
</mi>
</math>
is positive, then it readily follows from the recursion that
<math alttext="y_{n}" class="ltx_Math" display="inline" id="S3.p3.m5">
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</math>
is positive for all
<math alttext="n" class="ltx_Math" display="inline" id="S3.p3.m6">
<mi>
n
</mi>
</math>
. Hence,
<math alttext="\frac{1}{2}(y_{n+1}+y_{n})" class="ltx_Math" display="inline" id="S3.p3.m7">
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
+
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is also positive for all
<math alttext="n" class="ltx_Math" display="inline" id="S3.p3.m8">
<mi>
n
</mi>
</math>
. Now,
<math alttext="z_{2}=x/2" class="ltx_Math" display="inline" id="S3.p3.m9">
<mrow>
<msub>
<mi>
z
</mi>
<mn>
2
</mn>
</msub>
<mo>
=
</mo>
<mrow>
<mi>
x
</mi>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</mrow>
</math>
, so
<math alttext="z_{2}" class="ltx_Math" display="inline" id="S3.p3.m10">
<msub>
<mi>
z
</mi>
<mn>
2
</mn>
</msub>
</math>
is positive. Since the product of two positive quantities is positive, it follows by induction that
<math alttext="z_{n}" class="ltx_Math" display="inline" id="S3.p3.m11">
<msub>
<mi>
z
</mi>
<mi>
n
</mi>
</msub>
</math>
is positive for all
<math alttext="n" class="ltx_Math" display="inline" id="S3.p3.m12">
<mi>
n
</mi>
</math>
. By definition, this means that
<math alttext="y_{n+1}>y_{n}" class="ltx_Math" display="inline" id="S3.p3.m13">
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
>
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
</math>
for all
<math alttext="n" class="ltx_Math" display="inline" id="S3.p3.m14">
<mi>
n
</mi>
</math>
. Since we know that
<math alttext="y_{n}" class="ltx_Math" display="inline" id="S3.p3.m15">
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</math>
is less that
<math alttext="1" class="ltx_Math" display="inline" id="S3.p3.m16">
<mn>
1
</mn>
</math>
for all
<math alttext="n" class="ltx_Math" display="inline" id="S3.p3.m17">
<mi>
n
</mi>
</math>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
bounded
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/hahnbanachtheorem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/upperbound">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/IncreasingSequence.html">
increasing sequences
</a>
converge, it follows that the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
sequence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Sequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
convergent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Convergent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/convergentstoacontinuedfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/convergentseries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="S3.p4">
<p class="ltx_p">
If
<math alttext="x" class="ltx_Math" display="inline" id="S3.p4.m1">
<mi>
x
</mi>
</math>
is negative, then it can be shown that
<math alttext="y_{n}" class="ltx_Math" display="inline" id="S3.p4.m2">
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</math>
lies between
<math alttext="x/2" class="ltx_Math" display="inline" id="S3.p4.m3">
<mrow>
<mi>
x
</mi>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</math>
and
<math alttext="0" class="ltx_Math" display="inline" id="S3.p4.m4">
<mn>
0
</mn>
</math>
for all
<math alttext="n" class="ltx_Math" display="inline" id="S3.p4.m5">
<mi>
n
</mi>
</math>
. This is trivially true for
<math alttext="y_{1}" class="ltx_Math" display="inline" id="S3.p4.m6">
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
</math>
. Assume that it is true for a particular value of
<math alttext="n" class="ltx_Math" display="inline" id="S3.p4.m7">
<mi>
n
</mi>
</math>
. Then
<math alttext="y_{n}^{2}" class="ltx_Math" display="inline" id="S3.p4.m8">
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</math>
is positive and smaller than
<math alttext="-x" class="ltx_Math" display="inline" id="S3.p4.m9">
<mrow>
<mo>
-
</mo>
<mi>
x
</mi>
</mrow>
</math>
. Hence,
<math alttext="x+y_{n}^{2}" class="ltx_Math" display="inline" id="S3.p4.m10">
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</mrow>
</math>
lies between
<math alttext="x" class="ltx_Math" display="inline" id="S3.p4.m11">
<mi>
x
</mi>
</math>
and
<math alttext="0" class="ltx_Math" display="inline" id="S3.p4.m12">
<mn>
0
</mn>
</math>
, so
<math alttext="(x+y_{n}^{2})/2" class="ltx_Math" display="inline" id="S3.p4.m13">
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</math>
, which equals
<math alttext="y_{n+1}" class="ltx_Math" display="inline" id="S3.p4.m14">
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
</math>
, lies between
<math alttext="x/2" class="ltx_Math" display="inline" id="S3.p4.m15">
<mrow>
<mi>
x
</mi>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</math>
and
<math alttext="0" class="ltx_Math" display="inline" id="S3.p4.m16">
<mn>
0
</mn>
</math>
.
</p>
</div>
<div class="ltx_para" id="S3.p5">
<p class="ltx_p">
Consequently,
<math alttext="(y_{n+1}+y_{n})/2" class="ltx_Math" display="inline" id="S3.p5.m1">
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
+
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</math>
will lie between
<math alttext="x/2" class="ltx_Math" display="inline" id="S3.p5.m2">
<mrow>
<mi>
x
</mi>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</math>
and
<math alttext="0" class="ltx_Math" display="inline" id="S3.p5.m3">
<mn>
0
</mn>
</math>
. By the recursion for
<math alttext="z_{n}" class="ltx_Math" display="inline" id="S3.p5.m4">
<msub>
<mi>
z
</mi>
<mi>
n
</mi>
</msub>
</math>
, one therefore has
</p>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex31">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="|z_{n+1}|\leq\left|\frac{x}{2}\right|\,|z_{n}|" class="ltx_Math" display="block" id="S3.Ex31.m1">
<mrow>
<mrow>
<mo stretchy="false">
|
</mo>
<msub>
<mi>
z
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo stretchy="false">
|
</mo>
</mrow>
<mo>
≤
</mo>
<mrow>
<mrow>
<mo>
|
</mo>
<mfrac>
<mi>
x
</mi>
<mn>
2
</mn>
</mfrac>
<mo rspace="4.2pt">
|
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
|
</mo>
<msub>
<mi>
z
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
|
</mo>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
from which it follows that
</p>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex32">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="|z_{n}|\leq\left|\frac{x}{2}\right|^{n}|z_{1}|" class="ltx_Math" display="block" id="S3.Ex32.m1">
<mrow>
<mrow>
<mo stretchy="false">
|
</mo>
<msub>
<mi>
z
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
|
</mo>
</mrow>
<mo>
≤
</mo>
<mrow>
<msup>
<mrow>
<mo>
|
</mo>
<mfrac>
<mi>
x
</mi>
<mn>
2
</mn>
</mfrac>
<mo>
|
</mo>
</mrow>
<mi>
n
</mi>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
|
</mo>
<msub>
<mi>
z
</mi>
<mn>
1
</mn>
</msub>
<mo stretchy="false">
|
</mo>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
By definition of
<math alttext="z_{n}" class="ltx_Math" display="inline" id="S3.p5.m5">
<msub>
<mi>
z
</mi>
<mi>
n
</mi>
</msub>
</math>
, one has
</p>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex33">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="|y_{m}-y_{n}|=\left|\sum_{j=n+1}^{m}z_{j}\right|\leq\sum_{j=n+1}^{m}|z_{j}|%
\leq\sum_{j=n+1}^{m}\left|\frac{x}{2}\right|^{n}|z_{1}|=\frac{(|x|/2)^{m}-(|x|%
/2|)^{n}}{1-|x|/2}\,|z_{0}|." class="ltx_Math" display="block" id="S3.Ex33.m1">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
|
</mo>
<mrow>
<msub>
<mi>
y
</mi>
<mi>
m
</mi>
</msub>
<mo>
-
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mo stretchy="false">
|
</mo>
</mrow>
<mo>
=
</mo>
<mrow>
<mo>
|
</mo>
<mrow>
<munderover>
<mo largeop="true" movablelimits="false" symmetric="true">
∑
</mo>
<mrow>
<mi>
j
</mi>
<mo>
=
</mo>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
<mi>
m
</mi>
</munderover>
<msub>
<mi>
z
</mi>
<mi>
j
</mi>
</msub>
</mrow>
<mo>
|
</mo>
</mrow>
<mo>
≤
</mo>
<mrow>
<munderover>
<mo largeop="true" movablelimits="false" symmetric="true">
∑
</mo>
<mrow>
<mi>
j
</mi>
<mo>
=
</mo>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
<mi>
m
</mi>
</munderover>
<mrow>
<mo stretchy="false">
|
</mo>
<msub>
<mi>
z
</mi>
<mi>
j
</mi>
</msub>
<mo stretchy="false">
|
</mo>
</mrow>
</mrow>
<mo>
≤
</mo>
<mrow>
<munderover>
<mo largeop="true" movablelimits="false" symmetric="true">
∑
</mo>
<mrow>
<mi>
j
</mi>
<mo>
=
</mo>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
<mi>
m
</mi>
</munderover>
<mrow>
<msup>
<mrow>
<mo>
|
</mo>
<mfrac>
<mi>
x
</mi>
<mn>
2
</mn>
</mfrac>
<mo>
|
</mo>
</mrow>
<mi>
n
</mi>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
|
</mo>
<msub>
<mi>
z
</mi>
<mn>
1
</mn>
</msub>
<mo stretchy="false">
|
</mo>
</mrow>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mpadded width="+1.7pt">
<mfrac>
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mo stretchy="false">
|
</mo>
<mi>
x
</mi>
<mo stretchy="false">
|
</mo>
<mo>
/
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mi>
m
</mi>
</msup>
<mo>
-
</mo>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mo stretchy="false">
|
</mo>
<mi>
x
</mi>
<mo stretchy="false">
|
</mo>
<mo>
/
</mo>
<mn>
2
</mn>
<mo stretchy="false">
|
</mo>
<mo stretchy="false">
)
</mo>
</mrow>
<mi>
n
</mi>
</msup>
</mrow>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mrow>
<mrow>
<mo stretchy="false">
|
</mo>
<mi>
x
</mi>
<mo stretchy="false">
|
</mo>
</mrow>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</mrow>
</mfrac>
</mpadded>
<mo>
</mo>
<mrow>
<mo stretchy="false">
|
</mo>
<msub>
<mi>
z
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
|
</mo>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
If
<math alttext="x" class="ltx_Math" display="inline" id="S3.p5.m6">
<mi>
x
</mi>
</math>
lies between
<math alttext="-2" class="ltx_Math" display="inline" id="S3.p5.m7">
<mrow>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
</math>
and
<math alttext="0" class="ltx_Math" display="inline" id="S3.p5.m8">
<mn>
0
</mn>
</math>
, then
<math alttext="|x|/2" class="ltx_Math" display="inline" id="S3.p5.m9">
<mrow>
<mrow>
<mo stretchy="false">
|
</mo>
<mi>
x
</mi>
<mo stretchy="false">
|
</mo>
</mrow>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</math>
is smaller than
<math alttext="1" class="ltx_Math" display="inline" id="S3.p5.m10">
<mn>
1
</mn>
</math>
in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
absolute value
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AbsoluteValue.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/absolutevalueinavectorlattice">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/modulusofcomplexnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/valuation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/absolutevalue">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, so it follows from Cauchy’s criterion that the sequence converges.
</p>
</div>
<div class="ltx_para" id="S3.p6">
<p class="ltx_p">
It is not enough that the sequence converges; it must converge to the right value. This, however, is easily checked. Take the limit of both sides of the recursion:
</p>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex34">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\lim_{n\to\infty}y_{n+1}=\lim_{n\to\infty}\frac{1}{2}(x+y_{n}^{2})" class="ltx_Math" display="block" id="S3.Ex34.m1">
<mrow>
<mrow>
<munder>
<mo movablelimits="false">
lim
</mo>
<mrow>
<mi>
n
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</munder>
<mo>
</mo>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
</mrow>
<mo>
=
</mo>
<mrow>
<munder>
<mo movablelimits="false">
lim
</mo>
<mrow>
<mi>
n
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</munder>
<mo>
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msubsup>
<mi>
y
</mi>
<mi>
n
</mi>
<mn>
2
</mn>
</msubsup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Defining
<math alttext="y=\lim_{n\to\infty}y_{n}" class="ltx_Math" display="inline" id="S3.p6.m1">
<mrow>
<mi>
y
</mi>
<mo>
=
</mo>
<mrow>
<msub>
<mo>
lim
</mo>
<mrow>
<mi>
n
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</msub>
<mo>
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
</mrow>
</mrow>
</math>
, it follows that
</p>
<table class="ltx_equation ltx_eqn_table" id="S3.Ex35">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y=\frac{1}{2}(x+y^{2})." class="ltx_Math" display="block" id="S3.Ex35.m1">
<mrow>
<mrow>
<mi>
y
</mi>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Reading the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebraic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Algebraics.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraic1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://planetmath.org/complex">
argument
</a>
at the beginning of this entry backwards, one sees that
<math alttext="1-y" class="ltx_Math" display="inline" id="S3.p6.m2">
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mi>
y
</mi>
</mrow>
</math>
is indeed the square root of
<math alttext="1-x" class="ltx_Math" display="inline" id="S3.p6.m3">
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mi>
x
</mi>
</mrow>
</math>
as it is supposed to be.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/babylonianmethodofcomputingsquareroots">
Babylonian method of computing square roots
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
BabylonianMethodOfComputingSquareRoots
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:16:45
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:16:45
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
9
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Algorithm
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 01A17
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
BombellisMethodOfComputingSquareRoots
</td>
</tr>
</tbody>
</table>
</div>
</section>
</article>
</div>
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<html>
<head>
<title>
biogroupoids: mathematical models of species evolution
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<h1 class="ltx_title ltx_title_document">
biogroupoids: mathematical models of species evolution
</h1>
<span class="ltx_ERROR undefined">
\xyoption
</span>
<div class="ltx_para" id="p1">
<p class="ltx_p">
curve
</p>
</div>
<section class="ltx_subsection" id="S0.SS1">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
0.1
</span>
Introduction
</h2>
<div class="ltx_para" id="S0.SS1.p1">
<p class="ltx_p">
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/biogroupoidsmathematicalmodelsofspeciesevolution">
Biogroupoids
</a>
</em>
,
<math alttext="\mathcal{G_{B}}" class="ltx_Math" display="inline" id="S0.SS1.p1.m1">
<msub>
<mi class="ltx_font_mathcaligraphic">
𝒢
</mi>
<mi class="ltx_font_mathcaligraphic">
ℬ
</mi>
</msub>
</math>
, were introduced as
<a class="nnexus_concept" href="http://planetmath.org/arbsymmetryandgroupoidrepresentations">
mathematical representations
</a>
of
evolving biological species (
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
,
<a class="ltx_ref" href="#bib.bib2" title="">
2
</a>
]
</cite>
) generated by
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/weakhomotopyequivalence">
weakly equivalent
</a>
classes of living organisms
</em>
,
<math alttext="E_{O}" class="ltx_Math" display="inline" id="S0.SS1.p1.m2">
<msub>
<mi>
E
</mi>
<mi>
O
</mi>
</msub>
</math>
, specified by inter-breeding organisms;in this case, a
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/WeakEquivalence.html">
weak equivalence
</a>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
relation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relationonobjects">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/presentationofagroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
,
<math alttext="\sim_{w}" class="ltx_Math" display="inline" id="S0.SS1.p1.m3">
<msub>
<mo>
∼
</mo>
<mi>
w
</mi>
</msub>
</math>
, is defined on the set of evolving organisms modeled in terms of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functional
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/interpretationofintuitionisticlogicbymeansoffunctionals">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/intuitionisticlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functional">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
isomorphic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/isomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/grouphomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/isomorphicgroups">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://planetmath.org/geneticnets">
genome networks
</a>
</em>
,
<math alttext="G_{iso}^{N}" class="ltx_Math" display="inline" id="S0.SS1.p1.m4">
<msubsup>
<mi>
G
</mi>
<mrow>
<mi>
i
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<mi>
o
</mi>
</mrow>
<mi>
N
</mi>
</msubsup>
</math>
, such as those described by
<math alttext="LM_{n}" class="ltx_Math" display="inline" id="S0.SS1.p1.m5">
<mrow>
<mi>
L
</mi>
<mo>
</mo>
<msub>
<mi>
M
</mi>
<mi>
n
</mi>
</msub>
</mrow>
</math>
-logic networks in Łukasiewicz-Moisil,
<math alttext="{\mathcal{L}M}_{n}" class="ltx_Math" display="inline" id="S0.SS1.p1.m6">
<mrow>
<mi class="ltx_font_mathcaligraphic">
ℒ
</mi>
<mo>
</mo>
<msub>
<mi>
M
</mi>
<mi>
n
</mi>
</msub>
</mrow>
</math>
topoi (
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
]
</cite>
).
</p>
</div>
</section>
<section class="ltx_subsection" id="S0.SS2">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
0.2
</span>
AT-Formulation
</h2>
<div class="ltx_para" id="S0.SS2.p1">
<p class="ltx_p">
This
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
concept
</a>
allows an
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicTopology.html">
algebraic topology
</a>
formulation of the origin of species and biological evolution
both at organismal/organismic and biomolecular levels; it
<a class="nnexus_concept" href="http://planetmath.org/representablefunctor">
represents
</a>
a new approach to biological evolution
from the standpoint of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
super-complex systems
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/artificialintelligence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/noncommutativedynamicmodelingdiagrams">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/organismicsupercategoriesandsupercomplexsystemsbiodynamics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
biology.
</p>
</div>
</section>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Categories
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Category.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/category">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Higher Dimensional Algebra
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/ncategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/2groupoid">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and Łukasiewicz-Moisil Topos:
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Transformations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/functorialmorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/transformation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of Neuronal, Genetic and Neoplastic Networks.,
<em class="ltx_emph ltx_font_italic">
Axiomathes
</em>
,
<span class="ltx_text ltx_font_bold">
16
</span>
Nos. 1-2: 65-122.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007,
<a class="nnexus_concept" href="http://planetmath.org/categoricalquantumlmlogicalgebras">
Categorical Ontology
</a>
of Complex Spacetime
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Structures
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
: The Emergence of Life and Human Consciousness, Axiomathes,
<span class="ltx_text ltx_font_bold">
17
</span>
: 35-168.
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p2">
<table class="ltx_tabular ltx_align_middle">
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<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
biogroupoids: mathematical models of species evolution
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Canonical name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
BiogroupoidsMathematicalModelsOfSpeciesEvolution
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:11:47
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:11:47
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
48
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Topic
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 92B05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 03G20
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 92D15
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
mathematical model of species
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
origin of species
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
SupercategoriesOfComplexSystems
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
AlgebraicCategoryOfLMnLogicAlgebras
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
RosettaGroupoids
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
biogroupoid
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | Calculator | http://planetmath.org/Calculator | <!DOCTYPE html>
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calculator
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calculator
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<br class="ltx_break"/>
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A
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/calculator">
calculator
</a>
</span>
is an electronic, electrical or mechanical device (hardware) or a
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computer
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Program.html">
program
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(software) designed to perform a predefined set of
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arithmetic
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computations
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on numbers entered by the user and display the results.
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The abacus is sometimes given as an example of an early calculator; however, the completion of a calculation requires the active participation of the user and is easily subject to user error even when the device passes a diagnostic with flying colors, whereas a modern calculator in working order
<a class="nnexus_concept" href="http://planetmath.org/searchproblem">
calculates
</a>
and displays the correct result regardless of whether or not the user sticks around to see it (this is subject to certain caveats regarding precision, however). Of course it has always been the case in the history of calculators that users may enter incorrect inputs or misunderstand the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operation
</a>
<sup style="display: none;">
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of the device and thus get irrelevant results.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
One of the first mechanical calculators was designed by
<a class="nnexus_concept" href="http://planetmath.org/blaisepascal">
Blaise Pascal
</a>
in the 17th Century. It used a gear for each digit and the gears were
<a class="nnexus_concept" href="http://planetmath.org/connectedgraph">
connected
</a>
, it took up a little space on a desk. It was limited to
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and
<a class="nnexus_concept" href="http://planetmath.org/subtraction">
subtraction
</a>
. By the 1990s, electronic calculators were ubiquitous and small enough to put in a pocket.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
A basic calculator has about twenty keys: the digits 0 to 9, a
<a class="nnexus_concept" href="http://planetmath.org/decimalpoint">
decimal point
</a>
, sign change, addition, subtraction,
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
multiplication
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/division">
division
</a>
, equal or Enter key and a C key to clear error exceptions. Occasionally such calculators also have keys for
<a class="nnexus_concept" href="http://planetmath.org/percentage">
percentage
</a>
and square root. Another option sometimes found on basic calculators is a memory register and associated keys M+, MR and MC (for adding to memory register, recalling memory and clearing memory, respectively).
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
More advanced calculators include
<a class="nnexus_concept" href="http://planetmath.org/scientificcalculator">
scientific calculators
</a>
, graphing calculators, programmable calculators. Most calculators use standard
<a class="nnexus_concept" href="http://planetmath.org/infixnotation">
infix notation
</a>
, but there are
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
reverse Polish notation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ReversePolishNotation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reversepolishnotation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
calculators also available on the market.
</p>
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<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
calculator
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Calculator
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:39:16
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:39:16
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
4
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 01A07
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
CalculatorAndCASSupportForVariousPositionalBases
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | Characterisation | http://planetmath.org/Characterisation | <!DOCTYPE html>
<html>
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<title>
characterisation
</title>
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<h1 class="ltx_title ltx_title_document">
characterisation
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
In mathematics,
<span class="ltx_text ltx_font_italic">
characterisation
</span>
usually means a property or a condition to define a certain notion. A notion may, under some presumptions, have different ways to define it.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
For example, let
<math alttext="R" class="ltx_Math" display="inline" id="p2.m1">
<mi>
R
</mi>
</math>
be a
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CommutativeRing.html">
commutative ring
</a>
with
<a class="nnexus_concept" href="http://planetmath.org/unity">
non-zero unity
</a>
(the presumption). Then the following are
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
equivalent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Equivalent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/filterbasis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalenceofforcingnotions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalentmachines">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalencerelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
:
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
(1) All
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
finitely generated
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FinitelyGenerated.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/finitelygeneratedmodule">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/finitelygeneratedgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://planetmath.org/regularideal">
regular ideals
</a>
of
<math alttext="R" class="ltx_Math" display="inline" id="p3.m1">
<mi>
R
</mi>
</math>
are
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
invertible
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/fractionalidealofcommutativering">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inversefunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
(2) The
<math alttext="(a,\,b)(c,\,d)=(ac,\,bd,\,(a+b)(c+d))" class="ltx_Math" display="inline" id="p4.m1">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
a
</mi>
<mo rspace="4.2pt">
,
</mo>
<mi>
b
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
c
</mi>
<mo rspace="4.2pt">
,
</mo>
<mi>
d
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
a
</mi>
<mo>
</mo>
<mi>
c
</mi>
</mrow>
<mo rspace="4.2pt">
,
</mo>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mi>
d
</mi>
</mrow>
<mo rspace="4.2pt">
,
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
c
</mi>
<mo>
+
</mo>
<mi>
d
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
for multiplying ideals of
<math alttext="R" class="ltx_Math" display="inline" id="p4.m2">
<mi>
R
</mi>
</math>
is valid always when at least one of the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Element.html">
elements
</a>
<math alttext="a" class="ltx_Math" display="inline" id="p4.m3">
<mi>
a
</mi>
</math>
,
<math alttext="b" class="ltx_Math" display="inline" id="p4.m4">
<mi>
b
</mi>
</math>
,
<math alttext="c" class="ltx_Math" display="inline" id="p4.m5">
<mi>
c
</mi>
</math>
,
<math alttext="d" class="ltx_Math" display="inline" id="p4.m6">
<mi>
d
</mi>
</math>
of
<math alttext="R" class="ltx_Math" display="inline" id="p4.m7">
<mi>
R
</mi>
</math>
is not zero-divisor.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
(3) Every
<a class="nnexus_concept" href="http://planetmath.org/overring">
overring
</a>
of
<math alttext="R" class="ltx_Math" display="inline" id="p5.m1">
<mi>
R
</mi>
</math>
is
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
integrally closed
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/IntegrallyClosed.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integrallyclosed">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
Each of these conditions is
<a class="nnexus_concept" href="http://planetmath.org/necessaryandsufficient">
sufficient
</a>
(and necessary) for characterising and defining the Prüfer ring.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
characterisation
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Characterisation
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:22:28
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:22:28
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
18
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Characterization.html">
characterization
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
defining property
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
AlternativeDefinitionOfGroup
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
EquivalentFormulationsForContinuity
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
MultiplicationRuleGivesInverseIdeal
</td>
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0 | ConwaysChainedArrowNotation | http://planetmath.org/ConwaysChainedArrowNotation | <!DOCTYPE html>
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<head>
<title>
Conway’s chained arrow notation
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
Conway’s chained arrow notation
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<em class="ltx_emph ltx_font_italic">
Conway’s
<a class="nnexus_concept" href="http://planetmath.org/conwayschainedarrownotation">
chained arrow notation
</a>
</em>
is a way of writing numbers even larger than those provided by the up arrow notation. We define
<math alttext="m\rightarrow n\rightarrow p=m^{(p+2)}n=m\underbrace{\uparrow\cdots\uparrow}_{p}n" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mi>
m
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
<mo>
→
</mo>
<mi>
p
</mi>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
m
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
p
</mi>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
m
</mi>
<mo>
</mo>
<munder>
<munder accentunder="true">
<mrow>
<mi>
</mi>
<mo movablelimits="false">
↑
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo movablelimits="false">
↑
</mo>
<mi>
</mi>
</mrow>
<mo movablelimits="false">
⏟
</mo>
</munder>
<mi>
p
</mi>
</munder>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
</mrow>
</math>
and
<math alttext="m\rightarrow n=m\rightarrow n\rightarrow 1=m^{n}" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<mi>
m
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
<mo>
=
</mo>
<mi>
m
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
<mo>
→
</mo>
<mn>
1
</mn>
<mo>
=
</mo>
<msup>
<mi>
m
</mi>
<mi>
n
</mi>
</msup>
</mrow>
</math>
. Longer chains are evaluated by
</p>
</div>
<div class="ltx_para" id="p2">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="m\rightarrow\cdots\rightarrow n\rightarrow p\rightarrow 1=m\rightarrow\cdots%
\rightarrow n\rightarrow p" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mi>
m
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
<mo>
→
</mo>
<mi>
p
</mi>
<mo>
→
</mo>
<mn>
1
</mn>
<mo>
=
</mo>
<mi>
m
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
<mo>
→
</mo>
<mi>
p
</mi>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p3">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="m\rightarrow\cdots\rightarrow n\rightarrow 1\rightarrow q=m\rightarrow\cdots\rightarrow
n" class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mi>
m
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
<mo>
→
</mo>
<mn>
1
</mn>
<mo>
→
</mo>
<mi>
q
</mi>
<mo>
=
</mo>
<mi>
m
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
and
</p>
</div>
<div class="ltx_para" id="p5">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="m\rightarrow\cdots\rightarrow n\rightarrow p+1\rightarrow q+1=m\rightarrow%
\cdots\rightarrow n\rightarrow(m\rightarrow\cdots\rightarrow n\rightarrow p%
\rightarrow q+1)\rightarrow q" class="ltx_Math" display="block" id="S0.Ex3.m1">
<mrow>
<mi>
m
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
<mo>
→
</mo>
<mi>
p
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
<mo>
→
</mo>
<mi>
q
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
<mo>
=
</mo>
<mi>
m
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
m
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
→
</mo>
<mi>
n
</mi>
<mo>
→
</mo>
<mi>
p
</mi>
<mo>
→
</mo>
<mi>
q
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mi>
q
</mi>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
For example:
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex4">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow 3\rightarrow 2=" class="ltx_Math" display="inline" id="S0.Ex4.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
=
</mo>
<mi>
</mi>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex5">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow(3\rightarrow 2\rightarrow 2)\rightarrow 1=" class="ltx_Math" display="inline" id="S0.Ex5.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
1
</mn>
<mo>
=
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex6">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow(3\rightarrow 2\rightarrow 2)=" class="ltx_Math" display="inline" id="S0.Ex6.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
=
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex7">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow(3\rightarrow(3\rightarrow 1\rightarrow 2)%
\rightarrow 1)=" class="ltx_Math" display="inline" id="S0.Ex7.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
1
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
1
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
=
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex8">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow(3\rightarrow 3\rightarrow 1)=" class="ltx_Math" display="inline" id="S0.Ex8.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
1
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
=
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex9">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3^{3^{3}}=" class="ltx_Math" display="inline" id="S0.Ex9.m1">
<mrow>
<msup>
<mn>
3
</mn>
<msup>
<mn>
3
</mn>
<mn>
3
</mn>
</msup>
</msup>
<mo>
=
</mo>
<mi>
</mi>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex10">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3^{27}=7625597484987" class="ltx_Math" display="inline" id="S0.Ex10.m1">
<mrow>
<msup>
<mn>
3
</mn>
<mn>
27
</mn>
</msup>
<mo>
=
</mo>
<mn>
7625597484987
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
A much larger example is:
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex11">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow 2\rightarrow 4\rightarrow 4=" class="ltx_Math" display="inline" id="S0.Ex11.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mn>
4
</mn>
<mo>
→
</mo>
<mn>
4
</mn>
<mo>
=
</mo>
<mi>
</mi>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex12">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow 3\rightarrow
4%
)\rightarrow 3=" class="ltx_Math" display="inline" id="S0.Ex12.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
4
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo>
=
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex13">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow(3\rightarrow
2%
\rightarrow 2\rightarrow 4)\rightarrow 3)\rightarrow 3=" class="ltx_Math" display="inline" id="S0.Ex13.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mn>
4
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo>
=
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex14">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow(3\rightarrow
2%
\rightarrow(3\rightarrow 2\rightarrow 1\rightarrow 4)\rightarrow 3)\rightarrow
3%
)\rightarrow 3=" class="ltx_Math" display="inline" id="S0.Ex14.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mn>
1
</mn>
<mo>
→
</mo>
<mn>
4
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo>
=
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex15">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow(3\rightarrow
2%
\rightarrow(3\rightarrow 2)\rightarrow 3)\rightarrow 3)\rightarrow 3=" class="ltx_Math" display="inline" id="S0.Ex15.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo>
=
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex16">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow(3\rightarrow
2%
\rightarrow 9\rightarrow 3)\rightarrow 3)\rightarrow 3" class="ltx_Math" display="inline" id="S0.Ex16.m1">
<mrow>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
→
</mo>
<mn>
2
</mn>
<mo>
→
</mo>
<mn>
9
</mn>
<mo>
→
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
→
</mo>
<mn>
3
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
Clearly this is going to be a very large number. Note that, as large as it is, it is proceeding towards an eventual final evaluation, as evidenced by the fact that the final number in the chain is getting smaller.
</p>
<table class="ltx_tabular ltx_align_right ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
Conway’s chained arrow notation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
ConwaysChainedArrowNotation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:58:46
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:58:46
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
8
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
chained arrow notation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
chained arrow
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
chained-arrow
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
chained-arrow notation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Conway notation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
KnuthsUpArrowNotation
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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<div class="ltx_page_logo">
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0 | DecimalPlace | http://planetmath.org/DecimalPlace | <!DOCTYPE html>
<html>
<head>
<title>
decimal place
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
decimal place
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
A
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/decimalplace">
decimal place
</a>
</em>
of a number (a real number)
<math alttext="r" class="ltx_Math" display="inline" id="p2.m1">
<mi>
r
</mi>
</math>
is the position of a digit in its
<a class="nnexus_concept" href="http://planetmath.org/decimalexpansion">
decimal expansion
</a>
relative to the
<a class="nnexus_concept" href="http://planetmath.org/decimalpoint">
decimal point
</a>
. Let us write
<math alttext="r" class="ltx_Math" display="inline" id="p2.m2">
<mi>
r
</mi>
</math>
as a
<a class="nnexus_concept" href="http://planetmath.org/decimalfraction">
decimal number
</a>
:
</p>
<p class="ltx_p ltx_align_center">
<math alttext="\xymatrix@C=0pt@H=10pt{A_{n}A_{n-1}\ldots A_{1}&\bullet&D_{1}D_{2}\ldots D_{m}%
\ldots\inner@par&\save*\txt{thedecimalpoint}\restore\ar[u]&}" class="ltx_Math" display="inline" id="p2.m3">
<mrow>
<mrow>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\xymatrix
</mtext>
</merror>
<mo>
</mo>
<mi mathvariant="normal">
@
</mi>
<mo>
</mo>
<mi>
C
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
0
</mn>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi mathvariant="normal">
@
</mi>
<mo>
</mo>
<mi>
H
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<mrow>
<mrow>
<mrow>
<mn>
10
</mn>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<msub>
<mi>
A
</mi>
<mi>
n
</mi>
</msub>
<mo>
</mo>
<msub>
<mi>
A
</mi>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<mo>
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
</mo>
<msub>
<mi>
A
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<mi mathvariant="normal">
&
</mi>
</mrow>
<mo>
∙
</mo>
<mi mathvariant="normal">
&
</mi>
</mrow>
<mo>
</mo>
<msub>
<mi>
D
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<msub>
<mi>
D
</mi>
<mn>
2
</mn>
</msub>
<mo>
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
</mo>
<msub>
<mi>
D
</mi>
<mi>
m
</mi>
</msub>
<mo>
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
</mo>
<mi mathvariant="normal">
&
</mi>
<mo>
</mo>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\save
</mtext>
</merror>
</mrow>
<mo>
*
</mo>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\txt
</mtext>
</merror>
</mrow>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
m
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
n
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\restore
</mtext>
</merror>
<mo>
</mo>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\ar
</mtext>
</merror>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
u
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
<mo>
</mo>
<mi mathvariant="normal">
&
</mi>
</mrow>
</mrow>
</math>
</p>
<p class="ltx_p">
where each of the
<math alttext="A_{i}" class="ltx_Math" display="inline" id="p2.m4">
<msub>
<mi>
A
</mi>
<mi>
i
</mi>
</msub>
</math>
and
<math alttext="D_{j}" class="ltx_Math" display="inline" id="p2.m5">
<msub>
<mi>
D
</mi>
<mi>
j
</mi>
</msub>
</math>
is a digit in the decimal system (one of
<math alttext="0,1,2,3,4,5,6,7,8,9" class="ltx_Math" display="inline" id="p2.m6">
<mrow>
<mn>
0
</mn>
<mo>
,
</mo>
<mn>
1
</mn>
<mo>
,
</mo>
<mn>
2
</mn>
<mo>
,
</mo>
<mn>
3
</mn>
<mo>
,
</mo>
<mn>
4
</mn>
<mo>
,
</mo>
<mn>
5
</mn>
<mo>
,
</mo>
<mn>
6
</mn>
<mo>
,
</mo>
<mn>
7
</mn>
<mo>
,
</mo>
<mn>
8
</mn>
<mo>
,
</mo>
<mn>
9
</mn>
</mrow>
</math>
). In this decimal expansion,
</p>
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
the
<math alttext="i" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mi>
i
</mi>
</math>
-th decimal place to the left of the decimal point is the position of
<math alttext="A_{i}" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<msub>
<mi>
A
</mi>
<mi>
i
</mi>
</msub>
</math>
, and
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
the
<math alttext="j" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mi>
j
</mi>
</math>
-th decimal place to the right of the decimal place is the position of
<math alttext="D_{j}" class="ltx_Math" display="inline" id="I1.i2.p1.m2">
<msub>
<mi>
D
</mi>
<mi>
j
</mi>
</msub>
</math>
.
</p>
</div>
</li>
</ul>
<p class="ltx_p">
The digits
<math alttext="A_{i}" class="ltx_Math" display="inline" id="p2.m7">
<msub>
<mi>
A
</mi>
<mi>
i
</mi>
</msub>
</math>
and
<math alttext="D_{j}" class="ltx_Math" display="inline" id="p2.m8">
<msub>
<mi>
D
</mi>
<mi>
j
</mi>
</msub>
</math>
are called the
<em class="ltx_emph ltx_font_italic">
decimal place values
</em>
.
<math alttext="A_{i}" class="ltx_Math" display="inline" id="p2.m9">
<msub>
<mi>
A
</mi>
<mi>
i
</mi>
</msub>
</math>
is the decimal place value corresponding to the
<math alttext="i" class="ltx_Math" display="inline" id="p2.m10">
<mi>
i
</mi>
</math>
-th decimal place to the left of the decimal point, while
<math alttext="D_{j}" class="ltx_Math" display="inline" id="p2.m11">
<msub>
<mi>
D
</mi>
<mi>
j
</mi>
</msub>
</math>
is the decimal place value corresponding to the
<math alttext="j" class="ltx_Math" display="inline" id="p2.m12">
<mi>
j
</mi>
</math>
-th decimal place to the right of the decimal point.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
For decimal places closer to the decimal point, specific names are used. Below are some of the most commonly used names:
</p>
</div>
<div class="ltx_para ltx_centering" id="p4">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_l ltx_border_rr ltx_border_t">
name
</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t">
position from
</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t">
direction from
</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t">
example
</th>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_th ltx_th_column ltx_border_l ltx_border_rr">
</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r">
the decimal point
</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r">
the decimal point
</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r">
(position of 7)
</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_tt">
ones
</td>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_tt">
1st
</td>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_tt">
left
</td>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_tt">
<math alttext="7.2" class="ltx_Math" display="inline" id="p4.m1">
<mn>
7.2
</mn>
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tens
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2nd
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hundreds
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3rd
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thousands
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</td>
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</tbody>
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<p class="ltx_p">
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</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
m
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
e
</mi>
</mrow>
<mo>
,
</mo>
<mrow>
<mi>
w
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
e
</mi>
</mrow>
</mrow>
</math>
6
<math alttext="{{{islocatedonthenegativethirddecimalplace.\begin{flushright}\begin{tabular}[]{|%
ll|}\hline Title&decimal place\\
Canonical name&DecimalPlace\\
Date of creation&2013-03-22 17:27:24\\
Last modified on&2013-03-22 17:27:24\\
Owner&CWoo (3771)\\
Last modified by&CWoo (3771)\\
Numerical id&12\\
Author&CWoo (3771)\\
Entry type&Definition\\
Classification&msc 00A05\\
Related topic&MetricSystem\\
Defines&decimal place value\\
Defines&hundreds\\
Defines&thousands\\
Defines&tenths\\
Defines&hundredths\\
Defines&thousandths\\
Defines&ten thousandths\\
\hline}\end{tabular}}$}\end{flushright}\end{document}" class="ltx_Math" display="inline" id="p4.m17">
<mrow>
<mrow>
<mi>
i
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mi>
n
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
n
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
g
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
v
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
m
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
<mi>
l
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
e
</mi>
</mrow>
<mo>
.
</mo>
<mtable class="ltx_align_right" columnspacing="5pt" rowspacing="0pt">
<mtr>
<mtd class="ltx_border_l ltx_border_t" columnalign="left">
<mtext>
Title
</mtext>
</mtd>
<mtd class="ltx_border_r ltx_border_t" columnalign="left">
<mtext>
decimal place
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Canonical name
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
DecimalPlace
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Date of creation
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
2013-03-22 17:27:24
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Last modified on
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
2013-03-22 17:27:24
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Owner
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
CWoo (3771)
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Last modified by
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
CWoo (3771)
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Numerical id
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
12
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Author
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
CWoo (3771)
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Entry type
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
Definition
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Classification
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
msc 00A05
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Related topic
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
MetricSystem
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Defines
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
decimal place value
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Defines
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
hundreds
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Defines
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
thousands
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Defines
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
tenths
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Defines
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
hundredths
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l" columnalign="left">
<mtext>
Defines
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
thousandths
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_b ltx_border_l" columnalign="left">
<mtext>
Defines
</mtext>
</mtd>
<mtd class="ltx_border_b ltx_border_r" columnalign="left">
<mtext>
ten thousandths
</mtext>
</mtd>
</mtr>
</mtable>
</mrow>
</math>
</p>
</div>
</article>
</div>
<footer class="ltx_page_footer">
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|
0 | DecimalPoint | http://planetmath.org/DecimalPoint | <!DOCTYPE html>
<html>
<head>
<title>
decimal point
</title>
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
decimal point
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/decimalpoint">
decimal point
</a>
</span>
is a symbol separating those digits representing
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
powers of a base (usually base 10) on the left, and those representing fractional powers of a base (the base raised to a
<a class="nnexus_concept" href="http://planetmath.org/negativenumber">
negative number
</a>
) on the right. For example, in
<math alttext="\pi\approx 3.14" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mi>
π
</mi>
<mo>
≈
</mo>
<mn>
3.14
</mn>
</mrow>
</math>
, the 3 to the left of the decimal point corresponds to
<math alttext="3\times 10^{0}" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<mn>
3
</mn>
<mo>
×
</mo>
<msup>
<mn>
10
</mn>
<mn>
0
</mn>
</msup>
</mrow>
</math>
, while the 1 to the right of the decimal point corresponds to
<math alttext="1\times 10^{-1}" class="ltx_Math" display="inline" id="p1.m3">
<mrow>
<mn>
1
</mn>
<mo>
×
</mo>
<msup>
<mn>
10
</mn>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Most
<a class="nnexus_concept" href="http://planetmath.org/scientificcalculator">
scientific calculators
</a>
capable of displaying binary, octal and
<a class="nnexus_concept" href="http://planetmath.org/hexadecimal">
hexadecimal
</a>
limit numbers in those bases to integers, making moot the issue of what to call the decimal point in those bases.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
The decimal point is generally omitted for integers. However,
<a class="nnexus_concept" href="http://planetmath.org/mathematica">
Mathematica
</a>
will use the decimal point at the end of an integer to indicate the value has been computed using
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Floating-PointArithmetic.html">
floating-point arithmetic
</a>
and loss of precision is possible. For example,
<code class="ltx_verbatim ltx_font_typewriter">
(1/2)^(-1)
</code>
gives “2” as an answer but
<code class="ltx_verbatim ltx_font_typewriter">
.5^(-1)
</code>
gives “2.” for the answer. Even more pointedly,
<code class="ltx_verbatim ltx_font_typewriter">
1 + 1
</code>
gives “2” as the answer but
<code class="ltx_verbatim ltx_font_typewriter">
1. + 1.
</code>
gives “2.” as the answer.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
In the
<a class="nnexus_concept" href="http://planetmath.org/historyofmathematicsintheunitedstatesofamerica">
United States
</a>
, the decimal point is usually aligned with the bottom of the digit glyphs, while in the United Kingdom it is usually centered (and is distinguished from the central dot
<a class="nnexus_concept" href="http://planetmath.org/multiplicationoperator">
multiplication operator
</a>
purely on spacing). In Europe, a comma is used instead, so our example would be written
<math alttext="\pi\approx 3,\!14" class="ltx_Math" display="inline" id="p4.m1">
<mrow>
<mi>
π
</mi>
<mo>
≈
</mo>
<mrow>
<mn>
3
</mn>
<mo rspace="0.8pt">
,
</mo>
<mn>
14
</mn>
</mrow>
</mrow>
</math>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
decimal point
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
DecimalPoint
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:23:33
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:23:33
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A05
</td>
</tr>
</tbody>
</table>
</div>
</article>
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|
0 | Definition | http://planetmath.org/Definition | <!DOCTYPE html>
<html>
<head>
<title>
definition
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
definition
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
</span>
of a mathematical term is a meta-mathematical
<a class="nnexus_concept" href="http://planetmath.org/concretecategory">
construct
</a>
or statement that specifies as precisely as possible the meaning of that term. For example, a definition of “
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
sphenic number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SphenicNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sphenicnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
” is “a
<a class="nnexus_concept" href="http://planetmath.org/compositenumber">
composite
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
with three
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">
distinct prime factors
</a>
.” A mathematical
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
concept
</a>
is
<a class="nnexus_concept" href="http://planetmath.org/welldefined">
well-defined
</a>
if its content can be formulated independently of the form or the alternative representative which is used for defining it. Furthermore, one should distinguish between mathematical definitions and mathematical descriptions of a real system; mathematical definitions express completely and meaningfully a mathematical concept in terms of other, related mathematical concepts that have been already defined, and also
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Primary.html">
primary
</a>
or primitive concepts that can no longer be defined in terms of other mathematical concepts and logical operands. For example, the primitive concept of ‘
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
collection
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Collection.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/collection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Element.html">
elements
</a>
or members’ or ensemble, has no explicit definition even though it is employed to mathematically define the concept of set. A definition of a fundamental concept, such as set,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
category
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Category.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/category">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, topos,
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
topology
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
homology
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Homology.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homologyofachaincomplex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/collineation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
etc., also contains several axioms, or basic
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
assumptions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/deduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivationsinnaturaldeduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
/conditions imposed on the auxilliary concepts employed by such a fundamental concept definition. For example, the category of sheaves on a site is called a (Grothendieck) topos; however, a topos can also be defined directly by specifying only a few (
<a class="nnexus_concept" href="http://planetmath.org/topos">
Grothendieck topos
</a>
) axioms.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Therefore, ultimately, a mathematical definition depends on the choice of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
mathematical foundation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/bibliographyforaxiomaticsandmathematicsfoundationsincategories">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/analyticsandformallogicsinmetamathematics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
selected, e.g., set-theoretical, category-theoretical, or topos-theoretical, as well as the type of logic adopted, e.g.,
<a class="nnexus_concept" href="http://planetmath.org/boolean">
Boolean
</a>
, intuitionistic or
<a class="nnexus_concept" href="http://planetmath.org/bibliographyofmanyvaluedlogicsandapplications">
many-valued logic
</a>
. Thus, the general definition of a mathematical definition is not simply a mathematical concept, but it is instead a
<span class="ltx_text ltx_font_italic">
meta-mathematical construct
</span>
, or the
<math alttext="<construct>" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<mo>
<
</mo>
<mrow>
<mi>
c
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mi>
n
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
<mi>
u
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
<mo>
>
</mo>
</mrow>
</math>
of a construct. In the case of topos-theoretical
<a class="nnexus_concept" href="http://planetmath.org/axiomoffoundation">
foundations
</a>
the Brouwer-intuitionistic logic is explicitly assumed in the construction/definition of the topos. As an example, the
<a class="nnexus_concept" href="http://planetmath.org/categoryofsets">
category of sets
</a>
–subject to certain axioms, including the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
axiom of choice
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AxiomofChoice.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiomofchoice">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
– may be considered a
<a class="nnexus_concept" href="http://planetmath.org/canonical">
canonical
</a>
example of a Boolean topos, but it is not the only one possible, as different axioms may be selected to avoid several known antimonies in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
set theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SetTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/settheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Thus, a mathematical concept is well-defined only when its mathematical foundation
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Framework.html">
framework
</a>
is also specified either explicitly or by its context. For reasons related to apparent ‘simplicity’, many a mathematician prefers only Boolean logic and a set-theoretical foundation for definitions, in spite of severe limitations, known inherent
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
paradoxes
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Paradox.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/paradox">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
incompleteness
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Incompleteness.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/beyondformalismgodelsincompleteness">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Alternative definitions of the same concept often offer additional insights into the meaning(s) of the concept being defined, as well as added flexibility in solving problems and discovering proofs.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
A definition of a
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
constant
</a>
is an equation with a symbol (or some other notation) on the left and either an exact value or a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
formula
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Formula.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/firstorderlanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
for a value on the right. For example, the definition of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
golden ratio
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GoldenRatio.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/goldenratio">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\phi=\frac{1+\sqrt{5}}{2}." class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mi>
ϕ
</mi>
<mo>
=
</mo>
<mfrac>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<msqrt>
<mn>
5
</mn>
</msqrt>
</mrow>
<mn>
2
</mn>
</mfrac>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
A definition of a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is an equation, usually with function notation on the left (a symbol, with an
<a class="nnexus_concept" href="http://planetmath.org/argument">
argument
</a>
or a list of arguments in parentheses) and on the right a formula using the arguments to calculate the value of the function for those arguments. Optionally, the equation could be accompanied by statements of what acceptable arguments are. For example, Euler’s
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TotientFunction.html">
totient function
</a>
is defined as
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\phi(n)=n\prod_{p|n}\left(1-\frac{1}{p}\right)" class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
n
</mi>
<mo>
</mo>
<mrow>
<munder>
<mo largeop="true" movablelimits="false" symmetric="true">
∏
</mo>
<mrow>
<mi>
p
</mi>
<mo stretchy="false">
|
</mo>
<mi>
n
</mi>
</mrow>
</munder>
<mrow>
<mo>
(
</mo>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mfrac>
<mn>
1
</mn>
<mi>
p
</mi>
</mfrac>
</mrow>
<mo>
)
</mo>
</mrow>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
for
<math alttext="n\in\mathbb{Z}" class="ltx_Math" display="inline" id="p6.m1">
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℤ
</mi>
</mrow>
</math>
. To give one more example: the
<a class="nnexus_concept" href="http://planetmath.org/factorial">
factorial function
</a>
is defined as
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="n!=\prod_{i=1}^{n}i" class="ltx_Math" display="block" id="S0.Ex3.m1">
<mrow>
<mrow>
<mi>
n
</mi>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
<mo>
=
</mo>
<mrow>
<munderover>
<mo largeop="true" movablelimits="false" symmetric="true">
∏
</mo>
<mrow>
<mi>
i
</mi>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
<mi>
n
</mi>
</munderover>
<mi>
i
</mi>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
for
<math alttext="0<n\in\mathbb{Z}" class="ltx_Math" display="inline" id="p6.m2">
<mrow>
<mn>
0
</mn>
<mo>
<
</mo>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℤ
</mi>
</mrow>
</math>
. Euler’s
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
gamma function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/5#PT2">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://dlmf.nist.gov/5.2#E1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GammaFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/gammafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="\Gamma(x)" class="ltx_Math" display="inline" id="p6.m3">
<mrow>
<mi mathvariant="normal">
Γ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is sometimes said to extend the definition of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Factorial.html">
factorials
</a>
to any
<math alttext="x\in\mathbb{C}" class="ltx_Math" display="inline" id="p6.m4">
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
ℂ
</mi>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para ltx_align_right" id="p7">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
definition
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Canonical name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:32:19
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:32:19
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
18
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
mathematical description
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
meta-construct
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
SetTheory
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
CategoryTheory
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Related topic
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<td class="ltx_td ltx_align_left ltx_border_r">
Topos
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<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
AxiomOfChoice
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
mathematical definition
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | Division | http://planetmath.org/Division | <!DOCTYPE html>
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<title>
division
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
division
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/division">
Division
</a>
</span>
is the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
which assigns to every two numbers (or more generally, elements of a field)
<math alttext="a" class="ltx_Math" display="inline" id="p1.m1">
<mi>
a
</mi>
</math>
and
<math alttext="b" class="ltx_Math" display="inline" id="p1.m2">
<mi>
b
</mi>
</math>
their
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
quotient
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quotientoflanguages">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quotientgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
or ratio, provided that the latter,
<math alttext="b" class="ltx_Math" display="inline" id="p1.m3">
<mi>
b
</mi>
</math>
, is distinct from zero.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The
<span class="ltx_text ltx_font_italic">
quotient
</span>
(or
<span class="ltx_text ltx_font_italic">
ratio
</span>
)
<math alttext="\frac{a}{b}" class="ltx_Math" display="inline" id="p2.m1">
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
</math>
of
<math alttext="a" class="ltx_Math" display="inline" id="p2.m2">
<mi>
a
</mi>
</math>
and
<math alttext="b" class="ltx_Math" display="inline" id="p2.m3">
<mi>
b
</mi>
</math>
may be defined as such a number (or element of the field)
<math alttext="x" class="ltx_Math" display="inline" id="p2.m4">
<mi>
x
</mi>
</math>
that
<math alttext="b\cdot x=a" class="ltx_Math" display="inline" id="p2.m5">
<mrow>
<mrow>
<mi>
b
</mi>
<mo>
⋅
</mo>
<mi>
x
</mi>
</mrow>
<mo>
=
</mo>
<mi>
a
</mi>
</mrow>
</math>
. Thus,
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="b\cdot\frac{a}{b}=a," class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mrow>
<mi>
b
</mi>
<mo>
⋅
</mo>
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
</mrow>
<mo>
=
</mo>
<mi>
a
</mi>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
which is the “fundamental property of quotient”.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
The quotient of the numbers
<math alttext="a" class="ltx_Math" display="inline" id="p3.m1">
<mi>
a
</mi>
</math>
and
<math alttext="b" class="ltx_Math" display="inline" id="p3.m2">
<mi>
b
</mi>
</math>
(
<math alttext="\neq 0" class="ltx_Math" display="inline" id="p3.m3">
<mrow>
<mi>
</mi>
<mo>
≠
</mo>
<mn>
0
</mn>
</mrow>
</math>
) is a
uniquely determined number, since if one had
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{a}{b}=x\neq y=\frac{a}{b}," class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mrow>
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
<mo>
=
</mo>
<mi>
x
</mi>
<mo>
≠
</mo>
<mi>
y
</mi>
<mo>
=
</mo>
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
then we could write
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="b(x-y)=bx-by=a-a=0" class="ltx_Math" display="block" id="S0.Ex3.m1">
<mrow>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
-
</mo>
<mi>
y
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
-
</mo>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
a
</mi>
<mo>
-
</mo>
<mi>
a
</mi>
</mrow>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
from which the supposition
<math alttext="b\neq 0" class="ltx_Math" display="inline" id="p3.m4">
<mrow>
<mi>
b
</mi>
<mo>
≠
</mo>
<mn>
0
</mn>
</mrow>
</math>
would imply
<math alttext="x-y=0" class="ltx_Math" display="inline" id="p3.m5">
<mrow>
<mrow>
<mi>
x
</mi>
<mo>
-
</mo>
<mi>
y
</mi>
</mrow>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
, i.e.
<math alttext="x=y" class="ltx_Math" display="inline" id="p3.m6">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mi>
y
</mi>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
The explicit general
<a class="nnexus_concept" href="http://planetmath.org/expression">
expression
</a>
for
<math alttext="\frac{a}{b}" class="ltx_Math" display="inline" id="p4.m1">
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
</math>
is
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{a}{b}=b^{-1}\cdot a" class="ltx_Math" display="block" id="S0.Ex4.m1">
<mrow>
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
b
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
⋅
</mo>
<mi>
a
</mi>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
where
<math alttext="b^{-1}" class="ltx_Math" display="inline" id="p4.m2">
<msup>
<mi>
b
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
</math>
is the
<a class="nnexus_concept" href="http://planetmath.org/inversenumber">
inverse number
</a>
(the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
multiplicative inverse
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MultiplicativeInverse.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/ring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) of
<math alttext="a" class="ltx_Math" display="inline" id="p4.m3">
<mi>
a
</mi>
</math>
, because
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="b(b^{-1}a)=(bb^{-1})a=1a=a." class="ltx_Math" display="block" id="S0.Ex5.m1">
<mrow>
<mrow>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msup>
<mi>
b
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
</mo>
<mi>
a
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<msup>
<mi>
b
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mi>
a
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
1
</mn>
<mo>
</mo>
<mi>
a
</mi>
</mrow>
<mo>
=
</mo>
<mi>
a
</mi>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p5">
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
For positive numbers the quotient may be obtained by performing the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
division algorithm
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/longdivision">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divisionalgorithmforintegers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
with
<math alttext="a" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mi>
a
</mi>
</math>
and
<math alttext="b" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mi>
b
</mi>
</math>
. If
<math alttext="a>b>0" class="ltx_Math" display="inline" id="I1.i1.p1.m3">
<mrow>
<mi>
a
</mi>
<mo>
>
</mo>
<mi>
b
</mi>
<mo>
>
</mo>
<mn>
0
</mn>
</mrow>
</math>
, then
<math alttext="\frac{a}{b}" class="ltx_Math" display="inline" id="I1.i1.p1.m4">
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
</math>
indicates how many times
<math alttext="b" class="ltx_Math" display="inline" id="I1.i1.p1.m5">
<mi>
b
</mi>
</math>
fits in
<math alttext="a" class="ltx_Math" display="inline" id="I1.i1.p1.m6">
<mi>
a
</mi>
</math>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
The quotient of
<math alttext="a" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mi>
a
</mi>
</math>
and
<math alttext="b" class="ltx_Math" display="inline" id="I1.i2.p1.m2">
<mi>
b
</mi>
</math>
does not change if both numbers (
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Element.html">
elements
</a>
) are multiplied (or divided, which is called
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
reduction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/diamondlemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reducedword">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
) by any
<math alttext="k\neq 0" class="ltx_Math" display="inline" id="I1.i2.p1.m3">
<mrow>
<mi>
k
</mi>
<mo>
≠
</mo>
<mn>
0
</mn>
</mrow>
</math>
:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{ka}{kb}=(kb)^{-1}(ka)=b^{-1}k^{-1}ka=b^{-1}a=\frac{a}{b}" class="ltx_Math" display="block" id="S0.Ex6.m1">
<mrow>
<mfrac>
<mrow>
<mi>
k
</mi>
<mo>
</mo>
<mi>
a
</mi>
</mrow>
<mrow>
<mi>
k
</mi>
<mo>
</mo>
<mi>
b
</mi>
</mrow>
</mfrac>
<mo>
=
</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
k
</mi>
<mo>
</mo>
<mi>
b
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
k
</mi>
<mo>
</mo>
<mi>
a
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
b
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
</mo>
<msup>
<mi>
k
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
</mo>
<mi>
k
</mi>
<mo>
</mo>
<mi>
a
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
b
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
</mo>
<mi>
a
</mi>
</mrow>
<mo>
=
</mo>
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
So we have the method for getting the quotient of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complex numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complexnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{a}{b}=\frac{\bar{b}a}{\bar{b}b}," class="ltx_Math" display="block" id="S0.Ex7.m1">
<mrow>
<mrow>
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
<mo>
=
</mo>
<mfrac>
<mrow>
<mover accent="true">
<mi>
b
</mi>
<mo stretchy="false">
¯
</mo>
</mover>
<mo>
</mo>
<mi>
a
</mi>
</mrow>
<mrow>
<mover accent="true">
<mi>
b
</mi>
<mo stretchy="false">
¯
</mo>
</mover>
<mo>
</mo>
<mi>
b
</mi>
</mrow>
</mfrac>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
where
<math alttext="\bar{b}" class="ltx_Math" display="inline" id="I1.i2.p1.m4">
<mover accent="true">
<mi>
b
</mi>
<mo stretchy="false">
¯
</mo>
</mover>
</math>
is the
<a class="nnexus_concept" href="http://planetmath.org/complexconjugate">
complex conjugate
</a>
of
<math alttext="b" class="ltx_Math" display="inline" id="I1.i2.p1.m5">
<mi>
b
</mi>
</math>
, and the quotient of
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/SquareRootOfSquareRootBinomial
</span>
<a class="nnexus_concept" href="http://planetmath.org/squarerootofsquarerootbinomial">
square root polynomials
</a>
, e.g.
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{1}{5+2\sqrt{2}}=\frac{5-2\sqrt{2}}{(5-2\sqrt{2})(5+2\sqrt{2})}=\frac{5-2%
\sqrt{2}}{25-8}=\frac{5-2\sqrt{2}}{17};" class="ltx_Math" display="block" id="S0.Ex8.m1">
<mrow>
<mrow>
<mfrac>
<mn>
1
</mn>
<mrow>
<mn>
5
</mn>
<mo>
+
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msqrt>
<mn>
2
</mn>
</msqrt>
</mrow>
</mrow>
</mfrac>
<mo>
=
</mo>
<mfrac>
<mrow>
<mn>
5
</mn>
<mo>
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msqrt>
<mn>
2
</mn>
</msqrt>
</mrow>
</mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
5
</mn>
<mo>
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msqrt>
<mn>
2
</mn>
</msqrt>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
5
</mn>
<mo>
+
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msqrt>
<mn>
2
</mn>
</msqrt>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mfrac>
<mo>
=
</mo>
<mfrac>
<mrow>
<mn>
5
</mn>
<mo>
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msqrt>
<mn>
2
</mn>
</msqrt>
</mrow>
</mrow>
<mrow>
<mn>
25
</mn>
<mo>
-
</mo>
<mn>
8
</mn>
</mrow>
</mfrac>
<mo>
=
</mo>
<mfrac>
<mrow>
<mn>
5
</mn>
<mo>
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msqrt>
<mn>
2
</mn>
</msqrt>
</mrow>
</mrow>
<mn>
17
</mn>
</mfrac>
</mrow>
<mo>
;
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
in the first case one aspires after a real and in the second case after a
<a class="nnexus_concept" href="http://planetmath.org/rationalnumber">
rational
</a>
<a class="nnexus_concept" href="http://planetmath.org/fraction">
denominator
</a>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
The division is neither
<a class="nnexus_concept" href="http://planetmath.org/associative">
associative
</a>
nor
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
commutative
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/abeliangroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/commutativesemigroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/commutativelanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/commutative">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, but it is
<a class="nnexus_concept" href="http://planetmath.org/distributivity">
right distributive
</a>
over
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}" class="ltx_Math" display="block" id="S0.Ex9.m1">
<mrow>
<mfrac>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
<mi>
c
</mi>
</mfrac>
<mo>
=
</mo>
<mrow>
<mfrac>
<mi>
a
</mi>
<mi>
c
</mi>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mi>
b
</mi>
<mi>
c
</mi>
</mfrac>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
</li>
</ul>
</div>
<div class="ltx_para ltx_align_right" id="p6">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
division
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Division
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2014-08-08 17:51:29
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2014-08-08 17:51:29
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
29
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 12E99
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
InverseFormingInProportionToGroupOperation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
DivisionInGroup
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
ConjugationMnemonic
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Difference2
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
UniquenessOfDivisionAlgorithmInEuclideanDomain
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
quotient
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
ratio
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
fundamental property of quotient
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
reduction
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | FundamentalTheoremsInMathematics | http://planetmath.org/FundamentalTheoremsInMathematics | <!DOCTYPE html>
<html>
<head>
<title>
fundamental theorems in mathematics
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
fundamental theorems in mathematics
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
In different areas of mathematics, there are certain
<a class="nnexus_concept" href="http://planetmath.org/lemma">
theorems
</a>
called
<em class="ltx_emph ltx_font_italic">
fundamental
</em>
. We list a number of them below.
</p>
</div>
<div class="ltx_para" id="p2">
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
fundamental cohomology theorem
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental homomorphism theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalHomomorphismTheorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentalhomomorphismtheorem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofalgebra">
fundamental theorem of algebra
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental theorem of arithmetics
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofidealtheory">
fundamental theorem of ideal theory
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i6.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental theorem of calculus
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofcalculus">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremsofcalculusforlebesgueintegration">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i7.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofcalculusforkurzweilhenstockintegral">
fundamental theorem of calculus for Kurzweil-Henstock integral
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i8.p1">
<p class="ltx_p">
fundamental theorem of calculus for Lebesgue integration
</p>
</div>
</li>
<li class="ltx_item" id="I1.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i9.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofcalculusforriemannintegration">
fundamental theorem of calculus for Riemann integration
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i10.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/node/6745
</span>
fundamental theorem of calculus of variations
</p>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i11.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/cauchyintegraltheorem">
fundamental theorem of complex analysis
</a>
</p>
</div>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i12.p1">
<p class="ltx_p">
fundamental theorem of demography
</p>
</div>
</li>
<li class="ltx_item" id="I1.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i13.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremoffinitelygeneratedabeliangroups">
fundamental theorem of finitely generated Abelian groups
</a>
</p>
</div>
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<li class="ltx_item" id="I1.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i14.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental theorem of Galois theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalTheoremofGaloisTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
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</a>
</sup>
</p>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i15.p1">
<p class="ltx_p">
fundamental theorem of Gaussian quadrature
</p>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i16.p1">
<p class="ltx_p">
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fundamental theorem of integral calculus
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i17.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremonisogonallines">
fundamental theorem on isogonal lines
</a>
</p>
</div>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i18.p1">
<p class="ltx_p">
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</p>
</div>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i19.p1">
<p class="ltx_p">
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</p>
</div>
</li>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i20.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/node/12096
</span>
<a class="nnexus_concept" href="http://planetmath.org/gradienttheorem">
fundamental theorem of line integrals
</a>
</p>
</div>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i21.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental theorem of plane curves
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalTheoremofPlaneCurves.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
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</sup>
</p>
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<span class="ltx_tag ltx_tag_itemize">
•
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<div class="ltx_para" id="I1.i22.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental theorem of projective geometry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalTheoremofProjectiveGeometry.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
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<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
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<span class="ltx_tag ltx_tag_itemize">
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</a>
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</a>
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</sup>
</p>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i25.p1">
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</a>
</p>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i26.p1">
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<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
topology
</a>
</p>
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<span class="ltx_tag ltx_tag_itemize">
•
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fundamental theorem of topos theory
</p>
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<span class="ltx_tag ltx_tag_itemize">
•
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<div class="ltx_para" id="I1.i28.p1">
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fundamental theorem of transcendence
</a>
</p>
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<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i29.p1">
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</span>
fundamental theorem of vector calculus
</p>
</div>
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fundamental theorems in mathematics
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2013-03-22 19:11:13
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pahio (2872)
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11
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pahio (2872)
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0 | GrahamsNumber | http://planetmath.org/GrahamsNumber | <!DOCTYPE html>
<html>
<head>
<title>
Graham’s number
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
Graham’s number
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
Graham’s number
</span>
<math alttext="G" class="ltx_Math" display="inline" id="p1.m1">
<mi>
G
</mi>
</math>
is an
<a class="nnexus_concept" href="http://planetmath.org/upperbound">
upper bound
</a>
in a problem in
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RamseyTheory.html">
Ramsey theory
</a>
, first mentioned in a paper by
<a class="nnexus_concept" href="http://planetmath.org/ronaldgraham">
Ronald Graham
</a>
and B. Rothschild. The
<span class="ltx_text ltx_font_italic">
Guinness Book of World Records
</span>
calls it the largest number ever used in a mathematical proof. Graham’s number is too difficult to write in
<a class="nnexus_concept" href="http://planetmath.org/scientificnotation">
scientific notation
</a>
, so it is generally written using Knuth’s arrow-up notation. In Graham’s paper, the bound is given as
</p>
</div>
<div class="ltx_para" id="p2">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="6\leq N(2)\leq A(A(A(A(A(A(A(12,3),3),3),3),3),3),3)," class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mn>
6
</mn>
<mo>
≤
</mo>
<mrow>
<mi>
N
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
≤
</mo>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
12
</mn>
<mo>
,
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
,
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
,
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
,
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
,
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
,
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
,
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
where
<math alttext="N(2)" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mi>
N
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is the least
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
natural number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NaturalNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/naturalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
such that
<math alttext="n\geq N(2)" class="ltx_Math" display="inline" id="p3.m2">
<mrow>
<mi>
n
</mi>
<mo>
≥
</mo>
<mrow>
<mi>
N
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
implies that given any arbitrary
<math alttext="2" class="ltx_Math" display="inline" id="p3.m3">
<mn>
2
</mn>
</math>
-
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
coloring
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Coloring.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/coloring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the
<a class="nnexus_concept" href="http://planetmath.org/linesegment">
line segments
</a>
between pairs of vertices of an
<math alttext="n" class="ltx_Math" display="inline" id="p3.m4">
<mi>
n
</mi>
</math>
-dimensional box, there must exist a monochromatic
<a class="nnexus_concept" href="http://planetmath.org/specialelementsinarelationalgebra">
rectangle
</a>
in the box.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Here
<math alttext="A" class="ltx_Math" display="inline" id="p4.m1">
<mi>
A
</mi>
</math>
is the
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
defined by (this is a direct quote):
</p>
</div>
<div class="ltx_para" id="p5">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="A(1,n)=2^{n},A(m,2)=4,\ m\geq 1,\ n\geq 2,\\
A(m,n)=A(m-1,A(m,n-1)),\ m\geq 2,\ n\geq 3." class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mrow>
<mrow>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
1
</mn>
<mo>
,
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<msup>
<mn>
2
</mn>
<mi>
n
</mi>
</msup>
</mrow>
<mo>
,
</mo>
<mrow>
<mrow>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
m
</mi>
<mo>
,
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
4
</mn>
</mrow>
<mo rspace="7.5pt">
,
</mo>
<mrow>
<mrow>
<mi>
m
</mi>
<mo>
≥
</mo>
<mn>
1
</mn>
</mrow>
<mo rspace="7.5pt">
,
</mo>
<mrow>
<mrow>
<mi>
n
</mi>
<mo>
≥
</mo>
<mn>
2
</mn>
</mrow>
<mo>
,
</mo>
<mrow>
<mrow>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
m
</mi>
<mo>
,
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
m
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo>
,
</mo>
<mrow>
<mi>
A
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
m
</mi>
<mo>
,
</mo>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
<mo rspace="7.5pt">
,
</mo>
<mrow>
<mrow>
<mi>
m
</mi>
<mo>
≥
</mo>
<mn>
2
</mn>
</mrow>
<mo rspace="7.5pt">
,
</mo>
<mrow>
<mi>
n
</mi>
<mo>
≥
</mo>
<mn>
3
</mn>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
In the earlier paper Graham and Rothschild called this function
<math alttext="F" class="ltx_Math" display="inline" id="p6.m1">
<mi>
F
</mi>
</math>
instead of
<math alttext="A" class="ltx_Math" display="inline" id="p6.m2">
<mi>
A
</mi>
</math>
, and commented: “Clearly, there is some room for improvement here.”
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
In Knuth’s arrow-up notation, Graham’s number is still cumbersome to write: we define the
<a class="nnexus_concept" href="http://planetmath.org/recurrencerelation">
recurrence relation
</a>
<math alttext="g_{1}=3\uparrow\uparrow\uparrow\uparrow 3" class="ltx_Math" display="inline" id="p7.m1">
<mrow>
<msub>
<mi>
g
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<mn>
3
</mn>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
3
</mn>
</mrow>
</math>
and
<math alttext="g_{n}=3\uparrow^{g_{n-1}}3" class="ltx_Math" display="inline" id="p7.m2">
<mrow>
<msub>
<mi>
g
</mi>
<mi>
n
</mi>
</msub>
<mo>
=
</mo>
<mn>
3
</mn>
<msup>
<mo>
↑
</mo>
<msub>
<mi>
g
</mi>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msub>
</msup>
<mn>
3
</mn>
</mrow>
</math>
. Graham’s number is then
<math alttext="G=g_{64}" class="ltx_Math" display="inline" id="p7.m3">
<mrow>
<mi>
G
</mi>
<mo>
=
</mo>
<msub>
<mi>
g
</mi>
<mn>
64
</mn>
</msub>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
To help understand Graham’s number from the more familiar viewpoint of standard
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
exponentiation
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/wikipedia">
Wikipedia
</a>
offers the following chart:
</p>
</div>
<div class="ltx_para" id="p9">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="g_{1}=3\uparrow\uparrow\uparrow\uparrow 3=3\uparrow\uparrow\uparrow(3\uparrow%
\uparrow\uparrow 3)=3\uparrow\uparrow\uparrow\left(\begin{matrix}\underbrace{3%
^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}&\\
3^{3^{3}}&\text{threes}\end{matrix}\right)=\left.\begin{matrix}\underbrace{3^{%
3^{\cdot^{\cdot^{\cdot^{3}}}}}}&\\
\underbrace{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}&\text{threes}\\
\vdots&\vdots\\
\underbrace{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}&\text{threes}\\
3^{3^{3}}&\text{threes}\\
\end{matrix}\right\}\begin{matrix}&\\
\underbrace{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}&\mbox{ layers}\\
3^{3^{3}}&\text{ threes}\end{matrix}" class="ltx_Math" display="block" id="S0.Ex3.m1">
<mrow>
<msub>
<mi>
g
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<mn>
3
</mn>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
3
</mn>
<mo>
=
</mo>
<mn>
3
</mn>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
=
</mo>
<mn>
3
</mn>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mrow>
<mo>
(
</mo>
<mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt">
<mtr>
<mtd columnalign="center">
<munder accentunder="true">
<msup>
<mn>
3
</mn>
<msup>
<mn>
3
</mn>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<mn>
3
</mn>
</msup>
</msup>
</msup>
</msup>
</msup>
<mo>
⏟
</mo>
</munder>
</mtd>
<mtd columnalign="center">
<mi>
</mi>
</mtd>
</mtr>
<mtr>
<mtd columnalign="center">
<msup>
<mn>
3
</mn>
<msup>
<mn>
3
</mn>
<mn>
3
</mn>
</msup>
</msup>
</mtd>
<mtd columnalign="center">
<mtext>
threes
</mtext>
</mtd>
</mtr>
</mtable>
<mo>
)
</mo>
</mrow>
<mo>
=
</mo>
<mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt">
<mtr>
<mtd columnalign="center">
<munder accentunder="true">
<msup>
<mn>
3
</mn>
<msup>
<mn>
3
</mn>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<mn>
3
</mn>
</msup>
</msup>
</msup>
</msup>
</msup>
<mo>
⏟
</mo>
</munder>
</mtd>
<mtd columnalign="center">
<mi>
</mi>
</mtd>
</mtr>
<mtr>
<mtd columnalign="center">
<munder accentunder="true">
<msup>
<mn>
3
</mn>
<msup>
<mn>
3
</mn>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<mn>
3
</mn>
</msup>
</msup>
</msup>
</msup>
</msup>
<mo>
⏟
</mo>
</munder>
</mtd>
<mtd columnalign="center">
<mtext>
threes
</mtext>
</mtd>
</mtr>
<mtr>
<mtd columnalign="center">
<mi mathvariant="normal">
⋮
</mi>
</mtd>
<mtd columnalign="center">
<mi mathvariant="normal">
⋮
</mi>
</mtd>
</mtr>
<mtr>
<mtd columnalign="center">
<munder accentunder="true">
<msup>
<mn>
3
</mn>
<msup>
<mn>
3
</mn>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<mn>
3
</mn>
</msup>
</msup>
</msup>
</msup>
</msup>
<mo>
⏟
</mo>
</munder>
</mtd>
<mtd columnalign="center">
<mtext>
threes
</mtext>
</mtd>
</mtr>
<mtr>
<mtd columnalign="center">
<msup>
<mn>
3
</mn>
<msup>
<mn>
3
</mn>
<mn>
3
</mn>
</msup>
</msup>
</mtd>
<mtd columnalign="center">
<mtext>
threes
</mtext>
</mtd>
</mtr>
</mtable>
<mo>
}
</mo>
<mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt">
<mtr>
<mtd columnalign="center">
<mi>
</mi>
</mtd>
<mtd columnalign="center">
<mi>
</mi>
</mtd>
</mtr>
<mtr>
<mtd columnalign="center">
<munder accentunder="true">
<msup>
<mn>
3
</mn>
<msup>
<mn>
3
</mn>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<msup>
<mo>
⋅
</mo>
<mn>
3
</mn>
</msup>
</msup>
</msup>
</msup>
</msup>
<mo>
⏟
</mo>
</munder>
</mtd>
<mtd columnalign="center">
<mtext>
layers
</mtext>
</mtd>
</mtr>
<mtr>
<mtd columnalign="center">
<msup>
<mn>
3
</mn>
<msup>
<mn>
3
</mn>
<mn>
3
</mn>
</msup>
</msup>
</mtd>
<mtd columnalign="center">
<mtext>
threes
</mtext>
</mtd>
</mtr>
</mtable>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p10">
<p class="ltx_p">
We don’t know what the most significant base 10 digits of Graham’s number are, but we do know that the least significant digit is 7 (and of course 0 in base 3).
</p>
</div>
<div class="ltx_para" id="p11">
<p class="ltx_p">
Graham’s number has its own entry in Wells’s
<span class="ltx_text ltx_font_italic">
Dictionary of Curious and Interesting Numbers
</span>
, it is the very last entry, right after Skewes’ number, which it significantly dwarfs, and which was once also said to be the largest number ever used in a serious mathematical proof.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
M. P. Slone,
<a class="nnexus_concept" href="http://planetmath.org/planetmath">
PlanetMath
</a>
message, March 19, 2007.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
R. L. Graham and B. L. Rothschild, “Ramsey’s
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
for
<math alttext="n" class="ltx_Math" display="inline" id="bib.bib2.m1">
<mi>
n
</mi>
</math>
-
<a class="nnexus_concept" href="http://planetmath.org/parametre">
parameter
</a>
sets”,
<span class="ltx_text ltx_font_italic">
Trans. Amer. Math. Soc.
</span>
,
<span class="ltx_text ltx_font_bold">
159
</span>
(1971): 257 - 292
</span>
</li>
<li class="ltx_bibitem" id="bib.bib3">
<span class="ltx_bibtag ltx_role_refnum">
3
</span>
<span class="ltx_bibblock">
R. L. Graham and B. L. Rothschild. Ramsey theory.
<span class="ltx_text ltx_font_italic">
Studies in
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Combinatorics.html">
combinatorics
</a>
</span>
, ed. G.-C. Rota,
<a class="nnexus_concept" href="http://planetmath.org/mathematicalassociationofamerica">
Mathematical Association of America
</a>
, 1978.
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p12">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
Graham’s number
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
GrahamsNumber
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:15:15
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:15:15
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
4
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 05A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 68P30
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | Grapher | http://planetmath.org/Grapher | <!DOCTYPE html>
<html>
<head>
<title>
Grapher
</title>
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<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
Grapher
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/grapher">
Grapher
</a>
</span>
is a software graphing
<a class="nnexus_concept" href="http://planetmath.org/calculator">
calculator
</a>
that comes bundled with the Apple Mac OS Xoperating system. It can graph 2-dimensional equations like
<math alttext="y=\sin x" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mi>
y
</mi>
<mo>
=
</mo>
<mrow>
<mi>
sin
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
</math>
, as well as 3-dimensional equations like
<math alttext="z=\frac{y^{3}}{x^{2}+y^{2}}" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<mi>
z
</mi>
<mo>
=
</mo>
<mfrac>
<msup>
<mi>
y
</mi>
<mn>
3
</mn>
</msup>
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mfrac>
</mrow>
</math>
(Cartan’s umbrella, illustrated below).
</p>
</div>
<div class="ltx_para ltx_centering" id="p2">
<img alt="" class="ltx_graphics" id="p2.g1" src="C:TempGrapherScreenShot"/>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Equations to be graphed can be entered in Cartesian form or in
<a class="nnexus_concept" href="http://planetmath.org/parametre">
parametric form
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
polar coordinates
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PolarCoordinates.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polarcoordinates">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, logarithmic, etc. The program automatically “typesets” the formulas as they are entered, for example, converting
<code class="ltx_verbatim ltx_font_typewriter">
x^2
</code>
to
<math alttext="x^{2}" class="ltx_Math" display="inline" id="p3.m1">
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</math>
or
<code class="ltx_verbatim ltx_font_typewriter">
1/x
</code>
to
<math alttext="\frac{1}{x}" class="ltx_Math" display="inline" id="p3.m2">
<mfrac>
<mn>
1
</mn>
<mi>
x
</mi>
</mfrac>
</math>
. The program can create animations, and one can also move a graph or rotate it (just by dragging it, for 3D graphs; for 2D graphs one must first select the Move Tool).
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
In a default installation, the program is in the Utilities subfolder of the Applications folder. Mac OS 9 had a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
similar
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Similar.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/similarityingeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
program, called Graphing Calculator, which was also capable of graphing 2D and 3D equations as well as animations.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Maria Langer,
<span class="ltx_text ltx_font_italic">
Visual Quickstart Guide: Mac OS 9.1
</span>
. New York: Peachpit Press (2001): 107
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
David Pogue & Adam Goldstein,
<span class="ltx_text ltx_font_italic">
The Missing Manual: Switching to the Mac, Tiger Edition
</span>
. Sebastopol: O’Reilly Media, Inc. (2005): 452 - 453
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p5">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
Grapher
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Grapher
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:46:13
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:46:13
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
4
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 01A07
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | Hypothesis | http://planetmath.org/Hypothesis | <!DOCTYPE html>
<html>
<head>
<title>
hypothesis
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
hypothesis
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
In mathematics, a
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
hypothesis
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Hypothesis.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/hypothesis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/deduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
is an unproven statement which is supported by all the available data and by many weaker results. An unproven mathematical statement is usually called a “
<a class="nnexus_concept" href="http://planetmath.org/openquestion">
conjecture
</a>
”, and while experimentation can sometimes produce millions of examples to
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Support.html">
support
</a>
a conjecture, usually nothing short of a proof can convince experts in the field. But when a conjecture is supported not only but all the available data but also by numerous weaker results, it is upgraded in label to a hypothesis. The most famous conjecture in mathematics is the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Riemann hypothesis
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RiemannHypothesis.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/riemannzetafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, which despite many attempts at a proof, is supported by many related results. The
<a class="nnexus_concept" href="http://planetmath.org/convexityconjecture">
convexity conjecture
</a>
, on the other hand, is considered “incompatible” with the
<math alttext="n" class="ltx_Math" display="inline" id="p2.m1">
<mi>
n
</mi>
</math>
-tuples conjecture and more results appear to support the latter, thus neither is upgraded to hypothesis.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
R. Crandall & C. Pomerance,
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Prime Numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimeNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/prime">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
: A Computational Perspective
</span>
, Springer, NY, 2001: 1.2.4
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p3">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
hypothesis
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Hypothesis
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:15:18
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:15:18
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A05
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | Indeterminate | http://planetmath.org/Indeterminate | <!DOCTYPE html>
<html>
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<title>
indeterminate
</title>
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<h1 class="ltx_title ltx_title_document">
indeterminate
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
An
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
indeterminate
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Indeterminate.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/indeterminate">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
is simply a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
variable
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Variable.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/variable">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
that is not known or solvable. It is usually denoted by a mathematical alphabet (
<math alttext="x" class="ltx_Math" display="inline" id="p2.m1">
<mi>
x
</mi>
</math>
,
<math alttext="y" class="ltx_Math" display="inline" id="p2.m2">
<mi>
y
</mi>
</math>
,
<math alttext="z" class="ltx_Math" display="inline" id="p2.m3">
<mi>
z
</mi>
</math>
, or
<math alttext="\alpha" class="ltx_Math" display="inline" id="p2.m4">
<mi>
α
</mi>
</math>
,
<math alttext="\beta" class="ltx_Math" display="inline" id="p2.m5">
<mi>
β
</mi>
</math>
, etc…). It is important to distinguish between a variable and an indeterminate in that a variable is solvable, at least conditionally. To make this more precise, let’s see two examples:
</p>
</div>
<div class="ltx_para" id="p3">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
Let
<math alttext="x" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mi>
x
</mi>
</math>
be a variable such that
<math alttext="2+3x=a+bx" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
+
</mo>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
</mrow>
</math>
, where
<math alttext="a,b\in\mathbb{Q}" class="ltx_Math" display="inline" id="I1.i1.p1.m3">
<mrow>
<mrow>
<mi>
a
</mi>
<mo>
,
</mo>
<mi>
b
</mi>
</mrow>
<mo>
∈
</mo>
<mi>
ℚ
</mi>
</mrow>
</math>
. Then
<math alttext="x=(a-2)/(3-b)" class="ltx_Math" display="inline" id="I1.i1.p1.m4">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
a
</mi>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
/
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
3
</mn>
<mo>
-
</mo>
<mi>
b
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
. Here
<math alttext="x" class="ltx_Math" display="inline" id="I1.i1.p1.m5">
<mi>
x
</mi>
</math>
is solvable conditioned on the equation given. Any values of
<math alttext="a" class="ltx_Math" display="inline" id="I1.i1.p1.m6">
<mi>
a
</mi>
</math>
and
<math alttext="b\,(\neq 3)" class="ltx_Math" display="inline" id="I1.i1.p1.m7">
<mrow>
<mpadded width="+1.7pt">
<mi>
b
</mi>
</mpadded>
<mspace width="veryverythickmathspace">
</mspace>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
</mi>
<mo>
≠
</mo>
<mn>
3
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
will yield a value for
<math alttext="x" class="ltx_Math" display="inline" id="I1.i1.p1.m8">
<mi>
x
</mi>
</math>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
Let
<math alttext="x" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mi>
x
</mi>
</math>
be an indeterminate such that
<math alttext="2+3x=a+bx" class="ltx_Math" display="inline" id="I1.i2.p1.m2">
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
+
</mo>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
</mrow>
</math>
, where
<math alttext="a,\,b\in\mathbb{Q}" class="ltx_Math" display="inline" id="I1.i2.p1.m3">
<mrow>
<mrow>
<mi>
a
</mi>
<mo rspace="4.2pt">
,
</mo>
<mi>
b
</mi>
</mrow>
<mo>
∈
</mo>
<mi>
ℚ
</mi>
</mrow>
</math>
. Since
<math alttext="x" class="ltx_Math" display="inline" id="I1.i2.p1.m4">
<mi>
x
</mi>
</math>
can not be solved, we have
<math alttext="2=a" class="ltx_Math" display="inline" id="I1.i2.p1.m5">
<mrow>
<mn>
2
</mn>
<mo>
=
</mo>
<mi>
a
</mi>
</mrow>
</math>
and
<math alttext="3=b" class="ltx_Math" display="inline" id="I1.i2.p1.m6">
<mrow>
<mn>
3
</mn>
<mo>
=
</mo>
<mi>
b
</mi>
</mrow>
</math>
. Note that if
<math alttext="a" class="ltx_Math" display="inline" id="I1.i2.p1.m7">
<mi>
a
</mi>
</math>
and
<math alttext="b" class="ltx_Math" display="inline" id="I1.i2.p1.m8">
<mi>
b
</mi>
</math>
are previously assigned to be values other than 2 and 3 respectively, then
<math alttext="x" class="ltx_Math" display="inline" id="I1.i2.p1.m9">
<mi>
x
</mi>
</math>
is no longer an indeterminate.
</p>
</div>
</li>
</ol>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
indeterminate
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Indeterminate
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:47:33
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:47:33
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/parametre">
Parameter
</a>
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | IntegralSign | http://planetmath.org/IntegralSign | <!DOCTYPE html>
<html>
<head>
<title>
integral sign
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
integral sign
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The
<span class="ltx_text ltx_font_italic">
integral sign
</span>
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\int" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mo largeop="true" symmetric="true">
∫
</mo>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
is a stylised version of the
<span class="ltx_text ltx_font_italic">
long s
</span>
letter.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The long s is a typographic variant of lowercase s, being the only lowercase s in the Carolingian minuscule script. The
modern short (round) s appeared later to the ends of words, and has now replaced completely the long s in the antiqua script.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Gottfried Wilhelm Leibniz introduced the integral sign as the first letter s of the Latin word
<span class="ltx_text ltx_font_italic">
summa
</span>
(‘sum’). The long shape of
<math alttext="\displaystyle\int" class="ltx_Math" display="inline" id="p3.m1">
<mstyle displaystyle="true">
<mo largeop="true" symmetric="true">
∫
</mo>
</mstyle>
</math>
may be thought to symbolically depict the fact that
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/DefiniteIntegral
</span>
integral is a limiting case of sum.
<br class="ltx_break"/>
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
A variant
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\oint" class="ltx_Math" display="block" id="S0.Ex2.m1">
<mo largeop="true" symmetric="true">
∮
</mo>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
of the integral sign is used in integrals taken along a
<a class="nnexus_concept" href="http://planetmath.org/curve">
closed curve
</a>
in
<math alttext="\mathbb{R}^{2}" class="ltx_Math" display="inline" id="p4.m1">
<msup>
<mi>
ℝ
</mi>
<mn>
2
</mn>
</msup>
</math>
or about a closed surface in
<math alttext="\mathbb{R}^{3}" class="ltx_Math" display="inline" id="p4.m2">
<msup>
<mi>
ℝ
</mi>
<mn>
3
</mn>
</msup>
</math>
; see e.g.
<a class="nnexus_concept" href="http://planetmath.org/cauchyintegraltheorem">
Cauchy integral theorem
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/derivationofheatequation">
derivation of heat equation
</a>
.
<br class="ltx_break"/>
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
The
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
given after the integral sign, i.e. the function to be
<span class="ltx_text ltx_font_italic">
integrated
</span>
, is the
<span class="ltx_text ltx_font_italic">
integrand
</span>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
integral sign
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Canonical name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
IntegralSign
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:04:00
</td>
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<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:04:00
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A06
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
RiemannIntegral
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
RiemannStieltjesIntegral
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Integral2
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
integrand
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
integrate
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | Introducing0thPower | http://planetmath.org/Introducing0thPower | <!DOCTYPE html>
<html>
<head>
<title>
introducing 0th power
</title>
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
introducing 0th power
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Let
<math alttext="a" class="ltx_Math" display="inline" id="p1.m1">
<mi>
a
</mi>
</math>
be a number not equal to zero. Then for all
<math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</math>
, we have that
<math alttext="a^{n}" class="ltx_Math" display="inline" id="p1.m3">
<msup>
<mi>
a
</mi>
<mi>
n
</mi>
</msup>
</math>
is the product of
<math alttext="a" class="ltx_Math" display="inline" id="p1.m4">
<mi>
a
</mi>
</math>
with itself
<math alttext="n" class="ltx_Math" display="inline" id="p1.m5">
<mi>
n
</mi>
</math>
. Using the fact that the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
1 is a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
multiplicative identity
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/ring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/unity">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, (
<math alttext="a\cdot 1=a" class="ltx_Math" display="inline" id="p1.m6">
<mrow>
<mrow>
<mi>
a
</mi>
<mo>
⋅
</mo>
<mn>
1
</mn>
</mrow>
<mo>
=
</mo>
<mi>
a
</mi>
</mrow>
</math>
for any
<math alttext="a" class="ltx_Math" display="inline" id="p1.m7">
<mi>
a
</mi>
</math>
), we can write
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="a^{n}\cdot 1=a^{n}=a^{n+0}=a^{n}\cdot a^{0}," class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mrow>
<msup>
<mi>
a
</mi>
<mi>
n
</mi>
</msup>
<mo>
⋅
</mo>
<mn>
1
</mn>
</mrow>
<mo>
=
</mo>
<msup>
<mi>
a
</mi>
<mi>
n
</mi>
</msup>
<mo>
=
</mo>
<msup>
<mi>
a
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
0
</mn>
</mrow>
</msup>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
a
</mi>
<mi>
n
</mi>
</msup>
<mo>
⋅
</mo>
<msup>
<mi>
a
</mi>
<mn>
0
</mn>
</msup>
</mrow>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
where we have used the
<a class="nnexus_concept" href="http://planetmath.org/property">
properties
</a>
of
<a class="nnexus_concept" href="http://planetmath.org/exponent">
exponents
</a>
under
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
multiplication
</a>
. Now, after canceling a factor of
<math alttext="a^{n}" class="ltx_Math" display="inline" id="p1.m8">
<msup>
<mi>
a
</mi>
<mi>
n
</mi>
</msup>
</math>
from both sides of the above
<a class="nnexus_concept" href="http://planetmath.org/equation">
equation
</a>
, we derive that
<math alttext="a^{0}=1" class="ltx_Math" display="inline" id="p1.m9">
<mrow>
<msup>
<mi>
a
</mi>
<mn>
0
</mn>
</msup>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
for any non-zero number.
</p>
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<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/introducing0thpower">
introducing 0th power
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Introducing0thPower
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:24:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:24:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
8
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
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<td class="ltx_td ltx_align_left ltx_border_r">
Topic
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
EmptyProduct
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | KnuthsUpArrowNotation | http://planetmath.org/KnuthsUpArrowNotation | <!DOCTYPE html>
<html>
<head>
<title>
Knuth’s up arrow notation
</title>
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<div class="ltx_page_main">
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
Knuth’s up arrow notation
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<em class="ltx_emph ltx_font_italic">
Knuth’s up arrow noation
</em>
is a way of writing numbers which would be unwieldy in standard decimal notation. It expands on the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
exponential
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/definitionsintrigonometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/exponential">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
notation
<math alttext="m\uparrow n=m^{n}" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mi>
m
</mi>
<mo>
↑
</mo>
<mi>
n
</mi>
<mo>
=
</mo>
<msup>
<mi>
m
</mi>
<mi>
n
</mi>
</msup>
</mrow>
</math>
. Define
<math alttext="m\uparrow\uparrow 0=1" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<mi>
m
</mi>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
0
</mn>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
and
<math alttext="m\uparrow\uparrow n=m\uparrow(m\uparrow\uparrow[n-1])" class="ltx_Math" display="inline" id="p1.m3">
<mrow>
<mi>
m
</mi>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mi>
n
</mi>
<mo>
=
</mo>
<mi>
m
</mi>
<mo>
↑
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
m
</mi>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Obviously
<math alttext="m\uparrow\uparrow 1=m^{1}=m" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<mi>
m
</mi>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
1
</mn>
<mo>
=
</mo>
<msup>
<mi>
m
</mi>
<mn>
1
</mn>
</msup>
<mo>
=
</mo>
<mi>
m
</mi>
</mrow>
</math>
, so
<math alttext="3\uparrow\uparrow 2=3^{3\uparrow\uparrow 1}=3^{3}=27" class="ltx_Math" display="inline" id="p2.m2">
<mrow>
<mn>
3
</mn>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
2
</mn>
<mo>
=
</mo>
<msup>
<mn>
3
</mn>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mo>
↑
</mo>
<mo>
</mo>
<mrow>
<mi>
</mi>
<mo>
↑
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
</msup>
<mo>
=
</mo>
<msup>
<mn>
3
</mn>
<mn>
3
</mn>
</msup>
<mo>
=
</mo>
<mn>
27
</mn>
</mrow>
</math>
, but
<math alttext="2\uparrow\uparrow 3=2^{2\uparrow\uparrow 2}=2^{2^{2\uparrow\uparrow 1}}=2^{(2^%
{2})}=16" class="ltx_Math" display="inline" id="p2.m3">
<mrow>
<mn>
2
</mn>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
3
</mn>
<mo>
=
</mo>
<msup>
<mn>
2
</mn>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mo>
↑
</mo>
<mo>
</mo>
<mrow>
<mi>
</mi>
<mo>
↑
</mo>
<mn>
2
</mn>
</mrow>
</mrow>
</msup>
<mo>
=
</mo>
<msup>
<mn>
2
</mn>
<msup>
<mn>
2
</mn>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mo>
↑
</mo>
<mo>
</mo>
<mrow>
<mi>
</mi>
<mo>
↑
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
</msup>
</msup>
<mo>
=
</mo>
<msup>
<mn>
2
</mn>
<mrow>
<mo stretchy="false">
(
</mo>
<msup>
<mn>
2
</mn>
<mn>
2
</mn>
</msup>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<mo>
=
</mo>
<mn>
16
</mn>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
In general,
<math alttext="m\uparrow\uparrow n=m^{m^{\cdots^{m}}}" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mi>
m
</mi>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mi>
n
</mi>
<mo>
=
</mo>
<msup>
<mi>
m
</mi>
<msup>
<mi>
m
</mi>
<msup>
<mi mathvariant="normal">
⋯
</mi>
<mi>
m
</mi>
</msup>
</msup>
</msup>
</mrow>
</math>
, a tower of height
<math alttext="n" class="ltx_Math" display="inline" id="p3.m2">
<mi>
n
</mi>
</math>
.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Clearly, this process can be extended:
<math alttext="m\uparrow\uparrow\uparrow 0=1" class="ltx_Math" display="inline" id="p4.m1">
<mrow>
<mi>
m
</mi>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
0
</mn>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
and
<math alttext="m\uparrow\uparrow\uparrow n=m\uparrow\uparrow(m\uparrow\uparrow\uparrow[n-1])" class="ltx_Math" display="inline" id="p4.m2">
<mrow>
<mi>
m
</mi>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mi>
n
</mi>
<mo>
=
</mo>
<mi>
m
</mi>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
m
</mi>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
An alternate notation is to write
<math alttext="m^{(i)}n" class="ltx_Math" display="inline" id="p5.m1">
<mrow>
<msup>
<mi>
m
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
i
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
</math>
for
<math alttext="m\underbrace{\uparrow\cdots\uparrow}_{i-2\text{~{}times}}n" class="ltx_Math" display="inline" id="p5.m2">
<mrow>
<mi>
m
</mi>
<mo>
</mo>
<munder>
<munder accentunder="true">
<mrow>
<mi>
</mi>
<mo movablelimits="false">
↑
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo movablelimits="false">
↑
</mo>
<mi>
</mi>
</mrow>
<mo movablelimits="false">
⏟
</mo>
</munder>
<mrow>
<mi>
i
</mi>
<mo>
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mtext>
times
</mtext>
</mrow>
</mrow>
</munder>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
</math>
. (
<math alttext="i-2" class="ltx_Math" display="inline" id="p5.m3">
<mrow>
<mi>
i
</mi>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
</math>
times because then
<math alttext="m^{(2)}n=m\cdot n" class="ltx_Math" display="inline" id="p5.m4">
<mrow>
<mrow>
<msup>
<mi>
m
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
m
</mi>
<mo>
⋅
</mo>
<mi>
n
</mi>
</mrow>
</mrow>
</math>
and
<math alttext="m^{(1)}n=m+n" class="ltx_Math" display="inline" id="p5.m5">
<mrow>
<mrow>
<msup>
<mi>
m
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
1
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
m
</mi>
<mo>
+
</mo>
<mi>
n
</mi>
</mrow>
</mrow>
</math>
.) Then in general we can define
<math alttext="m^{(i)}n=m^{(i-1)}(m^{(i)}(n-1))" class="ltx_Math" display="inline" id="p5.m6">
<mrow>
<mrow>
<msup>
<mi>
m
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
i
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
m
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
i
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msup>
<mi>
m
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
i
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
To get a sense of how quickly these numbers grow,
<math alttext="3\uparrow\uparrow\uparrow 2=3\uparrow\uparrow 3" class="ltx_Math" display="inline" id="p6.m1">
<mrow>
<mn>
3
</mn>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
2
</mn>
<mo>
=
</mo>
<mn>
3
</mn>
<mo>
↑
</mo>
<mo>
↑
</mo>
<mn>
3
</mn>
</mrow>
</math>
is more than seven and a half trillion, and the numbers continue to grow much more than exponentially.
</p>
<table class="ltx_tabular ltx_align_right ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
Knuth’s up arrow notation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
KnuthsUpArrowNotation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:58:43
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:58:43
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/knuthsuparrownotation">
up-arrow
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
up arrow
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
up-arrow notation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
up arrow notation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Knuth notation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
ConwaysChainedArrowNotation
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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<div class="ltx_page_logo">
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|
0 | Lemma | http://planetmath.org/Lemma | <!DOCTYPE html>
<html>
<head>
<title>
lemma
</title>
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<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
lemma
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
There is no technical distinction a lemma, a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
proposition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/booleanvaluedfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/propositionallogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, and a
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
theorem
</a>
. A
<em class="ltx_emph ltx_font_italic">
lemma
</em>
is a proven statement, typically named a lemma to distinguish it as a truth used as a stepping stone to a larger result rather than an important statement in and of itself. Of course, some of the most powerful statements in mathematics are known as lemmas, including Zorn’s Lemma, Bezout’s Lemma, Gauss’ Lemma, Fatou’s lemma, etc., so one clearly can’t get too much simply by reading into a proposition’s name.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Even less
<a class="nnexus_concept" href="http://planetmath.org/welldefined">
well-defined
</a>
is the distinction between a proposition and a theorem. Many authors choose to name results only one or the other, or use both more or less interchangeably. A partially standard set of nomenclature is to use the
<em class="ltx_emph ltx_font_italic">
proposition
</em>
to denote a significant result that is still shy of deserving a proper name. In contrast, a
<em class="ltx_emph ltx_font_italic">
theorem
</em>
under this format would a major result, and would often be named in to mathematicians who worked on or solved the problem in question.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
The Greek word “lemma” itself means “anything which is received, such as a gift, profit, or a bribe.” According to
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
]
</cite>
, the plural ’Lemmas’ is commonly used. The correct Greek plural of lemma, however, is lemmata. The Greek “Theoria” means “view, or vision” and is clearly linguistically related to the word “theatre.” The apparent
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
relation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is that a theorem is a mathematical fact which you see to be true (and can now show others!).
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
A somewhat more distinct
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
concept
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Concept.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(though still subject to author discretion) is that of a
<em class="ltx_emph ltx_font_italic">
corollary
</em>
, which is a result that can be considered an immediate
<a class="nnexus_concept" href="http://planetmath.org/logicalimplication">
consequence
</a>
of a previous theorem (typically, the preceding theorem in the text).
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
Finally, it is worth observing that the above terms are occasionally used to refer to a statement of the prescribed form without reference to the actual truth of the result, e.g., as in the phrase “While the theorem itself is valid, the is actually false.” See this
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/ConverseTheorem
</span>
attached entry for more on this last example.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.
(pp. 16)
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p7">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
lemma
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Lemma
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:46:56
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:46:56
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
19
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
proposition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
theorem
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
corollary
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | MacOSCalculator | http://planetmath.org/MacOSCalculator | <!DOCTYPE html>
<html>
<head>
<title>
Mac OS Calculator
</title>
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<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
Mac OS Calculator
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/macoscalculator">
Mac OS Calculator
</a>
</span>
is a software calculator that comes bundled with the Apple Mac OS operating system. As late as Mac OS 9, the
<a class="nnexus_concept" href="http://planetmath.org/calculator">
Calculator
</a>
was a very basic calculator with only the
<a class="nnexus_concept" href="http://planetmath.org/operation">
arithmetic operations
</a>
. In Mac OS X, the Calculator program was upgraded, with not just the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of Scientific mode (a
<a class="nnexus_concept" href="http://planetmath.org/scientificcalculator">
scientific calculator
</a>
) but also Programmer mode with
<a class="nnexus_concept" href="http://planetmath.org/function">
functions
</a>
useful for
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computer
</a>
programmers, such as Unicode character lookup by numerical value. The currently displayed value is not lost at a change of mode.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Like the
<a class="nnexus_concept" href="http://planetmath.org/windowscalculator">
Windows Calculator
</a>
, for the Mac OS Calculator
<math alttext="0^{0}=1" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<msup>
<mn>
0
</mn>
<mn>
0
</mn>
</msup>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Maria Langer,
<span class="ltx_text ltx_font_italic">
Mac OS 8: Visual Quickstart Guide
</span>
Berkeley: Peachpit Press (1997): 108
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p3">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
Mac OS Calculator
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
MacOSCalculator
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:39:26
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:39:26
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 01A07
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mac OS 9 Calculator
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
Mac OS X Calculator
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | MetricSystem | http://planetmath.org/MetricSystem | <!DOCTYPE html>
<html>
<head>
<title>
metric system
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
metric system
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/metricsystem">
metric system
</a>
</span>
is a system of weights and measures first proposed in France and gradually coming closer to worldwide acceptance in which for each given
<a class="nnexus_concept" href="http://planetmath.org/dimension">
dimension
</a>
, each larger unit is ten times the smaller unit (or viceversa, each smaller unit is a tenth of the larger unit). This makes it easier and more convenient to convert larger units to smaller units and viceversa. In ancient systems prior to the metric system, units for a given dimension often related to one another by different scaling factors. For example, in Biblical times, a talent was 60 mina, a mina was 60 shekels and a shekel was 24 giru. To convert 17 talents to shekels was thus different from converting shekels to giru or talents to giru, etc. By contrast, to convert 17.29 kilograms to centigrams is a simple matter of moving the
<a class="nnexus_concept" href="http://planetmath.org/decimalpoint">
decimal point
</a>
to the right (and adding significant zeroes as necessary), and the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
difference
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/difference">
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</a>
</sup>
in converting kilograms to milligrams or hectograms entails only changes in where to move the decimal point and how far. Furthermore, the basic units are determined by measurements which are known to remain
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
constant
</a>
and double-checkable throughout the planet, and not ephemeral and hard-to-verify measurements (such as the shoe size of the current king). The system is further standardized by the use of
<a class="nnexus_concept" href="http://planetmath.org/consistent">
consistent
</a>
prefixes to add to the basic units to make them larger or smaller.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
The basic units are:
</p>
</div>
<div class="ltx_para" id="p4">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r">
<span class="ltx_text ltx_font_typewriter">
http://
<a class="nnexus_concept" href="http://planetmath.org/planetmath">
planetmath
</a>
.org/BasicLength
</span>
Length
</th>
<th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_r">
Meter
</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Mass
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Gram
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Time
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Second
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Temperature
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Kelvin
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
The standard prefixes are:
</p>
</div>
<div class="ltx_para" id="p6">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Kilo
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
1000.0000
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Hecto
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
100.0000
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Deca
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
10.0000
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
(none)
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
1.0000
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Deci
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
0.1000
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Centi
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
0.0100
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Milli
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
0.0010
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r">
Micro
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
0.0001
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
Measurement systems consistently based on 10 (rather than arbitrarily varying practical numbers like 12, 24 and 60) had been proposed since at least the 15th Century. It wasn’t until after the French Revolution that the proposals were taken seriously. Nowadays, the metric system has been adopted by scientists all over the world, but the general population of the
<a class="nnexus_concept" href="http://planetmath.org/historyofmathematicsintheunitedstatesofamerica">
United States
</a>
remains an important group of hold-outs.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
metric system
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
MetricSystem
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:29:37
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:29:37
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
DecimalPlace
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
<footer class="ltx_page_footer">
<div class="ltx_page_logo">
Generated on Tue Feb 6 22:18:22 2018 by
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|
0 | MockupOfABasicCalculator | http://planetmath.org/MockupOfABasicCalculator | <!DOCTYPE html>
<html>
<head>
<title>
mock-up of a basic calculator
</title>
<!--Generated on Tue Feb 6 22:18:24 2018 by LaTeXML (version 0.8.2) http://dlmf.nist.gov/LaTeXML/.-->
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<body>
<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
mock-up of a basic calculator
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
This mock-up of a basic
<a class="nnexus_concept" href="http://planetmath.org/calculator">
calculator
</a>
is realistic in that it has almost every
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
one can expect on a typical basic calculator. The layout will of course be different on an actual calculator. A calculator shaped like, say, a credit card will have more columns of buttons than rows.
</p>
</div>
<div class="ltx_para" id="p2">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
MC
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
OFF
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
C
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
ON
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
MR
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\div" class="ltx_Math" display="inline" id="p2.m1">
<mo>
÷
</mo>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
M+
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
7
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
8
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
9
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\times" class="ltx_Math" display="inline" id="p2.m2">
<mo>
×
</mo>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
STO
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
4
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
5
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="-" class="ltx_Math" display="inline" id="p2.m3">
<mo>
-
</mo>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="\sqrt{x}" class="ltx_Math" display="inline" id="p2.m4">
<msqrt>
<mi>
x
</mi>
</msqrt>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
1
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
2
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
3
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
+
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
%
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
0
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
.
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\pm" class="ltx_Math" display="inline" id="p2.m5">
<mo>
±
</mo>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
=
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
A basic calculator can be counted on to have buttons for the basic
<a class="nnexus_concept" href="http://planetmath.org/operation">
arithmetic operations
</a>
(
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/subtraction">
subtraction
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
multiplication
</a>
and
<a class="nnexus_concept" href="http://planetmath.org/division">
division
</a>
). A button for
<a class="nnexus_concept" href="http://planetmath.org/squareroot">
square root
</a>
is sometimes provided, its
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
operation
</a>
usually being “postfix”. The
<a class="nnexus_concept" href="http://planetmath.org/percentage">
percentage
</a>
key, when provided, does not always behave in a standard way. For example, on some calculators, a 15% tip on a meal costing
<math alttext="\$42.54" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mo>
$
</mo>
<mo>
</mo>
<mn>
42.54
</mn>
</mrow>
</math>
might be computed as
<math alttext="[4][2][.][5][4][+][1][5][\%][=]" class="ltx_Math" display="inline" id="p3.m2">
<mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
4
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
2
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mo>
.
</mo>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
5
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
4
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mo>
+
</mo>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
1
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
5
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mo lspace="0pt" rspace="3.5pt">
%
</mo>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mo>
=
</mo>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
</math>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
mock-up of a basic calculator
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
MockupOfABasicCalculator
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:53:48
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:53:48
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
4
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Example
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 01A07
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | MockupOfScientificCalculator | http://planetmath.org/MockupOfScientificCalculator | <!DOCTYPE html>
<html>
<head>
<title>
mock-up of scientific calculator
</title>
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
mock-up of scientific calculator
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
This mock-up of a
<a class="nnexus_concept" href="http://planetmath.org/scientificcalculator">
scientific calculator
</a>
is realistic in that it has almost every function one can expect on a typical scientific calculator. It is unrealistic in that each function gets its own key. Usually, scientific calculators have some kind of shift key (labeled “Shift”, “2nd” or something similar) and almost all the other buttons (including the digit buttons) have a second or even third use. Sometimes these shifts make sense (sine and arcsine on the same key, for example), sometimes less so (for example, the
<a class="nnexus_concept" href="http://planetmath.org/randomnumbers">
random number
</a>
generator on the key for
<math alttext="\pi" class="ltx_Math" display="inline" id="p1.m1">
<mi>
π
</mi>
</math>
or the
<a class="nnexus_concept" href="http://planetmath.org/decimalpoint">
decimal point
</a>
).
</p>
</div>
<div class="ltx_para" id="p2">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
BIN
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
EE
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
OFF
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
ON
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
OCT
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\frac{d}{c}" class="ltx_Math" display="inline" id="p2.m1">
<mfrac>
<mi>
d
</mi>
<mi>
c
</mi>
</mfrac>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
RAND
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
C
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
DEC
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="a\frac{b}{c}" class="ltx_Math" display="inline" id="p2.m2">
<mrow>
<mi>
a
</mi>
<mo>
</mo>
<mfrac>
<mi>
b
</mi>
<mi>
c
</mi>
</mfrac>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\Sigma+" class="ltx_Math" display="inline" id="p2.m3">
<mrow>
<mi mathvariant="normal">
Σ
</mi>
<mo>
+
</mo>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\Sigma-" class="ltx_Math" display="inline" id="p2.m4">
<mrow>
<mi mathvariant="normal">
Σ
</mi>
<mo>
-
</mo>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="y\sigma n-1" class="ltx_Math" display="inline" id="p2.m5">
<mrow>
<mrow>
<mi>
y
</mi>
<mo>
</mo>
<mi>
σ
</mi>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
HEX
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
nCr
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\hat{x}" class="ltx_Math" display="inline" id="p2.m6">
<mover accent="true">
<mi>
x
</mi>
<mo stretchy="false">
^
</mo>
</mover>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\hat{y}" class="ltx_Math" display="inline" id="p2.m7">
<mover accent="true">
<mi>
y
</mi>
<mo stretchy="false">
^
</mo>
</mover>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="y\sigma n" class="ltx_Math" display="inline" id="p2.m8">
<mrow>
<mi>
y
</mi>
<mo>
</mo>
<mi>
σ
</mi>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
x!
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
nPr
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\bar{x}" class="ltx_Math" display="inline" id="p2.m9">
<mover accent="true">
<mi>
x
</mi>
<mo stretchy="false">
¯
</mo>
</mover>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\bar{y}" class="ltx_Math" display="inline" id="p2.m10">
<mover accent="true">
<mi>
y
</mi>
<mo stretchy="false">
¯
</mo>
</mover>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="x\sigma n-1" class="ltx_Math" display="inline" id="p2.m11">
<mrow>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
σ
</mi>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="e^{x}" class="ltx_Math" display="inline" id="p2.m12">
<msup>
<mi>
e
</mi>
<mi>
x
</mi>
</msup>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\pi" class="ltx_Math" display="inline" id="p2.m13">
<mi>
π
</mi>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="P\to R" class="ltx_Math" display="inline" id="p2.m14">
<mrow>
<mi>
P
</mi>
<mo>
→
</mo>
<mi>
R
</mi>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="R\to P" class="ltx_Math" display="inline" id="p2.m15">
<mrow>
<mi>
R
</mi>
<mo>
→
</mo>
<mi>
P
</mi>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="x\sigma n" class="ltx_Math" display="inline" id="p2.m16">
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
σ
</mi>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
ln
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
DEG
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
GRAD
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
RAD
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
MR
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="10^{x}" class="ltx_Math" display="inline" id="p2.m17">
<msup>
<mn>
10
</mn>
<mi>
x
</mi>
</msup>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\sin^{-1}" class="ltx_Math" display="inline" id="p2.m18">
<msup>
<mi>
sin
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\cos^{-1}" class="ltx_Math" display="inline" id="p2.m19">
<msup>
<mi>
cos
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\tan^{-1}" class="ltx_Math" display="inline" id="p2.m20">
<msup>
<mi>
tan
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msup>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
M+
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
log
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
sin
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
cos
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
tan
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
STO
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="\sqrt[y]{x}" class="ltx_Math" display="inline" id="p2.m21">
<mroot>
<mi>
x
</mi>
<mi>
y
</mi>
</mroot>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
D
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
E
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
F
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
%
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="x^{y}" class="ltx_Math" display="inline" id="p2.m22">
<msup>
<mi>
x
</mi>
<mi>
y
</mi>
</msup>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
A
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
B
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
C
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\div" class="ltx_Math" display="inline" id="p2.m23">
<mo>
÷
</mo>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="\sqrt[3]{x}" class="ltx_Math" display="inline" id="p2.m24">
<mroot>
<mi>
x
</mi>
<mn>
3
</mn>
</mroot>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
7
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
8
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
9
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\times" class="ltx_Math" display="inline" id="p2.m25">
<mo>
×
</mo>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="x^{3}" class="ltx_Math" display="inline" id="p2.m26">
<msup>
<mi>
x
</mi>
<mn>
3
</mn>
</msup>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
4
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
5
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="-" class="ltx_Math" display="inline" id="p2.m27">
<mo>
-
</mo>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="\sqrt{x}" class="ltx_Math" display="inline" id="p2.m28">
<msqrt>
<mi>
x
</mi>
</msqrt>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
1
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
2
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
3
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
+
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="x^{2}" class="ltx_Math" display="inline" id="p2.m29">
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
0
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
.
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<math alttext="\pm" class="ltx_Math" display="inline" id="p2.m30">
<mo>
±
</mo>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
=
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r">
<math alttext="\frac{1}{x}" class="ltx_Math" display="inline" id="p2.m31">
<mfrac>
<mn>
1
</mn>
<mi>
x
</mi>
</mfrac>
</math>
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Not all scientific calculators support binary, octal and
<a class="nnexus_concept" href="http://planetmath.org/hexadecimal">
hexadecimal
</a>
display.
<a class="nnexus_concept" href="http://planetmath.org/fraction">
Fractions
</a>
display and conversion is another category of functions that is available on many, but not all, scientific calculators. The trigonometric and statistical functions, on the other hand, are always standard, even if the button labels aren’t always (mainly for the statistical functions). The
<a class="nnexus_concept" href="http://planetmath.org/percentage">
percentage
</a>
key is more of a rarity on scientific calculators, something reflected by the
<a class="nnexus_concept" href="http://planetmath.org/windowscalculator">
Windows Calculator
</a>
, which has percentage in Standard mode but not Scientific (puzzlingly, this is also true of the
<a class="nnexus_concept" href="http://planetmath.org/squareroot">
square root
</a>
key).
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Square root and cubic root are usually “postfix” operations, e.g., meaning that to compute
<math alttext="\sqrt{2209}" class="ltx_Math" display="inline" id="p4.m1">
<msqrt>
<mn>
2209
</mn>
</msqrt>
</math>
one would enter
<math alttext="[2][2][0][9][\sqrt{x}]" class="ltx_Math" display="inline" id="p4.m2">
<mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
2
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
2
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
0
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
9
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<msqrt>
<mi>
x
</mi>
</msqrt>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
</math>
. On the CVS-brand scientific calculator with 2-line display, however, that would result in a “syntax error”; the square root key has to be pushed before the digits of the operand.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/mockupofscientificcalculator">
mock-up of scientific calculator
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
MockupOfScientificCalculator
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:53:50
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:53:50
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
4
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Example
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 01A65
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
<footer class="ltx_page_footer">
<div class="ltx_page_logo">
Generated on Tue Feb 6 22:18:26 2018 by
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|
0 | NegativeNumber | http://planetmath.org/NegativeNumber | <!DOCTYPE html>
<html>
<head>
<title>
negative number
</title>
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<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
negative number
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/negativenumber">
negative number
</a>
</span>
is a number that is less than zero. It is the “opposite” of a positive number. It can be the result of a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
subtraction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/subtraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/additionandsubtractionformulasforsineandcosine">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in which the
<a class="nnexus_concept" href="http://planetmath.org/difference">
subtrahend
</a>
is greater than the minuend. In everyday life, negative numbers are most often used to indicate debts.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
For example, Bob’s checking account has $47.20 and Alice cashes the check he gave her for $80.00, and the bank gives the requested amount. Bob’s account is then down to
<math alttext="-32.80" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<mo>
-
</mo>
<mn>
32.80
</mn>
</mrow>
</math>
, and could be further indebted if the bank collects some sort of fee for giving more money than was available.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
In a number line, negative numbers are usually to the left of zero:
</p>
</div>
<div class="ltx_para ltx_centering" id="p4">
<svg fragid="p4.pic1" height="17.1pt" overflow="visible" version="1.1" viewbox="0 5.73157287525381 170.71 11.3884271247462" width="170.71">
<g transform="translate(0,17.1)">
<g transform="scale(1 -1)">
<g transform="translate(28.45,8.54)">
<path d="M -28.45,0 142.26,0" fill="none" stroke="black" stroke-width="0.8">
</path>
<g>
<circle cx="0" cy="0" fill="black" r="2">
</circle>
<circle cx="28.45" cy="0" fill="black" r="2">
</circle>
<circle cx="56.91" cy="0" fill="black" r="2">
</circle>
<circle cx="85.36" cy="0" fill="black" r="2">
</circle>
<circle cx="113.81" cy="0" fill="black" r="2">
</circle>
</g>
<g pos="a" transform="translate(0,-8.54)">
<text dominant-baseline="middle" fill="black" text-anchor="middle" transform="scale(1 -1)" x="0" y="0">
-2
</text>
</g>
<g pos="a" transform="translate(28.45,-8.54)">
<text dominant-baseline="middle" fill="black" text-anchor="middle" transform="scale(1 -1)" x="0" y="0">
-1
</text>
</g>
<g pos="a" transform="translate(56.91,-8.54)">
<text dominant-baseline="middle" fill="black" text-anchor="middle" transform="scale(1 -1)" x="0" y="0">
0
</text>
</g>
<g pos="a" transform="translate(85.36,-8.54)">
<text dominant-baseline="middle" fill="black" text-anchor="middle" transform="scale(1 -1)" x="0" y="0">
+1
</text>
</g>
<g pos="a" transform="translate(113.81,-8.54)">
<text dominant-baseline="middle" fill="black" text-anchor="middle" transform="scale(1 -1)" x="0" y="0">
+2
</text>
</g>
<g pos="l" transform="translate(-28.45,0)">
<text dominant-baseline="middle" fill="black" text-anchor="start" transform="scale(1 -1)" x="0" y="0">
.
</text>
</g>
<g pos="r" transform="translate(142.26,0)">
<text dominant-baseline="middle" fill="black" text-anchor="end" transform="scale(1 -1)" x="0" y="0">
.
</text>
</g>
</g>
</g>
</g>
</svg>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
In a 2-dimensional coordinate plane, negative numbers on the vertical axis are usually below zero. (In the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complex plane
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexPlane.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, technically, they may be above if desired).
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
Whereas giving a
<a class="nnexus_concept" href="http://planetmath.org/plussign">
plus sign
</a>
for positive numbers is optional, giving the minus sign for negative numbers is a must.
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/addition">
Addition
</a>
and subtraction of negative numbers is fairly straightforward and intuitive. Suppose Bob also gave Carol and Dick a check for $80.00 each and they both cash their checks after Alice cashed hers. This is
<math alttext="-32.80-80.00-80.00=-192.80" class="ltx_Math" display="inline" id="p7.m1">
<mrow>
<mrow>
<mrow>
<mo>
-
</mo>
<mn>
32.80
</mn>
</mrow>
<mo>
-
</mo>
<mn>
80.00
</mn>
<mo>
-
</mo>
<mn>
80.00
</mn>
</mrow>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mn>
192.80
</mn>
</mrow>
</mrow>
</math>
.
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
Multiplication
</a>
of a negative number by a positive number is also straightforward. Suppose Bob’s bank charges a fee of $25.00 for each instance of overdraft. This
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
translates
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Translate.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/euclideantransformation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to
<math alttext="-25.00\times 3=-75.00" class="ltx_Math" display="inline" id="p7.m2">
<mrow>
<mrow>
<mo>
-
</mo>
<mrow>
<mn>
25.00
</mn>
<mo>
×
</mo>
<mn>
3
</mn>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mn>
75.00
</mn>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
However, multiplication of a negative number by another negative number gives a positive number. In all honesty, I can’t think of a situation in everyday life in which it would be necessary to multiply two negative numbers. At any rate, the rule of sign changes has the consequence that
<math alttext="(-x)^{a}>0" class="ltx_Math" display="inline" id="p8.m1">
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mi>
x
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mi>
a
</mi>
</msup>
<mo>
>
</mo>
<mn>
0
</mn>
</mrow>
</math>
if
<math alttext="a" class="ltx_Math" display="inline" id="p8.m2">
<mi>
a
</mi>
</math>
is even and
<math alttext="(-x)^{a}<0" class="ltx_Math" display="inline" id="p8.m3">
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mi>
x
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mi>
a
</mi>
</msup>
<mo>
<
</mo>
<mn>
0
</mn>
</mrow>
</math>
if
<math alttext="a" class="ltx_Math" display="inline" id="p8.m4">
<mi>
a
</mi>
</math>
is odd. This comes in very handy in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
number theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NumberTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/numbertheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
for creating
<a class="nnexus_concept" href="http://planetmath.org/alternatingsum">
alternating sums
</a>
or checking the relative
<a class="nnexus_concept" href="http://planetmath.org/tensordensity">
density
</a>
of one kind of number to another. For example, for
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
squarefree
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Squarefree.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/squarefreenumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="n" class="ltx_Math" display="inline" id="p8.m5">
<mi>
n
</mi>
</math>
, the Möbius
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="\mu(n)=(-1)^{\omega(n)}" class="ltx_Math" display="inline" id="p8.m6">
<mrow>
<mrow>
<mi>
μ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mrow>
<mi>
ω
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</msup>
</mrow>
</math>
(where
<math alttext="\omega(n)" class="ltx_Math" display="inline" id="p8.m7">
<mrow>
<mi>
ω
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is the
<a class="nnexus_concept" href="http://planetmath.org/numberofdistinctprimefactorsfunction">
number of distinct prime factors function
</a>
).
</p>
</div>
<div class="ltx_para" id="p9">
<p class="ltx_p">
As a consequence of these sign changes, a positive real number
<math alttext="x^{2}" class="ltx_Math" display="inline" id="p9.m1">
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</math>
technically has two
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
square roots
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SquareRoot.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/squareroot">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<math alttext="x" class="ltx_Math" display="inline" id="p9.m2">
<mi>
x
</mi>
</math>
and
<math alttext="-x" class="ltx_Math" display="inline" id="p9.m3">
<mrow>
<mo>
-
</mo>
<mi>
x
</mi>
</mrow>
</math>
. The specific case of
<math alttext="x^{2}=25" class="ltx_Math" display="inline" id="p9.m4">
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
=
</mo>
<mn>
25
</mn>
</mrow>
</math>
was used in
<span class="ltx_text ltx_font_italic">
The Simpsons
</span>
episode “Girls Just Want to Have Sums,” (first aired April 30, 2006) in which Lisa Simpson dressed up as a boy to sneak into a math class. Asked for the solution, Lisa answers 5, but the teacher says this is wrong. Martin then gives the two correct answers, 5 and
<math alttext="-5" class="ltx_Math" display="inline" id="p9.m5">
<mrow>
<mo>
-
</mo>
<mn>
5
</mn>
</mrow>
</math>
. Even
<a class="nnexus_concept" href="http://planetmath.org/symboliccomputation">
computer algebra systems
</a>
, and certainly most
<a class="nnexus_concept" href="http://planetmath.org/scientificcalculator">
scientific calculators
</a>
, will only give the
<a class="nnexus_concept" href="http://planetmath.org/positive">
positive
</a>
answer.
</p>
</div>
<div class="ltx_para" id="p10">
<p class="ltx_p">
What numbers are the square roots of a negative real number? No such numbers exist. More precisely, they are
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
imaginary numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ImaginaryNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/imaginary">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
multiples
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Multiple.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divisibility">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
imaginary unit
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ImaginaryUnit.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/imaginaryunit">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. For example,
<math alttext="\sqrt{-25}=5i" class="ltx_Math" display="inline" id="p10.m1">
<mrow>
<msqrt>
<mrow>
<mo>
-
</mo>
<mn>
25
</mn>
</mrow>
</msqrt>
<mo>
=
</mo>
<mrow>
<mn>
5
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
</math>
or
<math alttext="-5i" class="ltx_Math" display="inline" id="p10.m2">
<mrow>
<mo>
-
</mo>
<mrow>
<mn>
5
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p11">
<p class="ltx_p">
A question that comes up much less often is: What is a negative number raised to a fractional power? As you may know, raising a positive number to the
<a class="nnexus_concept" href="http://planetmath.org/inversenumber">
reciprocal
</a>
of
<math alttext="n" class="ltx_Math" display="inline" id="p11.m1">
<mi>
n
</mi>
</math>
has the same effect as taking the
<math alttext="n" class="ltx_Math" display="inline" id="p11.m2">
<mi>
n
</mi>
</math>
th root of
<math alttext="n" class="ltx_Math" display="inline" id="p11.m3">
<mi>
n
</mi>
</math>
. We can plot the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
powers of, say, 2, with dots and then connect the dots with straight lines; the result, though not technically correct, would not be too far off the mark (which would be a
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SmoothCurve.html">
smooth curve
</a>
connecting the dots). From such a graphic we might draw the incorrect conclusion that
<math alttext="2^{1.5}=3" class="ltx_Math" display="inline" id="p11.m4">
<mrow>
<msup>
<mn>
2
</mn>
<mn>
1.5
</mn>
</msup>
<mo>
=
</mo>
<mn>
3
</mn>
</mrow>
</math>
, while the correct answer (the base 2
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Logarithm.html">
logarithm
</a>
of 3) is more like 1.5849625007211561815.
</p>
</div>
<div class="ltx_para" id="p12">
<p class="ltx_p">
Whether we connect the dots in a plot of
<math alttext="(-2)^{n}" class="ltx_Math" display="inline" id="p12.m1">
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mi>
n
</mi>
</msup>
</math>
with either straight lines or curves, we might be fooling ourselves. The incorrect conclusion of
<math alttext="(-2)^{1.5}=0" class="ltx_Math" display="inline" id="p12.m2">
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
1.5
</mn>
</msup>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
just doesn’t make sense.
</p>
</div>
<div class="ltx_para ltx_centering" id="p13">
<img alt="" class="ltx_graphics" id="p13.g1" src="NegTwoIntegerPowers"/>
</div>
<div class="ltx_para" id="p14">
<p class="ltx_p">
On most scientific calculators, trying to raise a negative number to a fractional power will result in an error exception (unless of course you enter it in such a way that the
<a class="nnexus_concept" href="http://planetmath.org/calculator">
calculator
</a>
interprets as merely a positive number raised to a fractional power and then multiplied by
<math alttext="-1" class="ltx_Math" display="inline" id="p14.m1">
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</math>
).
</p>
</div>
<div class="ltx_para" id="p15">
<p class="ltx_p">
Once again, imaginary numbers come to the rescue. Since the square root of a negative number is an imaginary number, it only makes sense to extend this to negative numbers raised to fractional powers. For example,
<math alttext="(-2)^{\frac{3}{2}}=-2i\sqrt{2}" class="ltx_Math" display="inline" id="p15.m1">
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mfrac>
<mn>
3
</mn>
<mn>
2
</mn>
</mfrac>
</msup>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<msqrt>
<mn>
2
</mn>
</msqrt>
</mrow>
</mrow>
</mrow>
</math>
(and of course the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complex conjugate
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.9#E11">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexConjugate.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complexconjugate">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of that).
</p>
</div>
<div class="ltx_para" id="p16">
<p class="ltx_p">
In Mathematica, I made the following plot of
<math alttext="(-2)^{n}" class="ltx_Math" display="inline" id="p16.m1">
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mi>
n
</mi>
</msup>
</math>
in steps of
<math alttext="\frac{1}{10}" class="ltx_Math" display="inline" id="p16.m2">
<mfrac>
<mn>
1
</mn>
<mn>
10
</mn>
</mfrac>
</math>
and then
<a class="nnexus_concept" href="http://planetmath.org/separatedscheme">
separated
</a>
the resulting
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complex numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complexnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
into their real and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
imaginary parts
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.9#E2">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ImaginaryPart.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para ltx_centering" id="p17">
<img alt="" class="ltx_graphics" id="p17.g1" src="NegTwoFractPowers2D"/>
</div>
<div class="ltx_para" id="p18">
<p class="ltx_p">
Not entirely convinced this is right, I tried a 3-dimensional plot (in the plot,
<math alttext="x" class="ltx_Math" display="inline" id="p18.m1">
<mi>
x
</mi>
</math>
runs from 1 to 2,
<math alttext="y" class="ltx_Math" display="inline" id="p18.m2">
<mi>
y
</mi>
</math>
runs from 1 to 40 and
<math alttext="z" class="ltx_Math" display="inline" id="p18.m3">
<mi>
z
</mi>
</math>
runs from
<math alttext="-20" class="ltx_Math" display="inline" id="p18.m4">
<mrow>
<mo>
-
</mo>
<mn>
20
</mn>
</mrow>
</math>
to 20):
</p>
</div>
<div class="ltx_para ltx_centering" id="p19">
<img alt="" class="ltx_graphics" id="p19.g1" src="NegTwoFractPowers"/>
</div>
<div class="ltx_para" id="p20">
<p class="ltx_p">
As a final sanity check, I compared this to a
<a class="nnexus_concept" href="http://planetmath.org/similarmatrix">
similar
</a>
plot of
<math alttext="2^{n}" class="ltx_Math" display="inline" id="p20.m1">
<msup>
<mn>
2
</mn>
<mi>
n
</mi>
</msup>
</math>
(with the axes oriented in the same way as in the previous plot):
</p>
</div>
<div class="ltx_para ltx_centering" id="p21">
<img alt="" class="ltx_graphics" id="p21.g1" src="PosTwoWImPlot"/>
</div>
<div class="ltx_para" id="p22">
<p class="ltx_p">
In the end, though the 3-dimensional plots may look “cooler,” the 2-dimensional plot is actually more enlightening, showing the powers of a negative number fall on a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
logarithmic spiral
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LogarithmicSpiral.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logarithmicspiral">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Screen name “Cromulent Kwyjibo”. Personal communication, June 8, 2007.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
Alberto A. Martínez,
<span class="ltx_text ltx_font_italic">
Negative Math: How Mathematical Rules Can Be Positively Bent
</span>
. Princeton and Oxford: Princeton University Press (2006)
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p23">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
negative number
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
NegativeNumber
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:13:34
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:13:34
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
12
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
ClassificationOfComplexNumbers
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | Parametre | http://planetmath.org/Parametre | <!DOCTYPE html>
<html>
<head>
<title>
parametre
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
parametre
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/parametre">
Parametre
</a>
</em>
means often a quantity which is considered as constant in a certain situation but which may take different values in other situations; so the parametre is a “
<a class="nnexus_concept" href="http://planetmath.org/variable">
variable
</a>
constant”. But in giving a curve or a surface in
<span class="ltx_text ltx_font_italic">
parametric form
</span>
, the parametres work as proper variables which determine the values of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
coordinates
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Coordinates.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/coordinatevector">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/frame">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the points; then we can describe the parametres as “auxiliary variables”.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The parametric
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex1">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle\begin{cases}x=a\cos{t}\\
y=a\sin{t}\end{cases}" class="ltx_Math" display="inline" id="S0.Ex1.m1">
<mrow>
<mo>
{
</mo>
<mtable columnspacing="5pt" rowspacing="0pt">
<mtr>
<mtd columnalign="left">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mi>
a
</mi>
<mo>
</mo>
<mrow>
<mi>
cos
</mi>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
</mrow>
</mrow>
</mtd>
<mtd>
</mtd>
</mtr>
<mtr>
<mtd columnalign="left">
<mrow>
<mi>
y
</mi>
<mo>
=
</mo>
<mrow>
<mi>
a
</mi>
<mo>
</mo>
<mrow>
<mi>
sin
</mi>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
</mrow>
</mrow>
</mtd>
<mtd>
</mtd>
</mtr>
</mtable>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
of the origin-centered circle of parametres:
<math alttext="a" class="ltx_Math" display="inline" id="p2.m1">
<mi>
a
</mi>
</math>
(the radius) is a variable constant which is held constant all the time when one considers one circle;
<math alttext="t" class="ltx_Math" display="inline" id="p2.m2">
<mi>
t
</mi>
</math>
is an auxiliary variable which has to get all real values (e.g. from the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
interval
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Interval.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/orderedgeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="[0,\,2\pi]" class="ltx_Math" display="inline" id="p2.m3">
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
0
</mn>
<mo>
,
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
<mo stretchy="false">
]
</mo>
</mrow>
</math>
) for obtaining all points of the perimetre.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
In the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
analytic geometry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AnalyticGeometry.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/analyticgeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, one speaks of the
<em class="ltx_emph ltx_font_italic">
parametre of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
parabola
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/conicsection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/parabola">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
(a.k.a.
<em class="ltx_emph ltx_font_italic">
latus rectum
</em>
): it means the chord of the parabola which is
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
perpendicular
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Perpendicular.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mutualpositionsofvectors">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/perpendicularityineuclideanplane">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dihedralangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to the axis and goes through the focus; it is the quantity
<math alttext="2p" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
p
</mi>
</mrow>
</math>
in the standard equation
<math alttext="x^{2}=2py" class="ltx_Math" display="inline" id="p3.m2">
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
</math>
of the parabola (
<math alttext="p" class="ltx_Math" display="inline" id="p3.m3">
<mi>
p
</mi>
</math>
is the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Distance.html">
distance
</a>
of the focus and the
<a class="nnexus_concept" href="http://planetmath.org/ruledsurface">
directrix
</a>
).
</p>
<table class="ltx_tabular ltx_align_right ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
parametre
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Parametre
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:06:59
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:06:59
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
17
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
parameter
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/indeterminate">
Indeterminate
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
DerivativeForParametricForm
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Curve
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
PerimeterOfAstroid
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
CissoidOfDiocles
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Variable
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
SurfaceNormal
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
auxiliary variable
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
parametric form
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
parametric presentation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
parameter of parabola
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | Percentage | http://planetmath.org/Percentage | <!DOCTYPE html>
<html>
<head>
<title>
percentage
</title>
<!--Generated on Tue Feb 6 22:18:31 2018 by LaTeXML (version 0.8.2) http://dlmf.nist.gov/LaTeXML/.-->
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
percentage
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/percentage">
percentage
</a>
</span>
is a ratio expressed in terms of a unit being 100. A percentage is usually denoted by the symbol “%.” For example,
<math alttext="20\%" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mn>
20
</mn>
<mo lspace="0pt" rspace="3.5pt">
%
</mo>
</mrow>
</math>
of
<math alttext="\$700.47" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<mo>
$
</mo>
<mo>
</mo>
<mn>
700.47
</mn>
</mrow>
</math>
is
<math alttext="\$175.12" class="ltx_Math" display="inline" id="p1.m3">
<mrow>
<mo>
$
</mo>
<mo>
</mo>
<mn>
175.12
</mn>
</mrow>
</math>
(using
<a class="nnexus_concept" href="http://planetmath.org/fixedpointarithmetic">
fixed point arithmetic
</a>
to two
<a class="nnexus_concept" href="http://planetmath.org/decimalplace">
decimal places
</a>
for display).
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
On most
<a class="nnexus_concept" href="http://planetmath.org/calculator">
calculators
</a>
, one sure way to
<a class="nnexus_concept" href="http://planetmath.org/searchproblem">
calculate
</a>
a percentage is by entering a
<a class="nnexus_concept" href="http://planetmath.org/decimalpoint">
decimal point
</a>
before the desired percentage and multiplying that by the amount one wishes to calculate the percentage of. Some calculators have a percentage key.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
When tipping at most restaurants in the
<a class="nnexus_concept" href="http://planetmath.org/historyofmathematicsintheunitedstatesofamerica">
United States
</a>
, it is customary to tip 15% of the check to the waiter for parties of as much as four people. One common shortcut is to divide by 10 (by moving the decimal point to the left) and then add half of that amount.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Note that the percentage symbol % (Shift-5 in most American keyboard layouts) is
<a class="nnexus_concept" href="http://planetmath.org/overload">
overloaded
</a>
in TeX as a comment start indicator and in
<a class="nnexus_concept" href="http://planetmath.org/mathematica">
Mathematica
</a>
as a shortcut for referring to the previous output.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
percentage
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Percentage
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:37:56
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:37:56
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompositeFan (12809)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompositeFan (12809)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompositeFan (12809)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A05
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | Proof | http://planetmath.org/Proof | <!DOCTYPE html>
<html>
<head>
<title>
proof
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
proof
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<span class="ltx_text ltx_font_italic">
proof
</span>
is an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
argument
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Argument.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/argument">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
designed to show that a statement (usually a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) is true.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The mathematical
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
concept
</a>
of proof differs from the scientific concept in that in mathematics, a proof is logically deduced from axioms or from other theorems which have also been logically deduced, whereas for a scientific proof a preponderance of evidence is
<a class="nnexus_concept" href="http://planetmath.org/necessaryandsufficient">
sufficient
</a>
. Thus, a valid mathematical proof assures there are no
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
counterexamples
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Counterexample.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/counterexample">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to the proven statement.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
There are several kinds of proofs, one commonly used one being
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
proof by contradiction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ProofbyContradiction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradiction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/deductiontheoremholdsforclassicalpropositionallogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. A proof by contradiction starts by assuming that the opposite of the theorem is true, and then proceeds to work out the
<a class="nnexus_concept" href="http://planetmath.org/logicalimplication">
consequences
</a>
of that
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
assumption
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/deduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivationsinnaturaldeduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
until encountering a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
contradiction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Contradiction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradictorystatement">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradiction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, thus proving the theorem.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
According to Paul Nahin, the most famous proof by contradiction is Euclid’s proof of the infinitude of primes, which starts by assuming that there is in fact a largest prime number (and thus the primes are finite). Proofs that a given number is irrational (such as
<math alttext="\pi" class="ltx_Math" display="inline" id="p4.m1">
<mi>
π
</mi>
</math>
or
<math alttext="\sqrt{5}" class="ltx_Math" display="inline" id="p4.m2">
<msqrt>
<mn>
5
</mn>
</msqrt>
</math>
) also tend to prove the irrationality of the number by at first assuming that the number is in fact
<a class="nnexus_concept" href="http://planetmath.org/rationalnumber">
rational
</a>
and that there are two integers which form a ratio for the given number.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
Another kind of proof is the proof by induction, which starts by showing the statement is true for a small case (such as
<math alttext="n=1" class="ltx_Math" display="inline" id="p5.m1">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
when dealing with integers) and that the statement is true for a larger case when it is true for the immediately smaller case (e.g., that if it’s true for
<math alttext="n" class="ltx_Math" display="inline" id="p5.m2">
<mi>
n
</mi>
</math>
it is also true for
<math alttext="n+1" class="ltx_Math" display="inline" id="p5.m3">
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
). Thus, showing that it is true for the small case proves that it is also true for the next larger case, and the next larger case after that, and therefore all the larger cases.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
A proof by construction shows that a specified object actually exists by showing how to construct that object. For example, to prove that it is possible to draw by
<a class="nnexus_concept" href="http://planetmath.org/compassandstraightedgeconstruction">
compass
</a>
and straightedge an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
isosceles triangle
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/IsoscelesTriangle.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/isoscelestriangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
with an angle that is half of any of the two other angles, a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
constructive proof
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ConstructiveProof.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/techniquesinmathematicalproofs">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
would give the instructions on how to draw such a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
triangle
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Triangle.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/triangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Paul J. Nahin,
<span class="ltx_text ltx_font_italic">
Dr. Euler’s Fabulous
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Formula
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Formula.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/firstorderlanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
: Cures Many Mathematical Ills
</span>
. Princeton: Princeton University Press (2006): 8
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
Thomas A. Whitelaw,
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/introduction">
Introduction
</a>
to Abstract Algebra
</span>
. New York: CRC Press (1995): 11
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p7">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
proof
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Proof
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:10:22
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:10:22
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
4
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A05
</td>
</tr>
</tbody>
</table>
</div>
</article>
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|
0 | Property | http://planetmath.org/Property | <!DOCTYPE html>
<html>
<head>
<title>
property
</title>
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
property
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Let
<math alttext="X" class="ltx_Math" display="inline" id="p1.m1">
<mi>
X
</mi>
</math>
be a set. A
<em class="ltx_emph ltx_font_italic">
property
</em>
<math alttext="p" class="ltx_Math" display="inline" id="p1.m2">
<mi>
p
</mi>
</math>
of
<math alttext="X" class="ltx_Math" display="inline" id="p1.m3">
<mi>
X
</mi>
</math>
is a
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="p\colon X\to\{\mathit{true},\mathit{false}\}." class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mi>
p
</mi>
<mo>
:
</mo>
<mrow>
<mi>
X
</mi>
<mo>
→
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mi>
𝑡𝑟𝑢𝑒
</mi>
<mo>
,
</mo>
<mi>
𝑓𝑎𝑙𝑠𝑒
</mi>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
An
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Element.html">
element
</a>
<math alttext="x\in X" class="ltx_Math" display="inline" id="p1.m4">
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
X
</mi>
</mrow>
</math>
is said to
<em class="ltx_emph ltx_font_italic">
have
</em>
or
<em class="ltx_emph ltx_font_italic">
does not have the property
</em>
<math alttext="p" class="ltx_Math" display="inline" id="p1.m5">
<mi>
p
</mi>
</math>
depending on whether
<math alttext="p(x)=\mathit{true}" class="ltx_Math" display="inline" id="p1.m6">
<mrow>
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mi>
𝑡𝑟𝑢𝑒
</mi>
</mrow>
</math>
or
<math alttext="p(x)=\mathit{false}" class="ltx_Math" display="inline" id="p1.m7">
<mrow>
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mi>
𝑓𝑎𝑙𝑠𝑒
</mi>
</mrow>
</math>
. Any property gives rise in a natural way to the set
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="X(p):=\{x\in X|\ x\text{ has property }p\}" class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mrow>
<mi>
X
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
:=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
X
</mi>
</mrow>
<mo rspace="7.5pt" stretchy="false">
|
</mo>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mtext>
has property
</mtext>
<mo>
</mo>
<mi>
p
</mi>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
and the corresponding
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/node/CharacteristicFunction
</span>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
characteristic function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CharacteristicFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/standardenumeration">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/booleanvaluedfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/characteristicfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="1_{X(p)}" class="ltx_Math" display="inline" id="p1.m8">
<msub>
<mn>
1
</mn>
<mrow>
<mi>
X
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</msub>
</math>
. The identification of
<math alttext="p" class="ltx_Math" display="inline" id="p1.m9">
<mi>
p
</mi>
</math>
with
<math alttext="X(p)\subseteq X" class="ltx_Math" display="inline" id="p1.m10">
<mrow>
<mrow>
<mi>
X
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
⊆
</mo>
<mi>
X
</mi>
</mrow>
</math>
enables us to think of a property of
<math alttext="X" class="ltx_Math" display="inline" id="p1.m11">
<mi>
X
</mi>
</math>
as a 1-ary, or a
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/relation">
unary relation
</a>
</em>
on
<math alttext="X" class="ltx_Math" display="inline" id="p1.m12">
<mi>
X
</mi>
</math>
. Therefore, one may treat all these notions equivalently.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Usually, a property
<math alttext="p" class="ltx_Math" display="inline" id="p2.m1">
<mi>
p
</mi>
</math>
of
<math alttext="X" class="ltx_Math" display="inline" id="p2.m2">
<mi>
X
</mi>
</math>
can be identified with a so-called
<em class="ltx_emph ltx_font_italic">
propositional function
</em>
, or a
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Predicate.html">
predicate
</a>
</em>
<math alttext="\varphi(v)" class="ltx_Math" display="inline" id="p2.m3">
<mrow>
<mi>
φ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
v
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
, where
<math alttext="v" class="ltx_Math" display="inline" id="p2.m4">
<mi>
v
</mi>
</math>
is a variable or a tuple of variables whose values range over
<math alttext="X" class="ltx_Math" display="inline" id="p2.m5">
<mi>
X
</mi>
</math>
. The values of a propositional function is a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
proposition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/propositionallogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/booleanvaluedfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, which can be interpreted as being either “true” or “false”, so that
<math alttext="X(p)=\{x\mid\varphi(x)\mbox{ is }\mathit{true}\}" class="ltx_Math" display="inline" id="p2.m6">
<mrow>
<mrow>
<mi>
X
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mi>
x
</mi>
<mo>
∣
</mo>
<mrow>
<mi>
φ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mtext>
is
</mtext>
<mo>
</mo>
<mi>
𝑡𝑟𝑢𝑒
</mi>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Below are a few examples:
</p>
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
Let
<math alttext="X=\mathbb{Z}" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mrow>
<mi>
X
</mi>
<mo>
=
</mo>
<mi>
ℤ
</mi>
</mrow>
</math>
. Let
<math alttext="\varphi(v)" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mrow>
<mi>
φ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
v
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
be the propositional function “
<math alttext="v" class="ltx_Math" display="inline" id="I1.i1.p1.m3">
<mi>
v
</mi>
</math>
is divisible by
<math alttext="3" class="ltx_Math" display="inline" id="I1.i1.p1.m4">
<mn>
3
</mn>
</math>
”. If
<math alttext="p" class="ltx_Math" display="inline" id="I1.i1.p1.m5">
<mi>
p
</mi>
</math>
is the property identified with
<math alttext="\varphi(v)" class="ltx_Math" display="inline" id="I1.i1.p1.m6">
<mrow>
<mi>
φ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
v
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
, then
<math alttext="X(p)=3\mathbb{Z}" class="ltx_Math" display="inline" id="I1.i1.p1.m7">
<mrow>
<mrow>
<mi>
X
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mi>
ℤ
</mi>
</mrow>
</mrow>
</math>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
Again, let
<math alttext="X=\mathbb{Z}" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mrow>
<mi>
X
</mi>
<mo>
=
</mo>
<mi>
ℤ
</mi>
</mrow>
</math>
. Let
<math alttext="\varphi(v_{1},v_{2}):=" class="ltx_Math" display="inline" id="I1.i2.p1.m2">
<mrow>
<mrow>
<mi>
φ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
v
</mi>
<mn>
1
</mn>
</msub>
<mo>
,
</mo>
<msub>
<mi>
v
</mi>
<mn>
2
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
:=
</mo>
<mi>
</mi>
</mrow>
</math>
“
<math alttext="v_{1}" class="ltx_Math" display="inline" id="I1.i2.p1.m3">
<msub>
<mi>
v
</mi>
<mn>
1
</mn>
</msub>
</math>
is divisible by
<math alttext="v_{2}" class="ltx_Math" display="inline" id="I1.i2.p1.m4">
<msub>
<mi>
v
</mi>
<mn>
2
</mn>
</msub>
</math>
” and
<math alttext="p" class="ltx_Math" display="inline" id="I1.i2.p1.m5">
<mi>
p
</mi>
</math>
the corresponding property. Then
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="X(p)=\{(m,n)\mid m=np\mbox{, for some }p\in\mathbb{Z}\}," class="ltx_Math" display="block" id="S0.Ex3.m1">
<mrow>
<mrow>
<mrow>
<mi>
X
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
m
</mi>
<mo>
,
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
∣
</mo>
<mrow>
<mi>
m
</mi>
<mo>
=
</mo>
<mrow>
<mi>
n
</mi>
<mo>
</mo>
<mi>
p
</mi>
<mo>
</mo>
<mtext>
, for some
</mtext>
<mo>
</mo>
<mi>
p
</mi>
</mrow>
<mo>
∈
</mo>
<mi>
ℤ
</mi>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
which is a subset of
<math alttext="X\times X" class="ltx_Math" display="inline" id="I1.i2.p1.m6">
<mrow>
<mi>
X
</mi>
<mo>
×
</mo>
<mi>
X
</mi>
</mrow>
</math>
. So
<math alttext="p" class="ltx_Math" display="inline" id="I1.i2.p1.m7">
<mi>
p
</mi>
</math>
is a property of
<math alttext="X\times X" class="ltx_Math" display="inline" id="I1.i2.p1.m8">
<mrow>
<mi>
X
</mi>
<mo>
×
</mo>
<mi>
X
</mi>
</mrow>
</math>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
The reflexive property of a
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BinaryRelation.html">
binary relation
</a>
on
<math alttext="X" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mi>
X
</mi>
</math>
can be identified with the propositional function
<math alttext="\varphi(V):=``\forall a\in X\mbox{, }(a,a)\in V" class="ltx_Math" display="inline" id="I1.i3.p1.m2">
<mrow>
<mrow>
<mi>
φ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
V
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
:=
</mo>
<mrow>
<mi mathvariant="normal">
`
</mi>
<mo>
</mo>
<mi mathvariant="normal">
`
</mi>
<mo>
</mo>
<mrow>
<mo>
∀
</mo>
<mi>
a
</mi>
</mrow>
</mrow>
<mo>
∈
</mo>
<mrow>
<mi>
X
</mi>
<mo>
</mo>
<mtext>
,
</mtext>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
a
</mi>
<mo>
,
</mo>
<mi>
a
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
∈
</mo>
<mi>
V
</mi>
</mrow>
</math>
”, and therefore
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="X(p)=\{R\subseteq X\times X\mid\varphi(R)\mbox{ is }\mathit{true}\}," class="ltx_Math" display="block" id="S0.Ex4.m1">
<mrow>
<mrow>
<mrow>
<mi>
X
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
R
</mi>
<mo>
⊆
</mo>
<mrow>
<mi>
X
</mi>
<mo>
×
</mo>
<mi>
X
</mi>
</mrow>
</mrow>
<mo>
∣
</mo>
<mrow>
<mi>
φ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
R
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mtext>
is
</mtext>
<mo>
</mo>
<mi>
𝑡𝑟𝑢𝑒
</mi>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
which is a subset of
<math alttext="2^{X\times X}" class="ltx_Math" display="inline" id="I1.i3.p1.m3">
<msup>
<mn>
2
</mn>
<mrow>
<mi>
X
</mi>
<mo>
×
</mo>
<mi>
X
</mi>
</mrow>
</msup>
</math>
. Thus,
<math alttext="p" class="ltx_Math" display="inline" id="I1.i3.p1.m4">
<mi>
p
</mi>
</math>
is a property of
<math alttext="2^{X\times X}" class="ltx_Math" display="inline" id="I1.i3.p1.m5">
<msup>
<mn>
2
</mn>
<mrow>
<mi>
X
</mi>
<mo>
×
</mo>
<mi>
X
</mi>
</mrow>
</msup>
</math>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
In point set
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
topology
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Topology.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topologicalspace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, we often encounter the
<a class="nnexus_concept" href="http://planetmath.org/finiteintersectionproperty">
finite intersection property
</a>
on a family of subsets of a given set
<math alttext="X" class="ltx_Math" display="inline" id="I1.i4.p1.m1">
<mi>
X
</mi>
</math>
. Let
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\varphi(\mathcal{V}):=``\forall n\in\mathbb{N},\forall E_{1}\in\mathcal{V},%
\ldots,\forall E_{n}\in\mathcal{V},\exists x\in X(x\in E_{1}\cap\cdots\cap E_{%
n})\mbox{''}" class="ltx_Math" display="block" id="S0.Ex5.m1">
<mrow>
<mi>
φ
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mi class="ltx_font_mathcaligraphic">
𝒱
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
:=
</mo>
<mi mathvariant="normal">
`
</mi>
<mi mathvariant="normal">
`
</mi>
<mo>
∀
</mo>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
<mo>
,
</mo>
<mo>
∀
</mo>
<msub>
<mi>
E
</mi>
<mn>
1
</mn>
</msub>
<mo>
∈
</mo>
<mi class="ltx_font_mathcaligraphic">
𝒱
</mi>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
,
</mo>
<mo>
∀
</mo>
<msub>
<mi>
E
</mi>
<mi>
n
</mi>
</msub>
<mo>
∈
</mo>
<mi class="ltx_font_mathcaligraphic">
𝒱
</mi>
<mo>
,
</mo>
<mo>
∃
</mo>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
X
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo>
∈
</mo>
<msub>
<mi>
E
</mi>
<mn>
1
</mn>
</msub>
<mo>
∩
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
∩
</mo>
<msub>
<mi>
E
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
<mtext>
”
</mtext>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
and
<math alttext="p" class="ltx_Math" display="inline" id="I1.i4.p1.m2">
<mi>
p
</mi>
</math>
the corresponding property, then
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="X(p)=\{\mathcal{F}\subseteq 2^{X}\mid\varphi(\mathcal{F})\mbox{ is }\mathit{%
true}\}," class="ltx_Math" display="block" id="S0.Ex6.m1">
<mrow>
<mrow>
<mrow>
<mi>
X
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi class="ltx_font_mathcaligraphic">
ℱ
</mi>
<mo>
⊆
</mo>
<msup>
<mn>
2
</mn>
<mi>
X
</mi>
</msup>
</mrow>
<mo>
∣
</mo>
<mrow>
<mi>
φ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi class="ltx_font_mathcaligraphic">
ℱ
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mtext>
is
</mtext>
<mo>
</mo>
<mi>
𝑡𝑟𝑢𝑒
</mi>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
which is a subset of
<math alttext="2^{2^{X}}" class="ltx_Math" display="inline" id="I1.i4.p1.m3">
<msup>
<mn>
2
</mn>
<msup>
<mn>
2
</mn>
<mi>
X
</mi>
</msup>
</msup>
</math>
. Thus
<math alttext="p" class="ltx_Math" display="inline" id="I1.i4.p1.m4">
<mi>
p
</mi>
</math>
is a property of
<math alttext="2^{2^{X}}" class="ltx_Math" display="inline" id="I1.i4.p1.m5">
<msup>
<mn>
2
</mn>
<msup>
<mn>
2
</mn>
<mi>
X
</mi>
</msup>
</msup>
</math>
.
</p>
</div>
</li>
</ul>
<table class="ltx_tabular ltx_align_right ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
property
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Canonical name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Property
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:01:29
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:01:29
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
drini (3)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
drini (3)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
15
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
drini (3)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
attribute
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
propositional function
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Subset
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
CharacteristicFunction
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
Relation
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
ClosureOfARelationWithRespectToAProperty
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
unary relation
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
predicate
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | RydiasMathemagicMinute | http://planetmath.org/RydiasMathemagicMinute | <!DOCTYPE html>
<html>
<head>
<title>
Rydia’s Mathemagic Minute
</title>
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<h1 class="ltx_title ltx_title_document">
Rydia’s Mathemagic Minute
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
Rydia’s Mathemagic Minute
</span>
is a mini-game from
<span class="ltx_text ltx_font_italic">
Final Fantasy XIII
</span>
in which Rydia presents the player a riddle in the form of four numbers, each in the range
<math alttext="-1<n<10" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo>
<
</mo>
<mi>
n
</mi>
<mo>
<
</mo>
<mn>
10
</mn>
</mrow>
</math>
which the player must then employ in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
conjunction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Conjunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/conjunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
with a selection of basic
<a class="nnexus_concept" href="http://planetmath.org/operation">
arithmetic operations
</a>
(
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/subtraction">
subtraction
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
multiplication
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/division">
division
</a>
) in order to obtain 10 as a result. For example, given {8, 8, 6, 4}, one possible
<a class="nnexus_concept" href="http://planetmath.org/equation">
solution
</a>
is
<math alttext="8\times 8-4\div 6" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<mrow>
<mn>
8
</mn>
<mo>
×
</mo>
<mn>
8
</mn>
</mrow>
<mo>
-
</mo>
<mrow>
<mn>
4
</mn>
<mo>
÷
</mo>
<mn>
6
</mn>
</mrow>
</mrow>
</math>
. If the player can solve the riddle, he is given another riddle of the same form, until exhausting the time limit of 90 seconds. A player may skip a riddle and move on to another one, but in so doing incurs a penalty in
<a class="nnexus_concept" href="http://planetmath.org/positive">
negative
</a>
points. Correctly answering five riddles in a row results in the player receives a time limit
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
extension
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/substructure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extension">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extensionofafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/concatenation">
Concatenation
</a>
is not allowed. For example, given {1, 7, 2, 9},
<math alttext="19-(2+7)" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<mn>
19
</mn>
<mo>
-
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
2
</mn>
<mo>
+
</mo>
<mn>
7
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is not a valid answer; the program in fact will not allow the player to even try concatenation. Answers involving
<a class="nnexus_concept" href="http://planetmath.org/negativenumber">
negative numbers
</a>
at intermediate steps are not valid either, though when they occur they’re displayed in blue (in lieu of a minus sign).
<a class="nnexus_concept" href="http://planetmath.org/fraction">
Fractions
</a>
are not at all allowed at intermediate steps, and when the player presents an
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
operation
</a>
that would result in such a result (e.g., dividing a number by a smaller
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
coprime
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Coprime.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/coprime">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
number), the
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computer
</a>
simply rejects the operation.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
The mini-game can be played either as a side quest on the full game, or by itself online at quare Enix Members.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
Rydia’s Mathemagic Minute
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
RydiasMathemagicMinute
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:57:16
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:57:16
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A05
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | SaddlePointApproximation | http://planetmath.org/SaddlePointApproximation | <!DOCTYPE html>
<html>
<head>
<title>
saddle point approximation
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
saddle point approximation
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The
<a class="nnexus_concept" href="http://planetmath.org/saddlepointapproximation">
saddle point approximation
</a>
(SPA), a.k.a. stationary phase approximation, is a widely used method in quantum field theory (QFT) and related fields.
Suppose we want to evaluate the following
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
integral
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.4#iv">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lebesgueintegral">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in the limit
<math alttext="\zeta\rightarrow\infty" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mi>
ζ
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</math>
:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.E1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="{\mathcal{I}}=\lim_{\zeta\rightarrow\infty}\int_{-\infty}^{\infty}{\mbox{d}}x%
\;{\mbox{e}}^{-\zeta f(x)}." class="ltx_Math" display="block" id="S0.E1.m1">
<mrow>
<mrow>
<mi class="ltx_font_mathcaligraphic">
ℐ
</mi>
<mo>
=
</mo>
<mrow>
<munder>
<mo movablelimits="false">
lim
</mo>
<mrow>
<mi>
ζ
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</munder>
<mo>
</mo>
<mrow>
<msubsup>
<mo largeop="true" symmetric="true">
∫
</mo>
<mrow>
<mo>
-
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
<mi mathvariant="normal">
∞
</mi>
</msubsup>
<mrow>
<mtext>
d
</mtext>
<mo>
</mo>
<mpadded width="+2.8pt">
<mi>
x
</mi>
</mpadded>
<mo>
</mo>
<msup>
<mtext>
e
</mtext>
<mrow>
<mo>
-
</mo>
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</msup>
</mrow>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(1)
</span>
</td>
</tr>
</table>
<p class="ltx_p">
The saddle point approximation can be applied if the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="f(x)" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
satisfies certain conditions. Assume that
<math alttext="f(x)" class="ltx_Math" display="inline" id="p1.m3">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
has a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
global minimum
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GlobalMinimum.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extremum">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="f(x_{0})=y_{min}" class="ltx_Math" display="inline" id="p1.m4">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<msub>
<mi>
y
</mi>
<mrow>
<mi>
m
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
</msub>
</mrow>
</math>
at
<math alttext="x=x_{0}" class="ltx_Math" display="inline" id="p1.m5">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
</math>
, which is sufficiently separated from other local minima and whose value is sufficiently smaller than the value of those.
Consider the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Taylor expansion
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TaylorExpansion.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/taylorseries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<math alttext="f(x)" class="ltx_Math" display="inline" id="p1.m6">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
about the point
<math alttext="x_{0}" class="ltx_Math" display="inline" id="p1.m7">
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</math>
:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="f(x)=f(x_{0})+\partial_{x}f(x){\Big{|}}_{x=x_{0}}(x-x_{0})+\frac{1}{2}{%
\partial_{x}}^{2}f(x){\Big{|}}_{x=x_{0}}(x-x_{0})^{2}+O(x^{3})." class="ltx_Math" display="block" id="S0.E2.m1">
<mrow>
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
+
</mo>
<mrow>
<msub>
<mrow>
<mrow>
<mrow>
<msub>
<mo>
∂
</mo>
<mi>
x
</mi>
</msub>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo fence="true" maxsize="160%" minsize="160%">
|
</mo>
</mrow>
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
</msub>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
-
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
+
</mo>
<mrow>
<msub>
<mrow>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<mrow>
<mmultiscripts>
<mo>
∂
</mo>
<mi>
x
</mi>
<none>
</none>
<none>
</none>
<mn>
2
</mn>
</mmultiscripts>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo fence="true" maxsize="160%" minsize="160%">
|
</mo>
</mrow>
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
</msub>
<mo>
</mo>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
-
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
+
</mo>
<mrow>
<mi>
O
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msup>
<mi>
x
</mi>
<mn>
3
</mn>
</msup>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(2)
</span>
</td>
</tr>
</table>
<p class="ltx_p">
Since
<math alttext="f(x_{0})" class="ltx_Math" display="inline" id="p1.m8">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is a (global)
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Minimum.html">
minimum
</a>
, it is clear that
<math alttext="f^{\prime}(x_{0})=0" class="ltx_Math" display="inline" id="p1.m9">
<mrow>
<mrow>
<msup>
<mi>
f
</mi>
<mo>
′
</mo>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
. Therefore
<math alttext="f(x)" class="ltx_Math" display="inline" id="p1.m10">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
may be approximated to quadratic order as
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="f(x)\approx f(x_{0})+\frac{1}{2}f^{\prime\prime}(x_{0})(x-x_{0})^{2}." class="ltx_Math" display="block" id="S0.E3.m1">
<mrow>
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
≈
</mo>
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
+
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<msup>
<mi>
f
</mi>
<mo>
′′
</mo>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
-
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(3)
</span>
</td>
</tr>
</table>
<p class="ltx_p">
The above assumptions on the minima of
<math alttext="f(x)" class="ltx_Math" display="inline" id="p1.m11">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
ensure that the dominant contribution to (
<a class="ltx_ref" href="#S0.E1" title="(1) ‣ saddle point approximation">
<span class="ltx_text ltx_ref_tag">
1
</span>
</a>
) in the limit
<math alttext="\zeta\rightarrow\infty" class="ltx_Math" display="inline" id="p1.m12">
<mrow>
<mi>
ζ
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</math>
will
come from the region of integration around
<math alttext="x_{0}" class="ltx_Math" display="inline" id="p1.m13">
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</math>
:
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx1">
<tbody id="S0.E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle{\mathcal{I}}" class="ltx_Math" display="inline" id="S0.E4.m1">
<mi class="ltx_font_mathcaligraphic">
ℐ
</mi>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle\approx\lim_{\zeta\rightarrow\infty}{\mbox{e}}^{-\zeta f(x_{0})}%
\int_{-\infty}^{\infty}{\mbox{d}}x\;{\mbox{e}}^{-\frac{\zeta}{2}f^{\prime%
\prime}(x_{0})(x-x_{0})^{2}}" class="ltx_Math" display="inline" id="S0.E4.m2">
<mrow>
<mi>
</mi>
<mo>
≈
</mo>
<mrow>
<munder>
<mo movablelimits="false">
lim
</mo>
<mrow>
<mi>
ζ
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</munder>
<mo>
</mo>
<mrow>
<msup>
<mtext>
e
</mtext>
<mrow>
<mo>
-
</mo>
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</msup>
<mo>
</mo>
<mrow>
<mstyle displaystyle="true">
<msubsup>
<mo largeop="true" symmetric="true">
∫
</mo>
<mrow>
<mo>
-
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
<mi mathvariant="normal">
∞
</mi>
</msubsup>
</mstyle>
<mrow>
<mtext>
d
</mtext>
<mo>
</mo>
<mpadded width="+2.8pt">
<mi>
x
</mi>
</mpadded>
<mo>
</mo>
<msup>
<mtext>
e
</mtext>
<mrow>
<mo>
-
</mo>
<mrow>
<mfrac>
<mi>
ζ
</mi>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<msup>
<mi>
f
</mi>
<mo>
′′
</mo>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
-
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
</mrow>
</mrow>
</msup>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(4)
</span>
</td>
</tr>
</tbody>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex1">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle\approx\lim_{\zeta\rightarrow\infty}{\mbox{e}}^{-\zeta f(x_{0})}%
\left(\frac{2\pi}{\zeta f^{\prime\prime}(x_{0})}\right)^{1/2}." class="ltx_Math" display="inline" id="S0.Ex1.m2">
<mrow>
<mrow>
<mi>
</mi>
<mo>
≈
</mo>
<mrow>
<munder>
<mo movablelimits="false">
lim
</mo>
<mrow>
<mi>
ζ
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</munder>
<mo>
</mo>
<mrow>
<msup>
<mtext>
e
</mtext>
<mrow>
<mo>
-
</mo>
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</msup>
<mo>
</mo>
<msup>
<mrow>
<mo>
(
</mo>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<msup>
<mi>
f
</mi>
<mo>
′′
</mo>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mfrac>
</mstyle>
<mo>
)
</mo>
</mrow>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
In the last step we have performed the Gaußian integral.
The next nonvanishing higher order correction to (
<a class="ltx_ref" href="#S0.E4" title="(4) ‣ saddle point approximation">
<span class="ltx_text ltx_ref_tag">
4
</span>
</a>
) stems from the quartic term of the expansion (
<a class="ltx_ref" href="#S0.E2" title="(2) ‣ saddle point approximation">
<span class="ltx_text ltx_ref_tag">
2
</span>
</a>
). This correction may be incorporated into (
<a class="ltx_ref" href="#S0.E4" title="(4) ‣ saddle point approximation">
<span class="ltx_text ltx_ref_tag">
4
</span>
</a>
) to yield (after expanding part of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
exponential
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Exponential.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/definitionsintrigonometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/exponential">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
):
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="{\mathcal{I}}\approx\lim_{\zeta\rightarrow\infty}{\mbox{e}}^{-\zeta f(x_{0})}%
\int_{-\infty}^{\infty}{\mbox{d}}x\;{\mbox{e}}^{-\frac{\zeta}{2}f^{\prime%
\prime}(x_{0})(x-x_{0})^{2}}\left(1-\frac{\zeta}{4!}(\partial_{x}^{4}f(x))|_{x%
=x_{0}}(x-x_{0})^{4}\right)." class="ltx_Math" display="block" id="S0.E5.m1">
<mrow>
<mrow>
<mi class="ltx_font_mathcaligraphic">
ℐ
</mi>
<mo>
≈
</mo>
<mrow>
<munder>
<mo movablelimits="false">
lim
</mo>
<mrow>
<mi>
ζ
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</munder>
<mo>
</mo>
<mrow>
<msup>
<mtext>
e
</mtext>
<mrow>
<mo>
-
</mo>
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</msup>
<mo>
</mo>
<mrow>
<msubsup>
<mo largeop="true" symmetric="true">
∫
</mo>
<mrow>
<mo>
-
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
<mi mathvariant="normal">
∞
</mi>
</msubsup>
<mrow>
<mtext>
d
</mtext>
<mo>
</mo>
<mpadded width="+2.8pt">
<mi>
x
</mi>
</mpadded>
<mo>
</mo>
<msup>
<mtext>
e
</mtext>
<mrow>
<mo>
-
</mo>
<mrow>
<mfrac>
<mi>
ζ
</mi>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<msup>
<mi>
f
</mi>
<mo>
′′
</mo>
</msup>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
-
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
</mrow>
</mrow>
</msup>
<mo>
</mo>
<mrow>
<mo>
(
</mo>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mrow>
<msub>
<mrow>
<mrow>
<mfrac>
<mi>
ζ
</mi>
<mrow>
<mn>
4
</mn>
<mo lspace="0pt" rspace="3.5pt">
!
</mo>
</mrow>
</mfrac>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<msubsup>
<mo>
∂
</mo>
<mi>
x
</mi>
<mn>
4
</mn>
</msubsup>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo fence="true" stretchy="false">
|
</mo>
</mrow>
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
</msub>
<mo>
</mo>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
-
</mo>
<msub>
<mi>
x
</mi>
<mn>
0
</mn>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
4
</mn>
</msup>
</mrow>
</mrow>
<mo>
)
</mo>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(5)
</span>
</td>
</tr>
</table>
<p class="ltx_p">
…to be continued with applications to physics…
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
saddle point approximation
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
SaddlePointApproximation
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:38:07
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:38:07
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msihl (2134)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msihl (2134)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msihl (2134)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Topic
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A79
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
stationary phase method
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | Singleton | http://planetmath.org/Singleton | <!DOCTYPE html>
<html>
<head>
<title>
singleton
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
singleton
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/singleton">
singleton
</a>
</span>
is a set containing a single element.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
singleton
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Singleton
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:13:41
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:13:41
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Koro (127)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Koro (127)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Koro (127)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
ATopologicalSpaceIsT_1IfAndOnlyIfEverySingletonIsClosed
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | StableSubspace | http://planetmath.org/StableSubspace | <!DOCTYPE html>
<html>
<head>
<title>
stable subspace
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
stable subspace
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A subset
<math alttext="S" class="ltx_Math" display="inline" id="p1.m1">
<mi>
S
</mi>
</math>
of a larger set
<math alttext="T" class="ltx_Math" display="inline" id="p1.m2">
<mi>
T
</mi>
</math>
is said to a
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/stablesubspace">
stable subset
</a>
</em>
for a function
<math alttext="f:T\to T" class="ltx_Math" display="inline" id="p1.m3">
<mrow>
<mi>
f
</mi>
<mo>
:
</mo>
<mrow>
<mi>
T
</mi>
<mo>
→
</mo>
<mi>
T
</mi>
</mrow>
</mrow>
</math>
<span class="ltx_text ltx_framed_underline">
iff
</span>
<math alttext="f(S)\subset S" class="ltx_Math" display="inline" id="p1.m4">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
S
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
⊂
</mo>
<mi>
S
</mi>
</mrow>
</math>
.
<br class="ltx_break"/>
Alternative phrasings with the same meaning are:
</p>
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<math alttext="f" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mi>
f
</mi>
</math>
is an invariant subset for
<math alttext="f" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mi>
f
</mi>
</math>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
<math alttext="f" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mi>
f
</mi>
</math>
stabilizes
<math alttext="S" class="ltx_Math" display="inline" id="I1.i2.p1.m2">
<mi>
S
</mi>
</math>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
<math alttext="S" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mi>
S
</mi>
</math>
is stable under (the action of)
<math alttext="f" class="ltx_Math" display="inline" id="I1.i3.p1.m2">
<mi>
f
</mi>
</math>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
<math alttext="S" class="ltx_Math" display="inline" id="I1.i4.p1.m1">
<mi>
S
</mi>
</math>
is
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Invariant.html">
invariant
</a>
under (the action of)
<math alttext="f" class="ltx_Math" display="inline" id="I1.i4.p1.m2">
<mi>
f
</mi>
</math>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
<math alttext="S" class="ltx_Math" display="inline" id="I1.i5.p1.m1">
<mi>
S
</mi>
</math>
is left stable by/under
<math alttext="f" class="ltx_Math" display="inline" id="I1.i5.p1.m2">
<mi>
f
</mi>
</math>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i6.p1">
<p class="ltx_p">
<math alttext="S" class="ltx_Math" display="inline" id="I1.i6.p1.m1">
<mi>
S
</mi>
</math>
is left invariant by/under
<math alttext="f" class="ltx_Math" display="inline" id="I1.i6.p1.m2">
<mi>
f
</mi>
</math>
</p>
</div>
</li>
</ul>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
stable subspace
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
StableSubspace
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:56:52
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:56:52
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
lalberti (18937)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
lalberti (18937)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
lalberti (18937)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/invariantsubspace">
invariant subspace
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
stable subset
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
invariant subset
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | Subsequence | http://planetmath.org/Subsequence | <!DOCTYPE html>
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<title>
subsequence
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
subsequence
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Given a
<a class="nnexus_concept" href="http://planetmath.org/sequence">
sequence
</a>
<math alttext="\{x_{n}\}_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="p1.m1">
<msub>
<mrow>
<mo stretchy="false">
{
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
}
</mo>
</mrow>
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</msub>
</math>
, any
<a class="nnexus_concept" href="http://planetmath.org/infinite">
infinite subset
</a>
of the sequence forms a
<a class="nnexus_concept" href="http://planetmath.org/subsequence">
subsequence
</a>
. We formalize this as follows:
</p>
</div>
<div class="ltx_theorem ltx_theorem_defn" id="Thmdefnx1">
<h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem">
Definition
</span>
.
</h6>
<div class="ltx_para" id="Thmdefnx1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
If
<math alttext="X" class="ltx_Math" display="inline" id="Thmdefnx1.p1.m1">
<mi>
X
</mi>
</math>
is a set and
<math alttext="\{a_{n}\}_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="Thmdefnx1.p1.m2">
<msub>
<mrow>
<mo mathvariant="normal" stretchy="false">
{
</mo>
<msub>
<mi>
a
</mi>
<mi>
n
</mi>
</msub>
<mo mathvariant="normal" stretchy="false">
}
</mo>
</mrow>
<mrow>
<mi>
n
</mi>
<mo mathvariant="normal">
∈
</mo>
<mi mathvariant="normal">
N
</mi>
</mrow>
</msub>
</math>
is a sequence in
<math alttext="X" class="ltx_Math" display="inline" id="Thmdefnx1.p1.m3">
<mi>
X
</mi>
</math>
, then a
<em class="ltx_emph ltx_font_upright">
subsequence
</em>
of
<math alttext="\{a_{n}\}" class="ltx_Math" display="inline" id="Thmdefnx1.p1.m4">
<mrow>
<mo mathvariant="normal" stretchy="false">
{
</mo>
<msub>
<mi>
a
</mi>
<mi>
n
</mi>
</msub>
<mo mathvariant="normal" stretchy="false">
}
</mo>
</mrow>
</math>
is a sequence of the form
<math alttext="\{a_{n_{r}}\}_{r\in\mathbb{N}}" class="ltx_Math" display="inline" id="Thmdefnx1.p1.m5">
<msub>
<mrow>
<mo mathvariant="normal" stretchy="false">
{
</mo>
<msub>
<mi>
a
</mi>
<msub>
<mi>
n
</mi>
<mi>
r
</mi>
</msub>
</msub>
<mo mathvariant="normal" stretchy="false">
}
</mo>
</mrow>
<mrow>
<mi>
r
</mi>
<mo mathvariant="normal">
∈
</mo>
<mi mathvariant="normal">
N
</mi>
</mrow>
</msub>
</math>
where
<math alttext="\{n_{r}\}_{r\in\mathbb{N}}" class="ltx_Math" display="inline" id="Thmdefnx1.p1.m6">
<msub>
<mrow>
<mo mathvariant="normal" stretchy="false">
{
</mo>
<msub>
<mi>
n
</mi>
<mi>
r
</mi>
</msub>
<mo mathvariant="normal" stretchy="false">
}
</mo>
</mrow>
<mrow>
<mi>
r
</mi>
<mo mathvariant="normal">
∈
</mo>
<mi mathvariant="normal">
N
</mi>
</mrow>
</msub>
</math>
is a
<a class="nnexus_concept" href="http://planetmath.org/increasingdecreasingmonotonefunction">
strictly increasing
</a>
sequence of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
natural numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NaturalNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/naturalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</span>
</p>
</div>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Equivalently,
<math alttext="\{y_{n}\}_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="p2.m1">
<msub>
<mrow>
<mo stretchy="false">
{
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
}
</mo>
</mrow>
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</msub>
</math>
is a subsequence of
<math alttext="\{x_{n}\}_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="p2.m2">
<msub>
<mrow>
<mo stretchy="false">
{
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
}
</mo>
</mrow>
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</msub>
</math>
if
</p>
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<math alttext="\{y_{n}\}_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<msub>
<mrow>
<mo stretchy="false">
{
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
}
</mo>
</mrow>
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</msub>
</math>
is a sequence of
<a class="nnexus_concept" href="http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9">
elements of
</a>
<math alttext="X" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mi>
X
</mi>
</math>
, and
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
there is a
<a class="nnexus_concept" href="http://mathworld.wolfram.com/StrictlyIncreasingFunction.html">
strictly increasing function
</a>
<math alttext="a:\mathbb{N}\to\mathbb{N}" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mrow>
<mi>
a
</mi>
<mo>
:
</mo>
<mrow>
<mi>
ℕ
</mi>
<mo>
→
</mo>
<mi>
ℕ
</mi>
</mrow>
</mrow>
</math>
such that
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{n}=x_{a(n)}\quad\text{ for all }n\in\mathbb{N}." class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mrow>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
<mo>
=
</mo>
<msub>
<mi>
x
</mi>
<mrow>
<mi>
a
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</msub>
</mrow>
<mo mathvariant="italic" separator="true">
</mo>
<mrow>
<mrow>
<mtext>
for all
</mtext>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
</li>
</ol>
</div>
<div class="ltx_theorem ltx_theorem_exa" id="Thmexax1">
<h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem">
Example
</span>
.
</h6>
<div class="ltx_para" id="Thmexax1.p1">
<p class="ltx_p">
Let
<math alttext="X=\mathbb{R}" class="ltx_Math" display="inline" id="Thmexax1.p1.m1">
<mrow>
<mi>
X
</mi>
<mo>
=
</mo>
<mi>
ℝ
</mi>
</mrow>
</math>
and let
<math alttext="\{x_{n}\}" class="ltx_Math" display="inline" id="Thmexax1.p1.m2">
<mrow>
<mo stretchy="false">
{
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
}
</mo>
</mrow>
</math>
be the sequence
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\left\{\frac{1}{n}\right\}_{n\in\mathbb{N}}=\left\{1,\frac{1}{2},\frac{1}{3},%
\frac{1}{4},\ldots\right\}." class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mrow>
<msub>
<mrow>
<mo>
{
</mo>
<mfrac>
<mn>
1
</mn>
<mi>
n
</mi>
</mfrac>
<mo>
}
</mo>
</mrow>
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</msub>
<mo>
=
</mo>
<mrow>
<mo>
{
</mo>
<mn>
1
</mn>
<mo>
,
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
,
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
3
</mn>
</mfrac>
<mo>
,
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
4
</mn>
</mfrac>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
}
</mo>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Then, the sequence
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\{y_{n}\}_{n\in\mathbb{N}}=\left\{\frac{1}{n^{2}}\right\}_{n\in\mathbb{N}}=%
\left\{1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\ldots\right\}" class="ltx_Math" display="block" id="S0.Ex3.m1">
<mrow>
<msub>
<mrow>
<mo stretchy="false">
{
</mo>
<msub>
<mi>
y
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
}
</mo>
</mrow>
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</msub>
<mo>
=
</mo>
<msub>
<mrow>
<mo>
{
</mo>
<mfrac>
<mn>
1
</mn>
<msup>
<mi>
n
</mi>
<mn>
2
</mn>
</msup>
</mfrac>
<mo>
}
</mo>
</mrow>
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</msub>
<mo>
=
</mo>
<mrow>
<mo>
{
</mo>
<mn>
1
</mn>
<mo>
,
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
4
</mn>
</mfrac>
<mo>
,
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
9
</mn>
</mfrac>
<mo>
,
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
16
</mn>
</mfrac>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
}
</mo>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
is a subsequence of
<math alttext="\{x_{n}\}" class="ltx_Math" display="inline" id="Thmexax1.p1.m3">
<mrow>
<mo stretchy="false">
{
</mo>
<msub>
<mi>
x
</mi>
<mi>
n
</mi>
</msub>
<mo stretchy="false">
}
</mo>
</mrow>
</math>
. The subsequence of natural numbers mentioned in the
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
is
<math alttext="\{n^{2}\}_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="Thmexax1.p1.m4">
<msub>
<mrow>
<mo stretchy="false">
{
</mo>
<msup>
<mi>
n
</mi>
<mn>
2
</mn>
</msup>
<mo stretchy="false">
}
</mo>
</mrow>
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<mi>
ℕ
</mi>
</mrow>
</msub>
</math>
and the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="a:\mathbb{N}\to\mathbb{N}" class="ltx_Math" display="inline" id="Thmexax1.p1.m5">
<mrow>
<mi>
a
</mi>
<mo>
:
</mo>
<mrow>
<mi>
ℕ
</mi>
<mo>
→
</mo>
<mi>
ℕ
</mi>
</mrow>
</mrow>
</math>
mentioned above is
<math alttext="a(n)=n^{2}" class="ltx_Math" display="inline" id="Thmexax1.p1.m6">
<mrow>
<mrow>
<mi>
a
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<msup>
<mi>
n
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</math>
.
</p>
</div>
</div>
<div class="ltx_para ltx_align_right" id="p3">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
subsequence
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Subsequence
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:56:34
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:56:34
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
alozano (2414)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
alozano (2414)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
alozano (2414)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A05
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | SumOfOddNumbers | http://planetmath.org/SumOfOddNumbers | <!DOCTYPE html>
<html>
<head>
<title>
sum of odd numbers
</title>
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
sum of odd numbers
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The sum of the first
<math alttext="n" class="ltx_Math" display="inline" id="p1.m1">
<mi>
n
</mi>
</math>
<a class="nnexus_concept" href="http://planetmath.org/positive">
positive
</a>
<a class="nnexus_concept" href="http://planetmath.org/evennumber">
odd integers
</a>
can be calculated by using the well-known of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
arithmetic progression
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/remainderarithmeticvsegyptianfractions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/arithmeticprogression">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, that the sum of its is equal to the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
arithmetic mean
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ArithmeticMean.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/arithmeticmean">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the first and the last , multiplied by the number of the :
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\underbrace{1+3+5+7+9+\cdots+(2n\!-\!1)}_{n}=n\cdot\frac{1\!+\!(2n\!-\!1)}{2}=%
n^{2}" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<munder>
<munder accentunder="true">
<mrow>
<mn>
1
</mn>
<mo movablelimits="false">
+
</mo>
<mn>
3
</mn>
<mo movablelimits="false">
+
</mo>
<mn>
5
</mn>
<mo movablelimits="false">
+
</mo>
<mn>
7
</mn>
<mo movablelimits="false">
+
</mo>
<mn>
9
</mn>
<mo movablelimits="false">
+
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo movablelimits="false">
+
</mo>
<mrow>
<mo movablelimits="false" stretchy="false">
(
</mo>
<mrow>
<mrow>
<mn>
2
</mn>
<mo movablelimits="false">
</mo>
<mpadded width="-1.7pt">
<mi>
n
</mi>
</mpadded>
</mrow>
<mo movablelimits="false" rspace="0.8pt">
-
</mo>
<mn>
1
</mn>
</mrow>
<mo movablelimits="false" stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo movablelimits="false">
⏟
</mo>
</munder>
<mi>
n
</mi>
</munder>
<mo>
=
</mo>
<mrow>
<mi>
n
</mi>
<mo>
⋅
</mo>
<mfrac>
<mrow>
<mpadded width="-1.7pt">
<mn>
1
</mn>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mpadded width="-1.7pt">
<mi>
n
</mi>
</mpadded>
</mrow>
<mo rspace="0.8pt">
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mn>
2
</mn>
</mfrac>
</mrow>
<mo>
=
</mo>
<msup>
<mi>
n
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Thus, the sum of the first
<math alttext="n" class="ltx_Math" display="inline" id="p1.m2">
<mi>
n
</mi>
</math>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
odd numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/OddNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/oddnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is
<math alttext="n^{2}" class="ltx_Math" display="inline" id="p1.m3">
<msup>
<mi>
n
</mi>
<mn>
2
</mn>
</msup>
</math>
(this result has been proved first time in 1575 by Francesco Maurolico).
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Below, the odd numbers have been set to form a
<a class="nnexus_concept" href="http://planetmath.org/triangle">
triangle
</a>
, each
<math alttext="n^{\rm{th}}" class="ltx_Math" display="inline" id="p2.m1">
<msup>
<mi>
n
</mi>
<mi>
th
</mi>
</msup>
</math>
row containing the next
<math alttext="n" class="ltx_Math" display="inline" id="p2.m2">
<mi>
n
</mi>
</math>
consecutive odd numbers. The arithmetic mean on the row is
<math alttext="n^{2}" class="ltx_Math" display="inline" id="p2.m3">
<msup>
<mi>
n
</mi>
<mn>
2
</mn>
</msup>
</math>
and the sum of its numbers is
<math alttext="n\cdot n^{2}=n^{3}" class="ltx_Math" display="inline" id="p2.m4">
<mrow>
<mrow>
<mi>
n
</mi>
<mo>
⋅
</mo>
<msup>
<mi>
n
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<msup>
<mi>
n
</mi>
<mn>
3
</mn>
</msup>
</mrow>
</math>
.
</p>
<table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S0.EGx1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex2">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle\begin{array}[]{cccccccccccccccccc}&&&&&&&&&1&&&&&&&&\\
&&&&&&&&3&&5&&&&&&&\\
&&&&&&&7&&9&&11&&&&&&\\
&&&&&&13&&15&&17&&19&&&&&\\
&&&&&21&&23&&25&&27&&29&&&&\\
&&&&31&&33&&35&&37&&39&&41&&&\\
&&&&&\vdots&&&&\vdots&&&&\vdots&&&&\\
\end{array}" class="ltx_Math" display="inline" id="S0.Ex2.m1">
<mtable columnspacing="5pt" rowspacing="0pt">
<mtr>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
1
</mn>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
</mtr>
<mtr>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
3
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
5
</mn>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
</mtr>
<mtr>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
7
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
9
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
11
</mn>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
</mtr>
<mtr>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
13
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
15
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
17
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
19
</mn>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
</mtr>
<mtr>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
21
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
23
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
25
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
27
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
29
</mn>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
</mtr>
<mtr>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
31
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
33
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
35
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
37
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
39
</mn>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mn>
41
</mn>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
</mtr>
<mtr>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mi mathvariant="normal">
⋮
</mi>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mi mathvariant="normal">
⋮
</mi>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd columnalign="center">
<mi mathvariant="normal">
⋮
</mi>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
<mtd>
</mtd>
</mtr>
</mtable>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/sumofoddnumbers">
sum of odd numbers
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
SumOfOddNumbers
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:38:35
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:38:35
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
15
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
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pahio (2872)
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msc 00A05
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msc 11B25
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NumberOdd
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0 | Supercomputers | http://planetmath.org/Supercomputers | <!DOCTYPE html>
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supercomputers
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<h1 class="ltx_title ltx_title_document">
supercomputers
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
Supercomputers
</h2>
<div class="ltx_para" id="S1.p1">
<p class="ltx_p">
The usual description of a
<em class="ltx_emph ltx_font_italic">
supercomputer
</em>
is as one of the most advanced computers in terms of execution, or processing speed (particularly speed of calculation), as well as permanent storage capacity that can be rapidly accessed in ‘real’ time. Supercomputer operating systems are often variants of UNIX or the popular Linux. The base
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
language
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/language">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/signature">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of supercomputer code is often
<math alttext="C" class="ltx_Math" display="inline" id="S1.p1.m1">
<mi>
C
</mi>
</math>
, and sometimes
<math alttext="Fortran" class="ltx_Math" display="inline" id="S1.p1.m2">
<mrow>
<mi>
F
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
</math>
, also using special libraries to share data between computing nodes or ‘cores’. Some websites currently restrict this description of a supercomputer to mainframe ones, by ‘defining’ a supercomputer as: “a mainframe computer that is among the largest, fastest, or most powerful of those available at a given time”. For example, the Los Alamos system, nicknamed the IBM’s ‘Roadrunner’ (a
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Cluster.html">
cluster
</a>
of 3240 computers, each with 40 processing cores) may be a challenge for the Cray XT5 supercomputer at Oak Ridge National Laboratory called the ‘Jaguar’. The former system, only the second one to break the petaflop/s barrier, has a top performance of 1.059 petaflop/s in running the Linpack benchmark application, with a petaflop/s representing one quadrillion floating point
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
per second.
</p>
</div>
<div class="ltx_para" id="S1.p2">
<p class="ltx_p">
Thus, a
<em class="ltx_emph ltx_font_italic">
supercomputer
</em>
can be currently, and narrowly, described in practice as any computer capable of one quadrillion floating point operations per second when running the Linpack benchmark application. Obviously, this is a somewhat arbitrary choice, but is
<a class="nnexus_concept" href="http://planetmath.org/necessaryandsufficient">
sufficient
</a>
as a practical example of a present day supercomputer.
</p>
</div>
<div class="ltx_para" id="S1.p3">
<p class="ltx_p">
Any
<em class="ltx_emph ltx_font_italic">
computer
</em>
, including the
<em class="ltx_emph ltx_font_italic">
supercomputer
</em>
, can be however defined in general as a sixtuple
<math alttext="[(A),\pi]" class="ltx_Math" display="inline" id="S1.p3.m1">
<mrow>
<mo stretchy="false">
[
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
A
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
,
</mo>
<mi>
π
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
</math>
where
<math alttext="(A)" class="ltx_Math" display="inline" id="S1.p3.m2">
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
A
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
is a quintuple
<a class="nnexus_concept" href="http://planetmath.org/automaton">
automaton
</a>
(or sequential machine) which is capable of making calculations, usually by executing a set of logical, sequential instructions or (software)
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Program.html">
program
</a>
(s)
<math alttext="\pi" class="ltx_Math" display="inline" id="S1.p3.m3">
<mi>
π
</mi>
</math>
that can be all defined recursively. Let us recall here also that an automaton,
<math alttext="A" class="ltx_Math" display="inline" id="S1.p3.m4">
<mi>
A
</mi>
</math>
, is a five-tuple
<math alttext="(S,\Sigma,\delta,I,F)" class="ltx_Math" display="inline" id="S1.p3.m5">
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
S
</mi>
<mo>
,
</mo>
<mi mathvariant="normal">
Σ
</mi>
<mo>
,
</mo>
<mi>
δ
</mi>
<mo>
,
</mo>
<mi>
I
</mi>
<mo>
,
</mo>
<mi>
F
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
, consisting of:
</p>
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
a non-empty set
<math alttext="S" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mi>
S
</mi>
</math>
of (sequential machine) states,
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
a non-empty set
<math alttext="\Sigma" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mi mathvariant="normal">
Σ
</mi>
</math>
of symbols; a pair
<math alttext="(s,\alpha)" class="ltx_Math" display="inline" id="I1.i2.p1.m2">
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
s
</mi>
<mo>
,
</mo>
<mi>
α
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
of a state
<math alttext="s\in S" class="ltx_Math" display="inline" id="I1.i2.p1.m3">
<mrow>
<mi>
s
</mi>
<mo>
∈
</mo>
<mi>
S
</mi>
</mrow>
</math>
and a symbol
<math alttext="\alpha\in\Sigma" class="ltx_Math" display="inline" id="I1.i2.p1.m4">
<mrow>
<mi>
α
</mi>
<mo>
∈
</mo>
<mi mathvariant="normal">
Σ
</mi>
</mrow>
</math>
is called a
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
configuration
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Configuration.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/unlimitedregistermachine">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
,
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
a rule
<math alttext="\delta" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mi>
δ
</mi>
</math>
associating every configuration
<math alttext="(s,\alpha)" class="ltx_Math" display="inline" id="I1.i3.p1.m2">
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
s
</mi>
<mo>
,
</mo>
<mi>
α
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
a subset
<math alttext="\delta(s,\alpha)\subseteq S" class="ltx_Math" display="inline" id="I1.i3.p1.m3">
<mrow>
<mrow>
<mi>
δ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
s
</mi>
<mo>
,
</mo>
<mi>
α
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
⊆
</mo>
<mi>
S
</mi>
</mrow>
</math>
of states;
<math alttext="\delta" class="ltx_Math" display="inline" id="I1.i3.p1.m4">
<mi>
δ
</mi>
</math>
is called a
<em class="ltx_emph ltx_font_italic">
next-state
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
relation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relationonobjects">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
, or a
<em class="ltx_emph ltx_font_italic">
transition relation
</em>
,
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
a non-empty set
<math alttext="I\subseteq S" class="ltx_Math" display="inline" id="I1.i4.p1.m1">
<mrow>
<mi>
I
</mi>
<mo>
⊆
</mo>
<mi>
S
</mi>
</mrow>
</math>
of
<em class="ltx_emph ltx_font_italic">
starting states
</em>
, and
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
a set
<math alttext="F\subseteq S" class="ltx_Math" display="inline" id="I1.i5.p1.m1">
<mrow>
<mi>
F
</mi>
<mo>
⊆
</mo>
<mi>
S
</mi>
</mrow>
</math>
of
<em class="ltx_emph ltx_font_italic">
final states
</em>
or
<em class="ltx_emph ltx_font_italic">
terminating states
</em>
.
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="S1.p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Remark:
</span>
</p>
</div>
<div class="ltx_para" id="S1.p5">
<p class="ltx_p">
A supercomputer architecture –such as, currently, a cluster of MIMD multiprocessors, each processor of which is a SIMD– can be thus regarded as a concrete realization of an automata
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
supercategory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/supercategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/nsupercategorytheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<a class="nnexus_concept" href="http://planetmath.org/recurrencerelation">
first order
</a>
<math alttext="\S_{1}" class="ltx_Math" display="inline" id="S1.p5.m1">
<msub>
<mi mathvariant="normal">
§
</mi>
<mn>
1
</mn>
</msub>
</math>
.
</p>
</div>
<div class="ltx_para" id="S1.p6">
<p class="ltx_p">
A supercomputer is capable of running many programs in parallel, and at extremely high speeds for most programs. Thus, any computer and supercomputer can be at least in principle simulated by an
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Universal Turing Machine
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/UniversalTuringMachine.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/artificialintelligence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universalturingmachine">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
(UTM). Quite remarkably, the UTM cannot do this for the thinking human brain, and thus the human brain cannot be
adequately described as ‘one ¡giant¿ supercomputer’.
</p>
</div>
<div class="ltx_para" id="S1.p7">
<p class="ltx_p">
There are also however some dissenting contenders who claim that
<span class="ltx_text ltx_font_typewriter">
http://www.nvidia.com/object/personal_supercomputing.html
</span>
personal supercomputing
is also possible, and that such personal supercomputers are also becoming available, which are not mainframes in the established sense of the word, (see for example:
<span class="ltx_text ltx_font_typewriter">
http://www.top500.org/
</span>
‘TOP500 Supercomputing Sites’). However, the top computing speed of such ‘personal supercomputers’ which are parallel processing
<a class="nnexus_concept" href="http://planetmath.org/categoryofautomata">
machines
</a>
is in the range of only 4 teraflop/s, still some 250 times faster than the fastest PCs.
Thus, one notes that an inexpensive model quad-core Xeon workstation running at 2.66 GHz will outperform a multimillion dollar Cray C90 supercomputer that was used in the early 1990’s much for the same tasks.
</p>
</div>
<div class="ltx_para" id="S1.p8">
<p class="ltx_p">
Whereas many ‘pure’ mathematicians or theoretical physicists may claim that
they do not need any supercomputer, this is not the case of the majority of applied mathematicians, computer scientists, mathematical and/or experimental physicists and engineers who find increasinly the need for having access to a supercomputer: the faster the device, and the more
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
accessible
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/simplifiedautomaton">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/kripkesemantics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
it is, the better the supercomputer is–one petaflop today, and hopefully one thousand petaflops tomorrow, with
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Petabyte.html">
petabyte
</a>
storage capability that’s not too slow to access.
</p>
</div>
<div class="ltx_para" id="S1.p9">
<p class="ltx_p">
Unclassified supercomputing is currently used “for highly calculation-intensive tasks such as problems involving quantum physics” (QCD, QED, QFT, AQFT), molecular dynamics and modeling/computing
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structures
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and properties of small
molecules, polymers, biopolymers/biological macromolecules (or even crystal lattices), physical
<a class="nnexus_concept" href="http://planetmath.org/bisimulation">
simulations
</a>
of all kinds
including fluid dynamics and aerodynamic testing, ‘long-term’ weather forecasting and climate research, theoretical nuclear fusion research, cryptographic
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Analysis.html">
analysis
</a>
, and lots more. Supercomputers are increasingly utilized at all major universities and scientific research laboratories for both academic and industrial purposes.
One also remembers that parallel processing was the key in the race for assembling the first human genome
map in two parallel projects that topped a billion dollars over a five year period; today, the cost of a single human ‘whole’ genome analysis has dropped by a factor of one million, and the analysis time has been
<a class="nnexus_concept" href="http://planetmath.org/reducedautomaton">
reduced
</a>
to one day.
</p>
</div>
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supercomputers
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50
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IndexOfCategories
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Defines
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supercomputer
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sequential machine
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0 | SurrealNumber | http://planetmath.org/SurrealNumber | <!DOCTYPE html>
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<title>
surreal number
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
surreal number
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
surreal numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SurrealNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/surrealnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
are a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
generalization
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/hilbertsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiomsystemforfirstorderlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the reals. Each surreal number consists of two parts (called the left and right), each of which is a set of surreal numbers. For any surreal number
<math alttext="N" class="ltx_Math" display="inline" id="p1.m1">
<mi>
N
</mi>
</math>
, these parts can be called
<math alttext="N_{L}" class="ltx_Math" display="inline" id="p1.m2">
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</math>
and
<math alttext="N_{R}" class="ltx_Math" display="inline" id="p1.m3">
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
</math>
. (This could be viewed as an
<a class="nnexus_concept" href="http://planetmath.org/orderedpair">
ordered pair
</a>
of sets, however the surreal numbers were intended to be a basis for mathematics, not something to be embedded in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
set theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SetTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/settheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.) A surreal number is written
<math alttext="N=\langle N_{L}\mid N_{R}\rangle" class="ltx_Math" display="inline" id="p1.m4">
<mrow>
<mi>
N
</mi>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
<mo>
∣
</mo>
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Not every number of this form is a surreal number. The surreal numbers
<a class="nnexus_concept" href="http://planetmath.org/satisfactionrelation">
satisfy
</a>
two additional
<a class="nnexus_concept" href="http://planetmath.org/property">
properties
</a>
. First, if
<math alttext="x\in N_{R}" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
</mrow>
</math>
and
<math alttext="y\in N_{L}" class="ltx_Math" display="inline" id="p2.m2">
<mrow>
<mi>
y
</mi>
<mo>
∈
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</mrow>
</math>
then
<math alttext="x\nleq y" class="ltx_Math" display="inline" id="p2.m3">
<mrow>
<mi>
x
</mi>
<mo>
≰
</mo>
<mi>
y
</mi>
</mrow>
</math>
. Secondly, they must be well founded. These properties are both satisfied by the following construction of the surreal numbers and the
<math alttext="\leq" class="ltx_Math" display="inline" id="p2.m4">
<mo>
≤
</mo>
</math>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
relation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
by mutual induction:
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
<math alttext="\langle\mid\rangle" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mo stretchy="false">
⟨
</mo>
<mo>
∣
</mo>
<mo stretchy="false">
⟩
</mo>
</mrow>
</math>
, which has both left and right parts empty, is
<math alttext="0" class="ltx_Math" display="inline" id="p3.m2">
<mn>
0
</mn>
</math>
.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Given two (possibly empty) sets of surreal numbers
<math alttext="R" class="ltx_Math" display="inline" id="p4.m1">
<mi>
R
</mi>
</math>
and
<math alttext="L" class="ltx_Math" display="inline" id="p4.m2">
<mi>
L
</mi>
</math>
such that for any
<math alttext="x\in R" class="ltx_Math" display="inline" id="p4.m3">
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
R
</mi>
</mrow>
</math>
and
<math alttext="y\in L" class="ltx_Math" display="inline" id="p4.m4">
<mrow>
<mi>
y
</mi>
<mo>
∈
</mo>
<mi>
L
</mi>
</mrow>
</math>
,
<math alttext="x\nleq y" class="ltx_Math" display="inline" id="p4.m5">
<mrow>
<mi>
x
</mi>
<mo>
≰
</mo>
<mi>
y
</mi>
</mrow>
</math>
,
<math alttext="\langle L\mid R\rangle" class="ltx_Math" display="inline" id="p4.m6">
<mrow>
<mo stretchy="false">
⟨
</mo>
<mi>
L
</mi>
<mo>
∣
</mo>
<mi>
R
</mi>
<mo stretchy="false">
⟩
</mo>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
Define
<math alttext="N\leq M" class="ltx_Math" display="inline" id="p5.m1">
<mrow>
<mi>
N
</mi>
<mo>
≤
</mo>
<mi>
M
</mi>
</mrow>
</math>
if there is no
<math alttext="x\in N_{L}" class="ltx_Math" display="inline" id="p5.m2">
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</mrow>
</math>
such that
<math alttext="M\leq x" class="ltx_Math" display="inline" id="p5.m3">
<mrow>
<mi>
M
</mi>
<mo>
≤
</mo>
<mi>
x
</mi>
</mrow>
</math>
and no
<math alttext="y\in M_{R}" class="ltx_Math" display="inline" id="p5.m4">
<mrow>
<mi>
y
</mi>
<mo>
∈
</mo>
<msub>
<mi>
M
</mi>
<mi>
R
</mi>
</msub>
</mrow>
</math>
such that
<math alttext="y\leq N" class="ltx_Math" display="inline" id="p5.m5">
<mrow>
<mi>
y
</mi>
<mo>
≤
</mo>
<mi>
N
</mi>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
This process can be continued transfinitely, to define
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinite
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinite.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/infinite">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and infinitesimal numbers. For
<a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">
instance
</a>
if
<math alttext="\mathbb{Z}" class="ltx_Math" display="inline" id="p6.m1">
<mi>
ℤ
</mi>
</math>
is the set of integers then
<math alttext="\omega=\langle\mathbb{Z}\mid\rangle" class="ltx_Math" display="inline" id="p6.m2">
<mrow>
<mi>
ω
</mi>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mi>
ℤ
</mi>
<mo>
∣
</mo>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
. Note that this does not make equality the same as
<a class="nnexus_concept" href="http://planetmath.org/multivaluedfunction">
identity
</a>
:
<math alttext="\langle 1\mid 1\rangle=\langle\mid\rangle" class="ltx_Math" display="inline" id="p6.m3">
<mrow>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mn>
1
</mn>
<mo>
∣
</mo>
<mn>
1
</mn>
<mo stretchy="false">
⟩
</mo>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mo>
∣
</mo>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
, for instance.
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
It can be shown that
<math alttext="N" class="ltx_Math" display="inline" id="p7.m1">
<mi>
N
</mi>
</math>
is “sandwiched” between the elements of
<math alttext="N_{L}" class="ltx_Math" display="inline" id="p7.m2">
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</math>
and
<math alttext="N_{R}" class="ltx_Math" display="inline" id="p7.m3">
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
</math>
: it is larger than any element of
<math alttext="N_{L}" class="ltx_Math" display="inline" id="p7.m4">
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</math>
and smaller than any element of
<math alttext="N_{R}" class="ltx_Math" display="inline" id="p7.m5">
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
</math>
.
</p>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of surreal numbers is defined by
</p>
</div>
<div class="ltx_para" id="p9">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="N+M=\langle\{N+x\mid x\in M_{L}\}\cup\{M+x\mid y\in N_{L}\}\mid\{N+x\mid x\in M%
_{R}\}\cup\{M+x\mid y\in N_{R}\}\rangle" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mi>
N
</mi>
<mo>
+
</mo>
<mi>
M
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mrow>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
N
</mi>
<mo>
+
</mo>
<mi>
x
</mi>
</mrow>
<mo>
∣
</mo>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<msub>
<mi>
M
</mi>
<mi>
L
</mi>
</msub>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
<mo>
∪
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
M
</mi>
<mo>
+
</mo>
<mi>
x
</mi>
</mrow>
<mo>
∣
</mo>
<mrow>
<mi>
y
</mi>
<mo>
∈
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
<mo>
∣
</mo>
<mrow>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
N
</mi>
<mo>
+
</mo>
<mi>
x
</mi>
</mrow>
<mo>
∣
</mo>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<msub>
<mi>
M
</mi>
<mi>
R
</mi>
</msub>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
<mo>
∪
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
M
</mi>
<mo>
+
</mo>
<mi>
x
</mi>
</mrow>
<mo>
∣
</mo>
<mrow>
<mi>
y
</mi>
<mo>
∈
</mo>
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p10">
<p class="ltx_p">
It follows that
<math alttext="-N=\langle-N_{R}\mid-N_{L}\rangle" class="ltx_Math" display="inline" id="p10.m1">
<mrow>
<mrow>
<mo>
-
</mo>
<mi>
N
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mrow>
<mo>
-
</mo>
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
</mrow>
<mo>
∣
</mo>
<mrow>
<mo>
-
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</mrow>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p11">
<p class="ltx_p">
The
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
of
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
multiplication
</a>
can be written more easily by defining
<math alttext="M\cdot N_{L}=\{M\cdot x\mid x\in N_{L}\}" class="ltx_Math" display="inline" id="p11.m1">
<mrow>
<mrow>
<mi>
M
</mi>
<mo>
⋅
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
M
</mi>
<mo>
⋅
</mo>
<mi>
x
</mi>
</mrow>
<mo>
∣
</mo>
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
</math>
and similarly for
<math alttext="N_{R}" class="ltx_Math" display="inline" id="p11.m2">
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
</math>
.
</p>
</div>
<div class="ltx_para" id="p12">
<p class="ltx_p">
Then
</p>
</div>
<div class="ltx_para" id="p13">
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex2">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle N\cdot M=" class="ltx_Math" display="inline" id="S0.Ex2.m1">
<mrow>
<mrow>
<mi>
N
</mi>
<mo>
⋅
</mo>
<mi>
M
</mi>
</mrow>
<mo>
=
</mo>
<mi>
</mi>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle\langle M\cdot N_{L}+N\cdot M_{L}-N_{L}\cdot M_{L},M\cdot N_{R}+N%
\cdot M_{R}-N_{R}\cdot M_{R}\mid" class="ltx_Math" display="inline" id="S0.Ex2.m2">
<mrow>
<mo stretchy="false">
⟨
</mo>
<mrow>
<mrow>
<mrow>
<mrow>
<mi>
M
</mi>
<mo>
⋅
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
</mrow>
<mo>
+
</mo>
<mrow>
<mi>
N
</mi>
<mo>
⋅
</mo>
<msub>
<mi>
M
</mi>
<mi>
L
</mi>
</msub>
</mrow>
</mrow>
<mo>
-
</mo>
<mrow>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
<mo>
⋅
</mo>
<msub>
<mi>
M
</mi>
<mi>
L
</mi>
</msub>
</mrow>
</mrow>
<mo>
,
</mo>
<mrow>
<mrow>
<mrow>
<mi>
M
</mi>
<mo>
⋅
</mo>
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
</mrow>
<mo>
+
</mo>
<mrow>
<mi>
N
</mi>
<mo>
⋅
</mo>
<msub>
<mi>
M
</mi>
<mi>
R
</mi>
</msub>
</mrow>
</mrow>
<mo>
-
</mo>
<mrow>
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
<mo>
⋅
</mo>
<msub>
<mi>
M
</mi>
<mi>
R
</mi>
</msub>
</mrow>
</mrow>
</mrow>
<mo fence="true">
∣
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex3">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle M\cdot N_{L}+N\cdot M_{R}-N_{L}\cdot M_{R},M\cdot N_{R}+N\cdot M%
_{L}-N_{R}\cdot M_{L}\rangle" class="ltx_Math" display="inline" id="S0.Ex3.m2">
<mrow>
<mi>
M
</mi>
<mo>
⋅
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
<mo>
+
</mo>
<mi>
N
</mi>
<mo>
⋅
</mo>
<msub>
<mi>
M
</mi>
<mi>
R
</mi>
</msub>
<mo>
-
</mo>
<msub>
<mi>
N
</mi>
<mi>
L
</mi>
</msub>
<mo>
⋅
</mo>
<msub>
<mi>
M
</mi>
<mi>
R
</mi>
</msub>
<mo>
,
</mo>
<mi>
M
</mi>
<mo>
⋅
</mo>
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
<mo>
+
</mo>
<mi>
N
</mi>
<mo>
⋅
</mo>
<msub>
<mi>
M
</mi>
<mi>
L
</mi>
</msub>
<mo>
-
</mo>
<msub>
<mi>
N
</mi>
<mi>
R
</mi>
</msub>
<mo>
⋅
</mo>
<msub>
<mi>
M
</mi>
<mi>
L
</mi>
</msub>
<mo stretchy="false">
⟩
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p14">
<p class="ltx_p">
The surreal numbers satisfy the axioms for a field under addition and multiplication (whether they really are a field is complicated by the fact that they are too large to be a set).
</p>
</div>
<div class="ltx_para" id="p15">
<p class="ltx_p">
The
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
of surreal mathematics are called the
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/OmnificInteger.html">
omnific integers
</a>
</em>
. In general
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PositiveInteger.html">
positive integers
</a>
<math alttext="n" class="ltx_Math" display="inline" id="p15.m1">
<mi>
n
</mi>
</math>
can always be written
<math alttext="\langle n-1\mid\rangle" class="ltx_Math" display="inline" id="p15.m2">
<mrow>
<mo stretchy="false">
⟨
</mo>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
<mo>
∣
</mo>
<mo stretchy="false">
⟩
</mo>
</mrow>
</math>
and so
<math alttext="-n=\langle\mid 1-n\rangle=\langle\mid(-n)+1\rangle" class="ltx_Math" display="inline" id="p15.m3">
<mrow>
<mo>
-
</mo>
<mi>
n
</mi>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mo>
∣
</mo>
<mn>
1
</mn>
<mo>
-
</mo>
<mi>
n
</mi>
<mo stretchy="false">
⟩
</mo>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mo>
∣
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mo>
-
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
+
</mo>
<mn>
1
</mn>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
. So for instance
<math alttext="1=\langle 0\mid\rangle" class="ltx_Math" display="inline" id="p15.m4">
<mrow>
<mn>
1
</mn>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mn>
0
</mn>
<mo>
∣
</mo>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p16">
<p class="ltx_p">
In general,
<math alttext="\langle a\mid b\rangle" class="ltx_Math" display="inline" id="p16.m1">
<mrow>
<mo stretchy="false">
⟨
</mo>
<mi>
a
</mi>
<mo>
∣
</mo>
<mi>
b
</mi>
<mo stretchy="false">
⟩
</mo>
</mrow>
</math>
is the simplest number between
<math alttext="a" class="ltx_Math" display="inline" id="p16.m2">
<mi>
a
</mi>
</math>
and
<math alttext="b" class="ltx_Math" display="inline" id="p16.m3">
<mi>
b
</mi>
</math>
. This can be easily used to define the dyadic fractions: for any integer
<math alttext="a" class="ltx_Math" display="inline" id="p16.m4">
<mi>
a
</mi>
</math>
,
<math alttext="a+\frac{1}{2}=\langle a\mid a+1\rangle" class="ltx_Math" display="inline" id="p16.m5">
<mrow>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mi>
a
</mi>
<mo>
∣
</mo>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
. Then
<math alttext="\frac{1}{2}=\langle 0\mid 1\rangle" class="ltx_Math" display="inline" id="p16.m6">
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mn>
0
</mn>
<mo>
∣
</mo>
<mn>
1
</mn>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
,
<math alttext="\frac{1}{4}=\langle 0\mid\frac{1}{2}\rangle" class="ltx_Math" display="inline" id="p16.m7">
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
4
</mn>
</mfrac>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mn>
0
</mn>
<mo>
∣
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo stretchy="false">
⟩
</mo>
</mrow>
</mrow>
</math>
, and so on. This can then be used to locate non-dyadic
<a class="nnexus_concept" href="http://planetmath.org/fraction">
fractions
</a>
by pinning them between a left part which gets infinitely close from below and a right part which gets infinitely close from above.
</p>
</div>
<div class="ltx_para" id="p17">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/ordinalarithmetic">
Ordinal arithmetic
</a>
can be defined starting with
<math alttext="\omega" class="ltx_Math" display="inline" id="p17.m1">
<mi>
ω
</mi>
</math>
as defined above and adding numbers such as
<math alttext="\langle\omega\mid\rangle=\omega+1" class="ltx_Math" display="inline" id="p17.m2">
<mrow>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mi>
ω
</mi>
<mo>
∣
</mo>
<mo stretchy="false">
⟩
</mo>
</mrow>
<mo>
=
</mo>
<mi>
ω
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
and so on. Similarly, a starting
<a class="nnexus_concept" href="http://planetmath.org/infinitesimal">
infinitesimal
</a>
can be found as
<math alttext="\langle 0\mid 1,\frac{1}{2},\frac{1}{4}\ldots\rangle=\frac{1}{\omega}" class="ltx_Math" display="inline" id="p17.m3">
<mrow>
<mrow>
<mo stretchy="false">
⟨
</mo>
<mn>
0
</mn>
<mo>
∣
</mo>
<mrow>
<mn>
1
</mn>
<mo>
,
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
,
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
4
</mn>
</mfrac>
<mo>
</mo>
<mi mathvariant="normal">
…
</mi>
</mrow>
</mrow>
<mo stretchy="false">
⟩
</mo>
</mrow>
<mo>
=
</mo>
<mfrac>
<mn>
1
</mn>
<mi>
ω
</mi>
</mfrac>
</mrow>
</math>
, and again more can be developed from there.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
surreal number
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
SurrealNumber
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:58:49
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:58:49
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
9
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Henry (455)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
omnific integers
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
<footer class="ltx_page_footer">
<div class="ltx_page_logo">
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0 | TimeInvariant | http://planetmath.org/TimeInvariant | <!DOCTYPE html>
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<title>
time invariant
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
time invariant
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
dynamical system
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DynamicalSystem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dynamicalsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/groupoidcdynamicalsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is
<span class="ltx_text ltx_font_bold">
time-invariant
</span>
if its generating formula is dependent on state only, and independent of time. A synonym for time-invariant is
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
autonomous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Autonomous.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/autonomoussystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. The complement of time-invariant is time-varying (or nonautonomous).
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
For example, the continuous-time system
<math alttext="\dot{x}=f(x,t)" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<mover accent="true">
<mi>
x
</mi>
<mo>
˙
</mo>
</mover>
<mo>
=
</mo>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo>
,
</mo>
<mi>
t
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
is time-invariant if and only if
<math alttext="f(x,t_{1})\equiv f(x,t_{2})" class="ltx_Math" display="inline" id="p2.m2">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo>
,
</mo>
<msub>
<mi>
t
</mi>
<mn>
1
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
≡
</mo>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo>
,
</mo>
<msub>
<mi>
t
</mi>
<mn>
2
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
for all valid states
<math alttext="x" class="ltx_Math" display="inline" id="p2.m3">
<mi>
x
</mi>
</math>
and times
<math alttext="t_{1}" class="ltx_Math" display="inline" id="p2.m4">
<msub>
<mi>
t
</mi>
<mn>
1
</mn>
</msub>
</math>
and
<math alttext="t_{2}" class="ltx_Math" display="inline" id="p2.m5">
<msub>
<mi>
t
</mi>
<mn>
2
</mn>
</msub>
</math>
. Thus
<math alttext="\dot{x}=\sin x" class="ltx_Math" display="inline" id="p2.m6">
<mrow>
<mover accent="true">
<mi>
x
</mi>
<mo>
˙
</mo>
</mover>
<mo>
=
</mo>
<mrow>
<mi>
sin
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
</math>
is time-invariant, while
<math alttext="\dot{x}=\frac{\sin x}{1+t}" class="ltx_Math" display="inline" id="p2.m7">
<mrow>
<mover accent="true">
<mi>
x
</mi>
<mo>
˙
</mo>
</mover>
<mo>
=
</mo>
<mfrac>
<mrow>
<mi>
sin
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mi>
t
</mi>
</mrow>
</mfrac>
</mrow>
</math>
is time-varying.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Likewise, the discrete-time system
<math alttext="x[n]=f[x,n]" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
n
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
x
</mi>
<mo>
,
</mo>
<mi>
n
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
</mrow>
</math>
is time-invariant (also called shift-invariant) if and only if
<math alttext="f[x,n_{1}]\equiv f[x,n_{2}]" class="ltx_Math" display="inline" id="p3.m2">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
x
</mi>
<mo>
,
</mo>
<msub>
<mi>
n
</mi>
<mn>
1
</mn>
</msub>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
<mo>
≡
</mo>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
x
</mi>
<mo>
,
</mo>
<msub>
<mi>
n
</mi>
<mn>
2
</mn>
</msub>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
</mrow>
</math>
for all valid states
<math alttext="x" class="ltx_Math" display="inline" id="p3.m3">
<mi>
x
</mi>
</math>
and time indices
<math alttext="n_{1}" class="ltx_Math" display="inline" id="p3.m4">
<msub>
<mi>
n
</mi>
<mn>
1
</mn>
</msub>
</math>
and
<math alttext="n_{2}" class="ltx_Math" display="inline" id="p3.m5">
<msub>
<mi>
n
</mi>
<mn>
2
</mn>
</msub>
</math>
. Thus
<math alttext="x[n]=2x[n-1]" class="ltx_Math" display="inline" id="p3.m6">
<mrow>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
n
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
x
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
</mrow>
</math>
is time-invariant, while
<math alttext="x[n]=2nx[n-1]" class="ltx_Math" display="inline" id="p3.m7">
<mrow>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
n
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
n
</mi>
<mo>
</mo>
<mi>
x
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
</mrow>
</math>
is time-varying.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
time invariant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
TimeInvariant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:02:14
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:02:14
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mathprof (13753)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mathprof (13753)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mathprof (13753)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
AutonomousSystem
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
time-invariant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
shift-invariant
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | Variable | http://planetmath.org/Variable | <!DOCTYPE html>
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<head>
<title>
variable
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
variable
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The word
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/variable">
variable
</a>
</em>
as used in mathematics (and in other scientific fields that use mathematics) is somewhat vague and may have different meanings depending on the
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
context
</a>
. Variables
are usually denoted by a single Roman or Greek letter, e.g.
<math alttext="x" class="ltx_Math" display="inline" id="p1.m1">
<mi>
x
</mi>
</math>
,
although sometimes a whole word or phrase can be used also.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Here is a list of some of the meanings of
<em class="ltx_emph ltx_font_italic">
variable
</em>
:
</p>
</div>
<div class="ltx_para" id="p3">
<dl class="ltx_description" id="I1">
<dt class="ltx_item" id="I1.ix1">
<span class="ltx_tag ltx_tag_description">
(i) As “mathematical” variables.
</span>
</dt>
<dd class="ltx_item">
<div class="ltx_para" id="I1.ix1.p1">
<p class="ltx_p">
These stand for a concrete object, for example, an element of the
<a class="nnexus_concept" href="http://planetmath.org/realnumber">
real numbers
</a>
That is, when we write the symbol
<math alttext="x" class="ltx_Math" display="inline" id="I1.ix1.p1.m1">
<mi>
x
</mi>
</math>
, it is a stand-in for various numbers:
e.g.
<math alttext="2,3,\pi,e,578.24" class="ltx_Math" display="inline" id="I1.ix1.p1.m2">
<mrow>
<mn>
2
</mn>
<mo>
,
</mo>
<mn>
3
</mn>
<mo>
,
</mo>
<mi>
π
</mi>
<mo>
,
</mo>
<mi>
e
</mi>
<mo>
,
</mo>
<mn>
578.24
</mn>
</mrow>
</math>
.
But we do not name these numbers specifically, because we may want to talk
about all these numbers at once, in a general statement,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, or proof
about numbers.
</p>
</div>
<div class="ltx_para" id="I1.ix1.p2">
<p class="ltx_p">
Sense (i) is probably the most common usage in mainstream mathematics.
</p>
</div>
</dd>
<dt class="ltx_item" id="I1.ix2">
<span class="ltx_tag ltx_tag_description">
(ii) As placeholders in functional notation.
</span>
</dt>
<dd class="ltx_item">
<div class="ltx_para" id="I1.ix2.p1">
<p class="ltx_p">
For example, we may be defining a
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
using the phrase
“define the function
<math alttext="f(z)=z^{2}+4" class="ltx_Math" display="inline" id="I1.ix2.p1.m1">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
z
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
z
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mn>
4
</mn>
</mrow>
</mrow>
</math>
for
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complex numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complexnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="z" class="ltx_Math" display="inline" id="I1.ix2.p1.m2">
<mi>
z
</mi>
</math>
.
This usage of a variable is slightly different from sense (i),
because our objective is to talk about the
<em class="ltx_emph ltx_font_italic">
function
</em>
<math alttext="f" class="ltx_Math" display="inline" id="I1.ix2.p1.m3">
<mi>
f
</mi>
</math>
,
<em class="ltx_emph ltx_font_italic">
and not its value
</em>
at a number
<math alttext="z" class="ltx_Math" display="inline" id="I1.ix2.p1.m4">
<mi>
z
</mi>
</math>
which is
<math alttext="f(z)" class="ltx_Math" display="inline" id="I1.ix2.p1.m5">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
z
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
. The notation “
<math alttext="f(z)=z^{2}+4" class="ltx_Math" display="inline" id="I1.ix2.p1.m6">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
z
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
z
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mn>
4
</mn>
</mrow>
</mrow>
</math>
”
is merely a much more convenient way of saying: “define the function
<math alttext="f" class="ltx_Math" display="inline" id="I1.ix2.p1.m7">
<mi>
f
</mi>
</math>
which takes a complex number, multiplies it by itself, and then adds four to it”.
It could also be rephrased this way:
“define a function
<math alttext="f" class="ltx_Math" display="inline" id="I1.ix2.p1.m8">
<mi>
f
</mi>
</math>
such that the statement
<math alttext="f(z)=z^{2}+4" class="ltx_Math" display="inline" id="I1.ix2.p1.m9">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
z
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
z
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mn>
4
</mn>
</mrow>
</mrow>
</math>
is true for all complex numbers
<math alttext="z" class="ltx_Math" display="inline" id="I1.ix2.p1.m10">
<mi>
z
</mi>
</math>
(in sense (i))”.
</p>
</div>
<div class="ltx_para" id="I1.ix2.p2">
<p class="ltx_p">
On the other hand, the symbol
<math alttext="f" class="ltx_Math" display="inline" id="I1.ix2.p2.m1">
<mi>
f
</mi>
</math>
, if we were to contemplate it as a “variable”,
arguably belongs to the sense (i); in this case we are talking about
<em class="ltx_emph ltx_font_italic">
some specific function
</em>
, not all functions.
</p>
</div>
</dd>
<dt class="ltx_item" id="I1.ix3">
<span class="ltx_tag ltx_tag_description">
(iii) As “formal” variables.
</span>
</dt>
<dd class="ltx_item">
<div class="ltx_para" id="I1.ix3.p1">
<p class="ltx_p">
For
<a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">
instance
</a>
, we may talk about a formal
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
polynomial
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/polynomial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomialring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="p(x)=1+x+x^{2}" class="ltx_Math" display="inline" id="I1.ix3.p1.m1">
<mrow>
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mi>
x
</mi>
<mo>
+
</mo>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mrow>
</math>
.
This is
<a class="nnexus_concept" href="http://planetmath.org/equivalentmachines">
similar
</a>
to sense (ii), but is not exactly the same.
The variable
<math alttext="x" class="ltx_Math" display="inline" id="I1.ix3.p1.m2">
<mi>
x
</mi>
</math>
here is not necessarily a complex number, or in any fixed
domain at all. It is a formal symbol, which we later replace by
actual elements of the real numbers, or matrices, etc. at our whim.
And
<math alttext="p" class="ltx_Math" display="inline" id="I1.ix3.p1.m3">
<mi>
p
</mi>
</math>
here is
<em class="ltx_emph ltx_font_italic">
not a function
</em>
; it is a polynomial.
</p>
</div>
<div class="ltx_para" id="I1.ix3.p2">
<p class="ltx_p">
The variables used in
<a class="nnexus_concept" href="http://planetmath.org/logicism">
formal logic
</a>
can also be considered to fall in sense (iii).
For example, we may have a set of variables
<math alttext="\{x,y,z\}" class="ltx_Math" display="inline" id="I1.ix3.p2.m1">
<mrow>
<mo stretchy="false">
{
</mo>
<mi>
x
</mi>
<mo>
,
</mo>
<mi>
y
</mi>
<mo>
,
</mo>
<mi>
z
</mi>
<mo stretchy="false">
}
</mo>
</mrow>
</math>
and a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
formula
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Formula.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/firstorderlanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
from the first-order language using such variables:
<math alttext="(\exists x((x\leq 0)\land\mathrm{R}(z))" class="ltx_Math" display="inline" id="I1.ix3.p2.m2">
<mrow>
<mo stretchy="false">
(
</mo>
<mo>
∃
</mo>
<mi>
x
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo>
≤
</mo>
<mn>
0
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
∧
</mo>
<mi mathvariant="normal">
R
</mi>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
z
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
</dd>
<dt class="ltx_item" id="I1.ix4">
<span class="ltx_tag ltx_tag_description">
(iv) As pieces of (experimental) data.
</span>
</dt>
<dd class="ltx_item">
<div class="ltx_para" id="I1.ix4.p1">
<p class="ltx_p">
Used in the sciences. One may say “at
<math alttext="t=4\>\mathrm{s}" class="ltx_Math" display="inline" id="I1.ix4.p1.m1">
<mrow>
<mi>
t
</mi>
<mo>
=
</mo>
<mrow>
<mpadded width="+2.2pt">
<mn>
4
</mn>
</mpadded>
<mo>
</mo>
<mi mathvariant="normal">
s
</mi>
</mrow>
</mrow>
</math>
,
<math alttext="x=23.1\>\mathrm{m}" class="ltx_Math" display="inline" id="I1.ix4.p1.m2">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mpadded width="+2.2pt">
<mn>
23.1
</mn>
</mpadded>
<mo>
</mo>
<mi mathvariant="normal">
m
</mi>
</mrow>
</mrow>
</math>
”
which may really mean: “at 4 seconds from the start of the experiment,
the object is 23.1 metres to the right of its initial position”.
</p>
</div>
<div class="ltx_para" id="I1.ix4.p2">
<p class="ltx_p">
So the symbols
<math alttext="t" class="ltx_Math" display="inline" id="I1.ix4.p2.m1">
<mi>
t
</mi>
</math>
and
<math alttext="x" class="ltx_Math" display="inline" id="I1.ix4.p2.m2">
<mi>
x
</mi>
</math>
are being used in the meaning
of “time” and “position” in general.
There may or may not be a functional relation between
the “variables”
<math alttext="t" class="ltx_Math" display="inline" id="I1.ix4.p2.m3">
<mi>
t
</mi>
</math>
and
<math alttext="x" class="ltx_Math" display="inline" id="I1.ix4.p2.m4">
<mi>
x
</mi>
</math>
. If there is, we might say
“
<math alttext="x" class="ltx_Math" display="inline" id="I1.ix4.p2.m5">
<mi>
x
</mi>
</math>
is a function of
<math alttext="t" class="ltx_Math" display="inline" id="I1.ix4.p2.m6">
<mi>
t
</mi>
</math>
”, and we can talk about
quantities such as
<math alttext="dx/dt" class="ltx_Math" display="inline" id="I1.ix4.p2.m7">
<mrow>
<mrow>
<mrow>
<mi>
d
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
/
</mo>
<mi>
d
</mi>
</mrow>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="I1.ix4.p3">
<p class="ltx_p">
If we want to talk about a specific (but unnamed) time,
we can use a notation such as “when
<math alttext="t=t_{0},\ldots" class="ltx_Math" display="inline" id="I1.ix4.p3.m1">
<mrow>
<mi>
t
</mi>
<mo>
=
</mo>
<mrow>
<msub>
<mi>
t
</mi>
<mn>
0
</mn>
</msub>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
</mrow>
</mrow>
</math>
”
for some variable
<math alttext="t_{0}" class="ltx_Math" display="inline" id="I1.ix4.p3.m2">
<msub>
<mi>
t
</mi>
<mn>
0
</mn>
</msub>
</math>
in sense (i).
</p>
</div>
<div class="ltx_para" id="I1.ix4.p4">
<p class="ltx_p">
The field of probability and statistics follows a similar
practice for what are termed “
<a class="nnexus_concept" href="http://planetmath.org/randomvariable">
random variables
</a>
”,
which are really functions defined on a measure space
<math alttext="\Omega" class="ltx_Math" display="inline" id="I1.ix4.p4.m1">
<mi mathvariant="normal">
Ω
</mi>
</math>
.
But in practice they are usually denoted with variable notation:
e.g. “the random variable
<math alttext="X" class="ltx_Math" display="inline" id="I1.ix4.p4.m2">
<mi>
X
</mi>
</math>
”, and a specific value of this
random variable
<math alttext="X" class="ltx_Math" display="inline" id="I1.ix4.p4.m3">
<mi>
X
</mi>
</math>
, at some unspecified
<math alttext="\omega\in\Omega" class="ltx_Math" display="inline" id="I1.ix4.p4.m4">
<mrow>
<mi>
ω
</mi>
<mo>
∈
</mo>
<mi mathvariant="normal">
Ω
</mi>
</mrow>
</math>
,
is denoted by
<math alttext="x" class="ltx_Math" display="inline" id="I1.ix4.p4.m5">
<mi>
x
</mi>
</math>
.
</p>
</div>
</dd>
<dt class="ltx_item" id="I1.ix5">
<span class="ltx_tag ltx_tag_description">
(v) As state variables in
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computer
</a>
algorithms.
</span>
</dt>
<dd class="ltx_item">
<div class="ltx_para" id="I1.ix5.p1">
<p class="ltx_p">
In this case, a variable
<math alttext="x" class="ltx_Math" display="inline" id="I1.ix5.p1.m1">
<mi>
x
</mi>
</math>
stands for a computer memory location.
Or in more abstract
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
language
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/language">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/signature">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<math alttext="x" class="ltx_Math" display="inline" id="I1.ix5.p1.m2">
<mi>
x
</mi>
</math>
is a name for a container
which may hold some object.
The contents of this container may change as time passes
or when it is modified by a program that the computer is executing.
</p>
</div>
<div class="ltx_para" id="I1.ix5.p2">
<p class="ltx_p">
In
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
formal language
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FormalLanguage.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/formalgrammar">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, putting a value in the container
is often denoted by notation like “
<math alttext="x\leftarrow 2" class="ltx_Math" display="inline" id="I1.ix5.p2.m1">
<mrow>
<mi>
x
</mi>
<mo>
←
</mo>
<mn>
2
</mn>
</mrow>
</math>
”.
</p>
</div>
</dd>
</dl>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Note that the above distinctions are not always clear-cut.
and the same symbol
<math alttext="x" class="ltx_Math" display="inline" id="p4.m1">
<mi>
x
</mi>
</math>
may be
used for different purposes at once,
which of course, may lead to confusion.
</p>
</div>
<div class="ltx_para ltx_align_right" id="p5">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
variable
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Variable
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:31:39
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:31:39
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
stevecheng (10074)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
stevecheng (10074)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
stevecheng (10074)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/parametre">
Parameter
</a>
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | VennDiagram | http://planetmath.org/VennDiagram | <!DOCTYPE html>
<html>
<head>
<title>
Venn diagram
</title>
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<h1 class="ltx_title ltx_title_document">
Venn diagram
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Note: Currently, overlapping do not seem to be showing up properly in html mode. Therefore, this entry is best viewed using page mode.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
A
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Venn diagram
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/VennDiagram.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/venndiagram">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
is a visual tool used in describing how two or more sets are logically related to one another. The simplest example is a Venn diagram for two sets. Each set is represented by a planar region bounded by a circle, so that the two regions overlap. In the
<a class="nnexus_concept" href="http://planetmath.org/commutativediagram">
diagram
</a>
below, set
<math alttext="A" class="ltx_Math" display="inline" id="p3.m1">
<mi>
A
</mi>
</math>
is represented by the reddish circular disc and set
<math alttext="B" class="ltx_Math" display="inline" id="p3.m2">
<mi>
B
</mi>
</math>
is represented by the bluish circular disc:
</p>
</div>
<div class="ltx_para ltx_centering" id="p4">
<svg fragid="p4.pic1" height="160.965787669306" overflow="visible" version="1.1" viewbox="-23.5728938346528 -23.5928938346529 215.72 137.372893834653" width="239.292893834653">
<g transform="translate(0,113.8)">
<g transform="scale(1 -1)">
<g>
<circle cx="56.91" cy="56.91" fill="none" r="56.91" stroke="black" stroke-width="0">
</circle>
<circle cx="113.81" cy="56.91" fill="none" r="56.91" stroke="black" stroke-width="0">
</circle>
<g transform="translate(28.45,56.91)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="A" class="ltx_Math" display="inline" id="p4.pic1.m1">
<mi>
A
</mi>
</math>
</foreignobject>
</g>
</g>
<g transform="translate(85.36,56.91)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="A\cap B" class="ltx_Math" display="inline" id="p4.pic1.m2">
<mrow>
<mi>
A
</mi>
<mo>
∩
</mo>
<mi>
B
</mi>
</mrow>
</math>
</foreignobject>
</g>
</g>
<g transform="translate(142.26,56.91)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="B" class="ltx_Math" display="inline" id="p4.pic1.m3">
<mi>
B
</mi>
</math>
</foreignobject>
</g>
</g>
<g>
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="." class="ltx_Math" display="inline" id="p4.pic1.m4">
<mrow>
<mi>
</mi>
<mo>
.
</mo>
</mrow>
</math>
</foreignobject>
</g>
</g>
<g transform="translate(170.72,113.81)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="." class="ltx_Math" display="inline" id="p4.pic1.m5">
<mrow>
<mi>
</mi>
<mo>
.
</mo>
</mrow>
</math>
</foreignobject>
</g>
</g>
</g>
</g>
</g>
</svg>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
The overlapping region
<a class="nnexus_concept" href="http://planetmath.org/representablefunctor">
represents
</a>
the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
intersection
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Intersection.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/partialorderingonsubobjectsofanobject">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/intersection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the sets
<math alttext="A" class="ltx_Math" display="inline" id="p5.m1">
<mi>
A
</mi>
</math>
and
<math alttext="B" class="ltx_Math" display="inline" id="p5.m2">
<mi>
B
</mi>
</math>
, denoted by
<math alttext="A\cap B" class="ltx_Math" display="inline" id="p5.m3">
<mrow>
<mi>
A
</mi>
<mo>
∩
</mo>
<mi>
B
</mi>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
Typically, the two sets are subsets of some bigger set
<math alttext="U" class="ltx_Math" display="inline" id="p6.m1">
<mi>
U
</mi>
</math>
, called a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
universe
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/universe">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universeofdiscourse">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. Therefore, the corresponding Venn diagram above is shown to be sitting inside a larger rectangular region representing the universe:
</p>
</div>
<div class="ltx_para ltx_centering" id="p7">
<svg fragid="p7.pic1" height="190.72" overflow="visible" version="1.1" viewbox="-5 -10.02 286.85 180.7" width="291.85">
<g transform="translate(0,170.7)">
<g transform="scale(1 -1)">
<g>
<path d="M 0,0 227.62,0 227.62,170.72 0,170.72 z" fill="none" stroke="black" stroke-width="0.8">
</path>
<circle cx="85.36" cy="85.36" fill="none" r="56.91" stroke="black" stroke-width="0">
</circle>
<circle cx="142.26" cy="85.36" fill="none" r="56.91" stroke="black" stroke-width="0">
</circle>
<g transform="translate(56.91,85.36)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="2" class="ltx_Math" display="inline" id="p7.pic1.m1">
<mn>
2
</mn>
</math>
</foreignobject>
</g>
</g>
<g transform="translate(113.81,85.36)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="1" class="ltx_Math" display="inline" id="p7.pic1.m2">
<mn>
1
</mn>
</math>
</foreignobject>
</g>
</g>
<g transform="translate(170.72,85.36)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="3" class="ltx_Math" display="inline" id="p7.pic1.m3">
<mn>
3
</mn>
</math>
</foreignobject>
</g>
</g>
<g transform="translate(199.17,142.26)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="4" class="ltx_Math" display="inline" id="p7.pic1.m4">
<mn>
4
</mn>
</math>
</foreignobject>
</g>
</g>
<g transform="translate(241.85,163.6)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="U" class="ltx_Math" display="inline" id="p7.pic1.m5">
<mi>
U
</mi>
</math>
</foreignobject>
</g>
</g>
<g>
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="." class="ltx_Math" display="inline" id="p7.pic1.m6">
<mrow>
<mi>
</mi>
<mo>
.
</mo>
</mrow>
</math>
</foreignobject>
</g>
</g>
<g transform="translate(227.62,170.72)">
<g transform="scale(1 -1) translate(-5,-10)">
<foreignobject height="20" overflow="visible" width="50">
<math alttext="." class="ltx_Math" display="inline" id="p7.pic1.m7">
<mrow>
<mi>
</mi>
<mo>
.
</mo>
</mrow>
</math>
</foreignobject>
</g>
</g>
</g>
</g>
</g>
</svg>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
Notice that the region representing the universe
<math alttext="U" class="ltx_Math" display="inline" id="p8.m1">
<mi>
U
</mi>
</math>
is partitioned into
<math alttext="4" class="ltx_Math" display="inline" id="p8.m2">
<mn>
4
</mn>
</math>
mutually exclusive regions:
</p>
<table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
Venn diagram
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
VennDiagram
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:45:43
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:45:43
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
16
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A06
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 03E99
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
<span class="ltx_ERROR undefined">
\@unrecurse
</span>
</th>
<td class="ltx_td ltx_border_t">
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | Welldefined | http://planetmath.org/Welldefined | <!DOCTYPE html>
<html>
<head>
<title>
well-defined
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
well-defined
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A mathematical
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
concept
</a>
is
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/welldefined">
well-defined
</a>
</em>
(German
<span class="ltx_text ltx_font_italic">
wohldefiniert
</span>
, French
<span class="ltx_text ltx_font_italic">
bien défini
</span>
), if its contents is
on the form or the alternative representative which is used for defining it.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
For example, in defining the
<span class="ltx_text ltx_font_typewriter">
http://
<a class="nnexus_concept" href="http://planetmath.org/planetmath">
planetmath
</a>
.org/FractionPower
</span>
power
<math alttext="x^{r}" class="ltx_Math" display="inline" id="p2.m1">
<msup>
<mi>
x
</mi>
<mi>
r
</mi>
</msup>
</math>
with
<math alttext="x" class="ltx_Math" display="inline" id="p2.m2">
<mi>
x
</mi>
</math>
a
<a class="nnexus_concept" href="http://planetmath.org/positive">
positive
</a>
real and
<math alttext="r" class="ltx_Math" display="inline" id="p2.m3">
<mi>
r
</mi>
</math>
a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
rational number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/egyptianfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/liberabaci">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/rationalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
we can freely choose the fraction form
<math alttext="\frac{m}{n}" class="ltx_Math" display="inline" id="p2.m4">
<mfrac>
<mi>
m
</mi>
<mi>
n
</mi>
</mfrac>
</math>
(
<math alttext="m\in\mathbb{Z}" class="ltx_Math" display="inline" id="p2.m5">
<mrow>
<mi>
m
</mi>
<mo>
∈
</mo>
<mi>
ℤ
</mi>
</mrow>
</math>
,
<math alttext="n\in\mathbb{Z}_{+}" class="ltx_Math" display="inline" id="p2.m6">
<mrow>
<mi>
n
</mi>
<mo>
∈
</mo>
<msub>
<mi>
ℤ
</mi>
<mo>
+
</mo>
</msub>
</mrow>
</math>
) of
<math alttext="r" class="ltx_Math" display="inline" id="p2.m7">
<mi>
r
</mi>
</math>
and take
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="x^{r}\;:=\;\sqrt[n]{x^{m}}" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mpadded width="+2.8pt">
<msup>
<mi>
x
</mi>
<mi>
r
</mi>
</msup>
</mpadded>
<mo rspace="5.3pt">
:=
</mo>
<mroot>
<msup>
<mi>
x
</mi>
<mi>
m
</mi>
</msup>
<mi>
n
</mi>
</mroot>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
and be sure that the value of
<math alttext="x^{r}" class="ltx_Math" display="inline" id="p2.m8">
<msup>
<mi>
x
</mi>
<mi>
r
</mi>
</msup>
</math>
does not depend on that choice (this is justified in the entry fraction power). So,
the
<math alttext="x^{r}" class="ltx_Math" display="inline" id="p2.m9">
<msup>
<mi>
x
</mi>
<mi>
r
</mi>
</msup>
</math>
is well-defined.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
In many
<a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">
instances
</a>
well-defined is a synonym for the formal
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
of a
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
between sets. For example,
the function
<math alttext="f(x):=x^{2}" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
:=
</mo>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</math>
is a well-defined function from the real numbers to the real numbers because
every input,
<math alttext="x" class="ltx_Math" display="inline" id="p3.m2">
<mi>
x
</mi>
</math>
, is assigned to precisely one output,
<math alttext="x^{2}" class="ltx_Math" display="inline" id="p3.m3">
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</math>
. However,
<math alttext="f(x):=\pm\sqrt{x}" class="ltx_Math" display="inline" id="p3.m4">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
:=
</mo>
<mrow>
<mo>
±
</mo>
<msqrt>
<mi>
x
</mi>
</msqrt>
</mrow>
</mrow>
</math>
is not well-defined
in that one input
<math alttext="x" class="ltx_Math" display="inline" id="p3.m5">
<mi>
x
</mi>
</math>
can be assigned any one of two possible outputs,
<math alttext="\sqrt{x}" class="ltx_Math" display="inline" id="p3.m6">
<msqrt>
<mi>
x
</mi>
</msqrt>
</math>
or
<math alttext="-\sqrt{x}" class="ltx_Math" display="inline" id="p3.m7">
<mrow>
<mo>
-
</mo>
<msqrt>
<mi>
x
</mi>
</msqrt>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
More subtle examples include
<a class="nnexus_concept" href="http://planetmath.org/expression">
expressions
</a>
such as
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="f\!\left(\frac{a}{b}\right)\;:=\;a\!+\!b,\quad\frac{a}{b}\in\mathbb{Q}." class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mrow>
<mrow>
<mrow>
<mpadded width="-1.7pt">
<mi>
f
</mi>
</mpadded>
<mo>
</mo>
<mrow>
<mo>
(
</mo>
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
<mo rspace="5.3pt">
)
</mo>
</mrow>
</mrow>
<mo rspace="5.3pt">
:=
</mo>
<mrow>
<mpadded width="-1.7pt">
<mi>
a
</mi>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<mi>
b
</mi>
</mrow>
</mrow>
<mo rspace="12.5pt">
,
</mo>
<mrow>
<mfrac>
<mi>
a
</mi>
<mi>
b
</mi>
</mfrac>
<mo>
∈
</mo>
<mi>
ℚ
</mi>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Certainly every input has an output, for instance,
<math alttext="f(1/2)=3" class="ltx_Math" display="inline" id="p4.m1">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
3
</mn>
</mrow>
</math>
. However, the expression is
<em class="ltx_emph ltx_font_italic">
not
</em>
well-defined since
<math alttext="1/2=2/4" class="ltx_Math" display="inline" id="p4.m2">
<mrow>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
/
</mo>
<mn>
4
</mn>
</mrow>
</mrow>
</math>
yet
<math alttext="f(1/2)=3" class="ltx_Math" display="inline" id="p4.m3">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
3
</mn>
</mrow>
</math>
while
<math alttext="f(2/4)=6" class="ltx_Math" display="inline" id="p4.m4">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
2
</mn>
<mo>
/
</mo>
<mn>
4
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
6
</mn>
</mrow>
</math>
and
<math alttext="3\neq 6" class="ltx_Math" display="inline" id="p4.m5">
<mrow>
<mn>
3
</mn>
<mo>
≠
</mo>
<mn>
6
</mn>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
One must question whether a function is well-defined whenever it is defined on a domain of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
equivalence classes
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/EquivalenceClass.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalenceclass">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalencerelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in such a manner that each output is determined for a representative of each equivalence class. For example, the
function
<math alttext="f(a/b):=a\!+\!b" class="ltx_Math" display="inline" id="p5.m1">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
a
</mi>
<mo>
/
</mo>
<mi>
b
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
:=
</mo>
<mrow>
<mpadded width="-1.7pt">
<mi>
a
</mi>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<mi>
b
</mi>
</mrow>
</mrow>
</math>
was defined using the representative
<math alttext="a/b" class="ltx_Math" display="inline" id="p5.m2">
<mrow>
<mi>
a
</mi>
<mo>
/
</mo>
<mi>
b
</mi>
</mrow>
</math>
of the equivalence class of fractions
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
equivalent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Equivalent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/filterbasis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalenceofforcingnotions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalentmachines">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to
<math alttext="a/b" class="ltx_Math" display="inline" id="p5.m3">
<mrow>
<mi>
a
</mi>
<mo>
/
</mo>
<mi>
b
</mi>
</mrow>
</math>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
well-defined
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Welldefined
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:31:32
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:31:32
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
9
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
well defined
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
function
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
WellDefinednessOfProductOfFinitelyGeneratedIdeals
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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|
0 | WindowsCalculator | http://planetmath.org/WindowsCalculator | <!DOCTYPE html>
<html>
<head>
<title>
Windows Calculator
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
Windows Calculator
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/windowscalculator">
Windows Calculator
</a>
</span>
is a software calculator that comes bundled with the Windows operating system. The basic mode is called “Standard” and is the default, Scientific mode has most of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
available on a typical
<a class="nnexus_concept" href="http://planetmath.org/scientificcalculator">
scientific calculator
</a>
. Note that switching between modes causes the loss of the current value displayed (unless of course that value is 0). For some reason, Standard mode has a
<a class="nnexus_concept" href="http://planetmath.org/squareroot">
square root
</a>
key but Scientific mode does not. As a workaround in scientific mode, one can enter, say,
<code class="ltx_verbatim ltx_font_typewriter">
[2] [x^y] [0] [.] [5]
</code>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/divisionbyzero">
Division by zero
</a>
causes an error condition that must be cleared with the C key on the displayed keyboard (or the Escape key on the
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computer
</a>
’s keyboard). Integer values smaller than
<math alttext="10^{32}" class="ltx_Math" display="inline" id="p2.m1">
<msup>
<mn>
10
</mn>
<mn>
32
</mn>
</msup>
</math>
can be displayed in all their digits. According to the Help, the Windows Calculator truncates
<math alttext="\pi" class="ltx_Math" display="inline" id="p2.m2">
<mi>
π
</mi>
</math>
to 32 digits, but
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
rational numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/egyptianfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/liberabaci">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/rationalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
are stored internally “as
<a class="nnexus_concept" href="http://planetmath.org/fraction">
fractions
</a>
”.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Like most scientific calculators, the Windows Calculator can display results in binary, octal and
<a class="nnexus_concept" href="http://planetmath.org/hexadecimal">
hexadecimal
</a>
, but is limited to
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
in those bases. Additionally,
<a class="nnexus_concept" href="http://planetmath.org/negativenumber">
negative numbers
</a>
are shown in two’s
<a class="nnexus_concept" href="http://planetmath.org/complement">
complement
</a>
(and the sign change key performs two’s complement on the displayed value). In those bases, the user can choose the data size: quadruple word (the default), double word, word or byte. Overflows don’t trigger any kind of exception or error notification, the
<a class="nnexus_concept" href="http://planetmath.org/calculator">
calculator
</a>
quietly discards the more significant digits and displays the least significant digits that will fit in the currently selected data size.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Like the
<a class="nnexus_concept" href="http://planetmath.org/macoscalculator">
Mac OS Calculator
</a>
, for the Windows Calculator
<math alttext="0^{0}=1" class="ltx_Math" display="inline" id="p4.m1">
<mrow>
<msup>
<mn>
0
</mn>
<mn>
0
</mn>
</msup>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
David A. Karp, Tim O’Reilly & Troy Mott,
<span class="ltx_text ltx_font_italic">
Windows XP in a Nutshell
</span>
Cambridge: O’Reilly (2002): 114 - 117
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p5">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
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Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
Windows Calculator
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
WindowsCalculator
</td>
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Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:39:22
</td>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
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2013-03-22 16:39:22
</td>
</tr>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
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Last modified by
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PrimeFan (13766)
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8
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Author
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PrimeFan (13766)
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Entry type
</th>
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Definition
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Classification
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msc 00A05
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0 | DoublingAndHalvingAlgorithmForIntegerMultiplication | http://planetmath.org/DoublingAndHalvingAlgorithmForIntegerMultiplication | <!DOCTYPE html>
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<title>
doubling and halving algorithm for integer multiplication
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
doubling and halving algorithm for integer multiplication
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Because multiplying and dividing by 2 is often easier for humans than multiplying and dividing by other numbers there is an algorithm for
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
multiplication
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of any two
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
that takes advantage of multiplication and
<a class="nnexus_concept" href="http://planetmath.org/division">
division
</a>
by 2.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Call the algorithm with two integers.
</p>
</div>
<div class="ltx_para" id="p3">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
Use one of the integers to start a column on the left and the other to start a column on the right. (Either number can be put in either column, there are very minor optimizations that are unlikely to make a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
difference
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/difference">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/setdifference">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in performance, such as not choosing for the left column numbers that end long
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Cunningham chains
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CunninghamChain.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cunninghamchain">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
).
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
Divide the previous integer on the left column by 2 and write the yield below it, ignoring any
<a class="nnexus_concept" href="http://planetmath.org/fractionalpart">
fractional part
</a>
there may be. Multiply the previous integer on the right column by 2 and write the
<a class="nnexus_concept" href="http://planetmath.org/product">
product
</a>
below.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
Repeat Step 2 until the yield on the left column is 1.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
For every
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
even number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/EvenNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/evennumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
on the left column, cross out the right column’s number of the same row.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
Add up the the numbers on the right column that haven’t been crossed out.
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
For example, to multiply 108 by 255:
</p>
</div>
<div class="ltx_para" id="p5">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
108
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
<math alttext="\not{255}" class="ltx_Math" display="inline" id="p5.m1">
<menclose notation="updiagonalstrike">
<mn>
255
</mn>
</menclose>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
54
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
<math alttext="\not{510}" class="ltx_Math" display="inline" id="p5.m2">
<menclose notation="updiagonalstrike">
<mn>
510
</mn>
</menclose>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
27
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
1020
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
13
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
2040
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
<math alttext="\not{4080}" class="ltx_Math" display="inline" id="p5.m3">
<menclose notation="updiagonalstrike">
<mn>
4080
</mn>
</menclose>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
3
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
8160
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
1
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
16320
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
27540
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
This works in any base (as long as one doesn’t get confused about parity in odd bases). For example, 18 times 24 in base 5:
</p>
</div>
<div class="ltx_para" id="p7">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
33
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
<math alttext="\not{44}" class="ltx_Math" display="inline" id="p7.m1">
<menclose notation="updiagonalstrike">
<mn>
44
</mn>
</menclose>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
14
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
143
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
4
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
<math alttext="\not{341}" class="ltx_Math" display="inline" id="p7.m2">
<menclose notation="updiagonalstrike">
<mn>
341
</mn>
</menclose>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
2
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
<math alttext="\not{1232}" class="ltx_Math" display="inline" id="p7.m3">
<menclose notation="updiagonalstrike">
<mn>
1232
</mn>
</menclose>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
3014
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
3212
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
Naturally one might wonder if this can be applied to binary and used by
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computers
</a>
. After all, halving and ignoring the fractional part is even easier: it’s just a matter of shifting the bits to the right, and it doesn’t matter what the computer does with the discarded bit (as long as it doesn’t put it back into the original byte or word in the most significant bit or the sign bit). Doubling is also easy, just a shift left, with the only concern being overflow.
</p>
</div>
<div class="ltx_para" id="p9">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1010
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
<math alttext="\not{111}" class="ltx_Math" display="inline" id="p9.m1">
<menclose notation="updiagonalstrike">
<mn>
111
</mn>
</menclose>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
101
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
1110
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
10
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
<math alttext="\not{11100}" class="ltx_Math" display="inline" id="p9.m2">
<menclose notation="updiagonalstrike">
<mn>
11100
</mn>
</menclose>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1
</th>
<td class="ltx_td ltx_align_right ltx_border_r">
111000
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
1000110
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p10">
<p class="ltx_p">
Of course this algorithm is not suitable for large integer multiplication as is required in the search for large prime numbers.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Paul Erdős & János Surányi
<span class="ltx_text ltx_font_italic">
Topics in the theory of numbers
</span>
New York: Springer (2003): 5
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
Ogilvy & Anderson,
<span class="ltx_text ltx_font_italic">
Excursions in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Number Theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NumberTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/numbertheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
. Oxford: Oxford University Press (1966). Reprinted New York: Dover (1988): 6 - 8
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p11">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/doublingandhalvingalgorithmforintegermultiplication">
doubling and halving algorithm for integer multiplication
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
DoublingAndHalvingAlgorithmForIntegerMultiplication
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:01:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:01:20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompositeFan (12809)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompositeFan (12809)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompositeFan (12809)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Algorithm
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A06
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 11B25
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | Multiplication | http://planetmath.org/Multiplication | <!DOCTYPE html>
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<title>
multiplication
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
multiplication
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Multiplication
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
is a mathematical
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in which two or more numbers are added up to themselves by a factor of other numbers. For example,
<math alttext="2\times 3=2+2+2=3+3=6" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
×
</mo>
<mn>
3
</mn>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
+
</mo>
<mn>
2
</mn>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
3
</mn>
<mo>
+
</mo>
<mn>
3
</mn>
</mrow>
<mo>
=
</mo>
<mn>
6
</mn>
</mrow>
</math>
. The numbers may be real,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
imaginary
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/imaginaries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/imaginary">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
or
<a class="nnexus_concept" href="http://planetmath.org/complex">
complex
</a>
, they may be
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
or
<a class="nnexus_concept" href="http://planetmath.org/fraction">
fractions
</a>
. Among
<a class="nnexus_concept" href="http://planetmath.org/realnumber">
real numbers
</a>
, if an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
odd number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/OddNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/evennumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/oddnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of multiplicands are
<a class="nnexus_concept" href="http://planetmath.org/positive">
negative
</a>
, the overall result is negative; if an
<a class="nnexus_concept" href="http://mathworld.wolfram.com/EvenNumber.html">
even number
</a>
of multiplicands are negative, the overall result is
<a class="nnexus_concept" href="http://planetmath.org/positiveelement">
positive
</a>
. Two examples:
<math alttext="(-3)\times(-5)=15" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
3
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
×
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
5
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
15
</mn>
</mrow>
</math>
;
<math alttext="(-2)\times(-3)\times(-5)=(-30)" class="ltx_Math" display="inline" id="p1.m3">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
×
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
3
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
×
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
5
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
30
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The usual
<a class="nnexus_concept" href="http://planetmath.org/operator">
operator
</a>
is the cross with its four arms of equal length pointing northeast, northwest, southeast and southwest:
<math alttext="\times" class="ltx_Math" display="inline" id="p2.m1">
<mo>
×
</mo>
</math>
. Other options are the central dot
<math alttext="\cdot" class="ltx_Math" display="inline" id="p2.m2">
<mo>
⋅
</mo>
</math>
and the
<a class="nnexus_concept" href="http://planetmath.org/tacitmultiplicationoperator">
tacit multiplication operator
</a>
. In many computer programming languages the asterisk is often used as it is almost always available on the keyboard (Shift-8 in most American layouts, as well as dedicated key if the keyboard has a numeric keypad), and this is the operator likely to be used in a computer implementation of a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
reverse Polish notation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ReversePolishNotation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reversepolishnotation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
calculator. In
<a class="nnexus_concept" href="http://planetmath.org/mathematica">
Mathematica
</a>
, the space can sometimes
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
as a
<a class="nnexus_concept" href="http://planetmath.org/multiplicationoperator">
multiplication operator
</a>
, but more experienced users warn novices not to rely on this feature.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Just as with
<a class="nnexus_concept" href="http://planetmath.org/addition">
addition
</a>
, multiplication is
<a class="nnexus_concept" href="http://planetmath.org/commutativelanguage">
commutative
</a>
:
<math alttext="xyz=xzy=yxz" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
y
</mi>
<mo>
</mo>
<mi>
z
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
z
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
y
</mi>
<mo>
</mo>
<mi>
x
</mi>
<mo>
</mo>
<mi>
z
</mi>
</mrow>
</mrow>
</math>
, etc.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
The iterative operator is the Greek capital letter pi:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\prod_{i=1}^{n}a_{i}," class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<munderover>
<mo largeop="true" movablelimits="false" symmetric="true">
∏
</mo>
<mrow>
<mi>
i
</mi>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
<mi>
n
</mi>
</munderover>
<msub>
<mi>
a
</mi>
<mi>
i
</mi>
</msub>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
which is a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
compact
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/topologyofthecomplexplane">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/compact">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
way of writing
<math alttext="a_{1}\times a_{2}\times\ldots\times a_{n}" class="ltx_Math" display="inline" id="p4.m1">
<mrow>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
<mo>
×
</mo>
<msub>
<mi>
a
</mi>
<mn>
2
</mn>
</msub>
<mo>
×
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
×
</mo>
<msub>
<mi>
a
</mi>
<mi>
n
</mi>
</msub>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
Multiplication of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complex numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complexnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is helped by the following
<a class="nnexus_concept" href="http://planetmath.org/multivaluedfunction">
identity
</a>
:
<math alttext="(a+bi)\times(x+yi)=(ax-by)+(ay+bx)i" class="ltx_Math" display="inline" id="p5.m1">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
×
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<mrow>
<mi>
y
</mi>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<mi>
a
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
-
</mo>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
+
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<mi>
a
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
<mo>
+
</mo>
<mrow>
<mi>
b
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
</mrow>
</math>
. To give three examples:
<math alttext="(17+29i)(11+38i)=-915+965i" class="ltx_Math" display="inline" id="p5.m2">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
17
</mn>
<mo>
+
</mo>
<mrow>
<mn>
29
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
11
</mn>
<mo>
+
</mo>
<mrow>
<mn>
38
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<mo>
-
</mo>
<mn>
915
</mn>
</mrow>
<mo>
+
</mo>
<mrow>
<mn>
965
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
</mrow>
</math>
(the result has both real and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ImaginaryPart.html">
imaginary parts
</a>
),
<math alttext="(1+2i)(1-2i)=5" class="ltx_Math" display="inline" id="p5.m3">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
5
</mn>
</mrow>
</math>
(the result is a
<a class="nnexus_concept" href="http://planetmath.org/valuation">
real prime
</a>
) and
<math alttext="(4+7i)(7+4i)=65i" class="ltx_Math" display="inline" id="p5.m4">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
4
</mn>
<mo>
+
</mo>
<mrow>
<mn>
7
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mn>
7
</mn>
<mo>
+
</mo>
<mrow>
<mn>
4
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
65
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</mrow>
</math>
(the result has only an imaginary part).
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
multiplication
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Multiplication
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:35:37
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:35:37
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A06
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 11B25
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/product">
Product
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
ProductOfNegativeNumbers
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
FactorsWithMinusSign
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | PlusSign | http://planetmath.org/PlusSign | <!DOCTYPE html>
<html>
<head>
<title>
plus sign
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
plus sign
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
There are two main uses of the
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/plussign">
plus sign
</a>
</span>
“
<math alttext="+" class="ltx_Math" display="inline" id="p1.m1">
<mo>
+
</mo>
</math>
” (which is a simplified form of “&”) in the mathematics and the applying sciences:
</p>
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
The original use is as the sign for the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
binary operation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BinaryOperation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/binaryoperation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
of numbers and other elements of rings and algebras, vectors, etc.:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="a\!+\!b\;:=\mbox{\, the sum of\, }a\mbox{\, and\, }b" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mpadded width="-1.7pt">
<mi>
a
</mi>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<mpadded width="+2.8pt">
<mi>
b
</mi>
</mpadded>
</mrow>
<mo>
:=
</mo>
<mrow>
<mtext>
the sum of
</mtext>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mtext>
and
</mtext>
<mo>
</mo>
<mi>
b
</mi>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
There is also a special use for the unary operation
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/identitymap">
identity mapping
</a>
</span>
concerning numbers and other ring
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Element.html">
elements
</a>
:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="+a\;:=\;a\mbox{\; (for all }a)" class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mo>
+
</mo>
<mpadded width="+2.8pt">
<mi>
a
</mi>
</mpadded>
<mo rspace="5.3pt">
:=
</mo>
<mi>
a
</mi>
<mtext>
(for all
</mtext>
<mi>
a
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
</li>
</ul>
<table class="ltx_tabular ltx_align_right ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
plus sign
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
PlusSign
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:35:43
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:35:43
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A06
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
plus
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Sum
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
SumOfSeries
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
SignumFunction
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
OppositeNumber
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
ProductOfNegativeNumbers
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | SimpleInterest | http://planetmath.org/SimpleInterest | <!DOCTYPE html>
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<title>
simple interest
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
simple interest
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Suppose a bank account is opened at time
<math alttext="0" class="ltx_Math" display="inline" id="p1.m1">
<mn>
0
</mn>
</math>
and
<math alttext="M_{0}" class="ltx_Math" display="inline" id="p1.m2">
<msub>
<mi>
M
</mi>
<mn>
0
</mn>
</msub>
</math>
is deposited into the account. A
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/simpleinterest">
simple interest
</a>
</em>
is
<a class="nnexus_concept" href="http://planetmath.org/interest">
interest
</a>
with the following characteristics:
</p>
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
it is earned at subsequent time periods
<math alttext="t,2t,\ldots" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mrow>
<mi>
t
</mi>
<mo>
,
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
</mrow>
</math>
, where
<math alttext="t" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mi>
t
</mi>
</math>
is the length of the initial time interval (1 for 1 month, 12 for 1 year, etc…)
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
the interest earned at the end of each time period is the same regardless of the time period
</p>
</div>
</li>
</ol>
<p class="ltx_p">
The following table illustrates the structure of the simple interest.
</p>
</div>
<div class="ltx_para ltx_centering" id="p2">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t">
time period at
</th>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
principal
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
interest
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
interest accrued
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_tt">
<math alttext="0" class="ltx_Math" display="inline" id="p2.m1">
<mn>
0
</mn>
</math>
</th>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt">
<math alttext="M_{0}" class="ltx_Math" display="inline" id="p2.m2">
<msub>
<mi>
M
</mi>
<mn>
0
</mn>
</msub>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt">
<math alttext="0" class="ltx_Math" display="inline" id="p2.m3">
<mn>
0
</mn>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt">
<math alttext="0" class="ltx_Math" display="inline" id="p2.m4">
<mn>
0
</mn>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t">
<math alttext="t" class="ltx_Math" display="inline" id="p2.m5">
<mi>
t
</mi>
</math>
</th>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="M_{0}" class="ltx_Math" display="inline" id="p2.m6">
<msub>
<mi>
M
</mi>
<mn>
0
</mn>
</msub>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="i" class="ltx_Math" display="inline" id="p2.m7">
<mi>
i
</mi>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="i" class="ltx_Math" display="inline" id="p2.m8">
<mi>
i
</mi>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t">
<math alttext="2t" class="ltx_Math" display="inline" id="p2.m9">
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
</math>
</th>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="M_{0}" class="ltx_Math" display="inline" id="p2.m10">
<msub>
<mi>
M
</mi>
<mn>
0
</mn>
</msub>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="i" class="ltx_Math" display="inline" id="p2.m11">
<mi>
i
</mi>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="2i" class="ltx_Math" display="inline" id="p2.m12">
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t">
<math alttext="3t" class="ltx_Math" display="inline" id="p2.m13">
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
</math>
</th>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="M_{0}" class="ltx_Math" display="inline" id="p2.m14">
<msub>
<mi>
M
</mi>
<mn>
0
</mn>
</msub>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="i" class="ltx_Math" display="inline" id="p2.m15">
<mi>
i
</mi>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="3i" class="ltx_Math" display="inline" id="p2.m16">
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t">
<math alttext="\vdots" class="ltx_Math" display="inline" id="p2.m17">
<mi mathvariant="normal">
⋮
</mi>
</math>
</th>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="\vdots" class="ltx_Math" display="inline" id="p2.m18">
<mi mathvariant="normal">
⋮
</mi>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="\vdots" class="ltx_Math" display="inline" id="p2.m19">
<mi mathvariant="normal">
⋮
</mi>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<math alttext="\vdots" class="ltx_Math" display="inline" id="p2.m20">
<mi mathvariant="normal">
⋮
</mi>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_rr ltx_border_t">
<math alttext="nt" class="ltx_Math" display="inline" id="p2.m21">
<mrow>
<mi>
n
</mi>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
</math>
</th>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<math alttext="M_{0}" class="ltx_Math" display="inline" id="p2.m22">
<msub>
<mi>
M
</mi>
<mn>
0
</mn>
</msub>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<math alttext="i" class="ltx_Math" display="inline" id="p2.m23">
<mi>
i
</mi>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<math alttext="ni" class="ltx_Math" display="inline" id="p2.m24">
<mrow>
<mi>
n
</mi>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</math>
</td>
</tr>
</tbody>
</table>
<p class="ltx_p">
<math alttext="{}\end{center}\inner@par The``total^{\prime\prime}interest" class="ltx_Math" display="inline" id="p2.m25">
<mrow>
<mi>
T
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi mathvariant="normal">
`
</mi>
<mo>
</mo>
<mi mathvariant="normal">
`
</mi>
<mo>
</mo>
<mi>
t
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<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
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t
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<msup>
<mi>
l
</mi>
<mo>
′′
</mo>
</msup>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
n
</mi>
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</mo>
<mi>
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<mo>
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<mi>
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<mo>
</mo>
<mi>
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<mi>
t
</mi>
</mrow>
</math>
i(nt)
<math alttext="earned(accrued)attheendoftime" class="ltx_Math" display="inline" id="p2.m26">
<mrow>
<mi>
e
</mi>
<mo>
</mo>
<mi>
a
</mi>
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</mo>
<mi>
r
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</mo>
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</mo>
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</mo>
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<mo>
</mo>
<mrow>
<mo stretchy="false">
(
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<mrow>
<mi>
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</mi>
<mo>
</mo>
<mi>
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</mrow>
<mo stretchy="false">
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</mrow>
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</mrow>
</math>
nt
<math alttext="is" class="ltx_Math" display="inline" id="p2.m27">
<mrow>
<mi>
i
</mi>
<mo>
</mo>
<mi>
s
</mi>
</mrow>
</math>
ni
<math alttext=".Iftheaccountisclosedandthemoneywithdrawnattheendof" class="ltx_Math" display="inline" id="p2.m28">
<mrow>
<mo>
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</mo>
<mi>
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</mi>
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</mrow>
</math>
nt
<math alttext=",andthetotalamountofmoneyreceivedis" class="ltx_Math" display="inline" id="p2.m29">
<mrow>
<mo>
,
</mo>
<mi>
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</mi>
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</mrow>
</math>
<math alttext="M(nt)=M_{0}+ni." class="ltx_Math" display="inline" id="p2.m30">
<mrow>
<mrow>
<mrow>
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</mi>
<mo>
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<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
n
</mi>
<mo>
</mo>
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</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msub>
<mi>
M
</mi>
<mn>
0
</mn>
</msub>
<mo>
+
</mo>
<mrow>
<mi>
n
</mi>
<mo>
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<mi>
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</mi>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
<math alttext="\inner@par
Theinterestrateassociatedwiththesimpleinterestaspresentedabovebetweentwotimeperiods%
,say" class="ltx_Math" display="inline" id="p2.m31">
<mrow>
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</mrow>
</mrow>
</math>
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<math alttext="and" class="ltx_Math" display="inline" id="p2.m32">
<mrow>
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</mi>
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<mi>
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</mrow>
</math>
bt
<math alttext=",isgivenby" class="ltx_Math" display="inline" id="p2.m33">
<mrow>
<mo>
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</mo>
<mi>
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</math>
<math alttext="r(at,bt)=\frac{1}{M_{0}}\frac{i(bt)-i(at)}{bt-at}=\frac{i}{M_{0}t}," class="ltx_Math" display="inline" id="p2.m34">
<mrow>
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<mrow>
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<mn>
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</mn>
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</mi>
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</mn>
</msub>
</mfrac>
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</mo>
<mfrac>
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</mi>
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<msub>
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</mi>
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</mn>
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t
</mi>
</mrow>
</mfrac>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
<math alttext="whichdoesnotdependonthechoiceof" class="ltx_Math" display="inline" id="p2.m35">
<mrow>
<mi>
w
</mi>
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<math alttext="and" class="ltx_Math" display="inline" id="p2.m36">
<mrow>
<mi>
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</math>
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<math alttext=".Inotherwords,theoriginalprincipal" class="ltx_Math" display="inline" id="p2.m37">
<mrow>
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</math>
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<math alttext=",theamountofinterest" class="ltx_Math" display="inline" id="p2.m38">
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</math>
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<math alttext=",andthelengthoftheinitialtimeinterval" class="ltx_Math" display="inline" id="p2.m39">
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</mo>
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</mrow>
</math>
t
<math alttext="{{{areenoughtodeterminetheinterestrate.\inner@par\textbf{Remark}.\begin{itemize}
\itemize@item The expression for the effective interest rate for simple %
interest is a bit more complicated:
$$\operatorname{eff.}r(at,bt)=\frac{1}{M(at)}\frac{i(bt)-i(at)}{bt-at}=
\frac{1}{M_{0}+ai}\frac{i}{t},$$
which decreases with increasing $a$. Imagine as $a$ becomes very large, the %
increase in interest has practically no impact on the ``accumulated'' %
principal $M(at)$.
\itemize@item More generally, we say that an interest is \emph{simple} if its %
interest rate $r$ is constant with respect to time $t$. Solving
$$r=\frac{1}{M_{0}}\frac{i(t)-i(0)}{t-0}$$
for $i(t)$, we get $i(t)=M_{0}rt$, or that the accrued interest is a linear %
function of $t$. It grows directly proportionally with respect to time.
\end{itemize}\begin{flushright}\begin{tabular}[]{|ll|}\hline Title&simple %
interest\\
Canonical name&SimpleInterest\\
Date of creation&2013-03-22 16:40:06\\
Last modified on&2013-03-22 16:40:06\\
Owner&CWoo (3771)\\
Last modified by&CWoo (3771)\\
Numerical id&7\\
Author&CWoo (3771)\\
Entry type&Example\\
Classification&msc 00A06\\
Classification&msc 00A69\\
Classification&msc 91B28\\
Related topic&CompoundInterest\\
Related topic&InterestRate\\
\hline}\end{tabular}}$}\end{flushright}\end{document}" class="ltx_Math" display="inline" id="p2.m40">
<mrow>
<mrow>
<mi>
a
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
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e
</mi>
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</mo>
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e
</mi>
<mo>
</mo>
<mi>
n
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mi>
u
</mi>
<mo>
</mo>
<mi>
g
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
o
</mi>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
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</mi>
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</mo>
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</mi>
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</mo>
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</mi>
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</mo>
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</mi>
<mo>
</mo>
<mi>
m
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
n
</mi>
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</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
n
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
<mi>
a
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
e
</mi>
</mrow>
<mo>
.
</mo>
<mtext>
𝐑𝐞𝐦𝐚𝐫𝐤
</mtext>
<mo>
.
</mo>
<mrow>
<mtext>
•
The expression for the effective interest rate for simple interest is a bit more complicated:
eff.r(at,bt)=
1
M(at)
i(bt)-i(at)
bt-at
=
1
M
0
+ai
i
t
,
which decreases with increasing
a
. Imagine as
a
becomes very large, the increase in interest has practically no impact on the “accumulated” principal
M
(
a
t
)
.
•
More generally, we say that an interest is simple if its interest rate
r
is constant with respect to time
t
. Solving
r=
1
M
0
i(t)-i(0)
t-0
for
i
(
t
)
, we get
=
i
(
t
)
M
0
r
t
, or that the accrued interest is a linear function of
t
. It grows directly proportionally with respect to time.
</mtext>
<mo>
</mo>
<mtable class="ltx_align_right ltx_guessed_headers" columnspacing="5pt" rowspacing="0pt">
<mtr>
<mtd class="ltx_border_l ltx_border_t ltx_th_row" columnalign="left">
<mtext>
Title
</mtext>
</mtd>
<mtd class="ltx_border_r ltx_border_t" columnalign="left">
<mtext>
simple interest
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Canonical name
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
SimpleInterest
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Date of creation
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
2013-03-22 16:40:06
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Last modified on
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
2013-03-22 16:40:06
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Owner
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
CWoo (3771)
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Last modified by
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
CWoo (3771)
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Numerical id
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
7
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Author
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
CWoo (3771)
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Entry type
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
Example
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Classification
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
msc 00A06
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Classification
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
msc 00A69
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Classification
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
msc 91B28
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Related topic
</mtext>
</mtd>
<mtd class="ltx_border_r" columnalign="left">
<mtext>
CompoundInterest
</mtext>
</mtd>
</mtr>
<mtr>
<mtd class="ltx_border_b ltx_border_l ltx_th_row" columnalign="left">
<mtext>
Related topic
</mtext>
</mtd>
<mtd class="ltx_border_b ltx_border_r" columnalign="left">
<mtext>
InterestRate
</mtext>
</mtd>
</mtr>
</mtable>
</mrow>
</mrow>
</math>
</p>
</div>
</article>
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subtraction
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
subtraction
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/subtraction">
Subtraction
</a>
</span>
is a mathematical
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in which the value of a number is decreased by the values of one or more other numbers. Subtraction can be seen as a kind of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
with
<a class="nnexus_concept" href="http://planetmath.org/negativenumber">
negative numbers
</a>
. For example,
<math alttext="7-4=7+(-4)=3" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mrow>
<mn>
7
</mn>
<mo>
-
</mo>
<mn>
4
</mn>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
7
</mn>
<mo>
+
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
4
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
3
</mn>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The usual
<a class="nnexus_concept" href="http://planetmath.org/operator">
operator
</a>
looks like a dash:
<math alttext="-" class="ltx_Math" display="inline" id="p2.m1">
<mo>
-
</mo>
</math>
. This operator is used in standard
<a class="nnexus_concept" href="http://planetmath.org/infixnotation">
infix notation
</a>
as well as in
<a class="nnexus_concept" href="http://planetmath.org/polishnotation">
Polish notation
</a>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
reverse Polish notation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ReversePolishNotation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reversepolishnotation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Besides the possibility of overflow or underflow, subtraction presents no problems for
<a class="nnexus_concept" href="http://planetmath.org/fixedpointarithmetic">
fixed point arithmetic
</a>
provided the operands are representable in
<a class="nnexus_concept" href="http://planetmath.org/fixedpoint">
fixed point
</a>
to begin with.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
subtraction
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Subtraction
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:35:31
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:35:31
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A06
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 11B25
</td>
</tr>
</tbody>
</table>
</div>
</article>
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|
0 | TableOfDivisionUpTo12 | http://planetmath.org/TableOfDivisionUpTo12 | <!DOCTYPE html>
<html>
<head>
<title>
table of division up to 12
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
table of division up to 12
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
In this table of
<a class="nnexus_concept" href="http://planetmath.org/division">
division
</a>
, the column operand is first and the row operand is second. For the sake of compactness, overlines have been used over repeating decimals when the operands are coprime to each other and to 10.
</p>
</div>
<div class="ltx_para" id="p2">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
<math alttext="\div" class="ltx_Math" display="inline" id="p2.m1">
<mo>
÷
</mo>
</math>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
1
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
3
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
4
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
8
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
9
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
10
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
11
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
12
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
1
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m2">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.5" class="ltx_Math" display="inline" id="p2.m3">
<mn>
0.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{3}" class="ltx_Math" display="inline" id="p2.m4">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.25" class="ltx_Math" display="inline" id="p2.m5">
<mn>
0.25
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.2" class="ltx_Math" display="inline" id="p2.m6">
<mn>
0.2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.1\overline{6}" class="ltx_Math" display="inline" id="p2.m7">
<mrow>
<mn>
0.1
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{142857}" class="ltx_Math" display="inline" id="p2.m8">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
142857
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.125" class="ltx_Math" display="inline" id="p2.m9">
<mn>
0.125
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{1}" class="ltx_Math" display="inline" id="p2.m10">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
1
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.1" class="ltx_Math" display="inline" id="p2.m11">
<mn>
0.1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{09}" class="ltx_Math" display="inline" id="p2.m12">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
09
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.08\overline{3}" class="ltx_Math" display="inline" id="p2.m13">
<mrow>
<mn>
0.08
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
2
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2" class="ltx_Math" display="inline" id="p2.m14">
<mn>
2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m15">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{6}" class="ltx_Math" display="inline" id="p2.m16">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.5" class="ltx_Math" display="inline" id="p2.m17">
<mn>
0.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.4" class="ltx_Math" display="inline" id="p2.m18">
<mn>
0.4
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{3}" class="ltx_Math" display="inline" id="p2.m19">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{285714}" class="ltx_Math" display="inline" id="p2.m20">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
285714
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.25" class="ltx_Math" display="inline" id="p2.m21">
<mn>
0.25
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{2}" class="ltx_Math" display="inline" id="p2.m22">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
2
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.2" class="ltx_Math" display="inline" id="p2.m23">
<mn>
0.2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{18}" class="ltx_Math" display="inline" id="p2.m24">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
18
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.1\overline{6}" class="ltx_Math" display="inline" id="p2.m25">
<mrow>
<mn>
0.1
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
3
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="3" class="ltx_Math" display="inline" id="p2.m26">
<mn>
3
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.5" class="ltx_Math" display="inline" id="p2.m27">
<mn>
1.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m28">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.75" class="ltx_Math" display="inline" id="p2.m29">
<mn>
0.75
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.6" class="ltx_Math" display="inline" id="p2.m30">
<mn>
0.6
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.5" class="ltx_Math" display="inline" id="p2.m31">
<mn>
0.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{428571}" class="ltx_Math" display="inline" id="p2.m32">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
428571
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.375" class="ltx_Math" display="inline" id="p2.m33">
<mn>
0.375
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{3}" class="ltx_Math" display="inline" id="p2.m34">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.3" class="ltx_Math" display="inline" id="p2.m35">
<mn>
0.3
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{27}" class="ltx_Math" display="inline" id="p2.m36">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
27
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.25" class="ltx_Math" display="inline" id="p2.m37">
<mn>
0.25
</mn>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
4
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="4" class="ltx_Math" display="inline" id="p2.m38">
<mn>
4
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2" class="ltx_Math" display="inline" id="p2.m39">
<mn>
2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{3}" class="ltx_Math" display="inline" id="p2.m40">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m41">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.8" class="ltx_Math" display="inline" id="p2.m42">
<mn>
0.8
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{6}" class="ltx_Math" display="inline" id="p2.m43">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{571428}" class="ltx_Math" display="inline" id="p2.m44">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
571428
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.5" class="ltx_Math" display="inline" id="p2.m45">
<mn>
0.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{4}" class="ltx_Math" display="inline" id="p2.m46">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
4
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.4" class="ltx_Math" display="inline" id="p2.m47">
<mn>
0.4
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{36}" class="ltx_Math" display="inline" id="p2.m48">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
36
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{3}" class="ltx_Math" display="inline" id="p2.m49">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
5
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="5" class="ltx_Math" display="inline" id="p2.m50">
<mn>
5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2.5" class="ltx_Math" display="inline" id="p2.m51">
<mn>
2.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{6}" class="ltx_Math" display="inline" id="p2.m52">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.25" class="ltx_Math" display="inline" id="p2.m53">
<mn>
1.25
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m54">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.8\overline{3}" class="ltx_Math" display="inline" id="p2.m55">
<mrow>
<mn>
0.8
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.714285" class="ltx_Math" display="inline" id="p2.m56">
<mn>
0.714285
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.625" class="ltx_Math" display="inline" id="p2.m57">
<mn>
0.625
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.5" class="ltx_Math" display="inline" id="p2.m58">
<mn>
0.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.5" class="ltx_Math" display="inline" id="p2.m59">
<mn>
0.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.45" class="ltx_Math" display="inline" id="p2.m60">
<mn>
0.45
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.41\overline{6}" class="ltx_Math" display="inline" id="p2.m61">
<mrow>
<mn>
0.41
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
6
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="6" class="ltx_Math" display="inline" id="p2.m62">
<mn>
6
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="3" class="ltx_Math" display="inline" id="p2.m63">
<mn>
3
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2" class="ltx_Math" display="inline" id="p2.m64">
<mn>
2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.5" class="ltx_Math" display="inline" id="p2.m65">
<mn>
1.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.2" class="ltx_Math" display="inline" id="p2.m66">
<mn>
1.2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m67">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{857142}" class="ltx_Math" display="inline" id="p2.m68">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
857142
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.75" class="ltx_Math" display="inline" id="p2.m69">
<mn>
0.75
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{6}" class="ltx_Math" display="inline" id="p2.m70">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.6" class="ltx_Math" display="inline" id="p2.m71">
<mn>
0.6
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{54}" class="ltx_Math" display="inline" id="p2.m72">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
54
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.5" class="ltx_Math" display="inline" id="p2.m73">
<mn>
0.5
</mn>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
7
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="7" class="ltx_Math" display="inline" id="p2.m74">
<mn>
7
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="3.5" class="ltx_Math" display="inline" id="p2.m75">
<mn>
3.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2.\overline{3}" class="ltx_Math" display="inline" id="p2.m76">
<mrow>
<mn>
2
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.75" class="ltx_Math" display="inline" id="p2.m77">
<mn>
1.75
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.4" class="ltx_Math" display="inline" id="p2.m78">
<mn>
1.4
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.1\overline{6}" class="ltx_Math" display="inline" id="p2.m79">
<mrow>
<mn>
1.1
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m80">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.875" class="ltx_Math" display="inline" id="p2.m81">
<mn>
0.875
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{7}" class="ltx_Math" display="inline" id="p2.m82">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
7
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.7" class="ltx_Math" display="inline" id="p2.m83">
<mn>
0.7
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{63}" class="ltx_Math" display="inline" id="p2.m84">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
63
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.58\overline{3}" class="ltx_Math" display="inline" id="p2.m85">
<mrow>
<mn>
0.58
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
8
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="8" class="ltx_Math" display="inline" id="p2.m86">
<mn>
8
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="4" class="ltx_Math" display="inline" id="p2.m87">
<mn>
4
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2.\overline{6}" class="ltx_Math" display="inline" id="p2.m88">
<mrow>
<mn>
2
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2" class="ltx_Math" display="inline" id="p2.m89">
<mn>
2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.6" class="ltx_Math" display="inline" id="p2.m90">
<mn>
1.6
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{3}" class="ltx_Math" display="inline" id="p2.m91">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{142857}" class="ltx_Math" display="inline" id="p2.m92">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
142857
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m93">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{8}" class="ltx_Math" display="inline" id="p2.m94">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
8
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.8" class="ltx_Math" display="inline" id="p2.m95">
<mn>
0.8
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{72}" class="ltx_Math" display="inline" id="p2.m96">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
72
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{6}" class="ltx_Math" display="inline" id="p2.m97">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
9
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="9" class="ltx_Math" display="inline" id="p2.m98">
<mn>
9
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="4.5" class="ltx_Math" display="inline" id="p2.m99">
<mn>
4.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="3" class="ltx_Math" display="inline" id="p2.m100">
<mn>
3
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2.25" class="ltx_Math" display="inline" id="p2.m101">
<mn>
2.25
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.8" class="ltx_Math" display="inline" id="p2.m102">
<mn>
1.8
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.5" class="ltx_Math" display="inline" id="p2.m103">
<mn>
1.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{285714}" class="ltx_Math" display="inline" id="p2.m104">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
285714
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.125" class="ltx_Math" display="inline" id="p2.m105">
<mn>
1.125
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m106">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.9" class="ltx_Math" display="inline" id="p2.m107">
<mn>
0.9
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{81}" class="ltx_Math" display="inline" id="p2.m108">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
81
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.75" class="ltx_Math" display="inline" id="p2.m109">
<mn>
0.75
</mn>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
10
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="10" class="ltx_Math" display="inline" id="p2.m110">
<mn>
10
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="5" class="ltx_Math" display="inline" id="p2.m111">
<mn>
5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="3.\overline{3}" class="ltx_Math" display="inline" id="p2.m112">
<mrow>
<mn>
3
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2.5" class="ltx_Math" display="inline" id="p2.m113">
<mn>
2.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2" class="ltx_Math" display="inline" id="p2.m114">
<mn>
2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{6}" class="ltx_Math" display="inline" id="p2.m115">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{428571}" class="ltx_Math" display="inline" id="p2.m116">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
428571
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.25" class="ltx_Math" display="inline" id="p2.m117">
<mn>
1.25
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{1}" class="ltx_Math" display="inline" id="p2.m118">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
1
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m119">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.\overline{90}" class="ltx_Math" display="inline" id="p2.m120">
<mrow>
<mn>
0
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
90
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.8\overline{3}" class="ltx_Math" display="inline" id="p2.m121">
<mrow>
<mn>
0.8
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
11
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="11" class="ltx_Math" display="inline" id="p2.m122">
<mn>
11
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="5.5" class="ltx_Math" display="inline" id="p2.m123">
<mn>
5.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="3.\overline{6}" class="ltx_Math" display="inline" id="p2.m124">
<mrow>
<mn>
3
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2.75" class="ltx_Math" display="inline" id="p2.m125">
<mn>
2.75
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2.2" class="ltx_Math" display="inline" id="p2.m126">
<mn>
2.2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.8\overline{3}" class="ltx_Math" display="inline" id="p2.m127">
<mrow>
<mn>
1.8
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{571428}" class="ltx_Math" display="inline" id="p2.m128">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
571428
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.375" class="ltx_Math" display="inline" id="p2.m129">
<mn>
1.375
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{2}" class="ltx_Math" display="inline" id="p2.m130">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
2
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.1" class="ltx_Math" display="inline" id="p2.m131">
<mn>
1.1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m132">
<mn>
1
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="0.91\overline{6}" class="ltx_Math" display="inline" id="p2.m133">
<mrow>
<mn>
0.91
</mn>
<mo>
</mo>
<mover accent="true">
<mn>
6
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r">
12
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="12" class="ltx_Math" display="inline" id="p2.m134">
<mn>
12
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="6" class="ltx_Math" display="inline" id="p2.m135">
<mn>
6
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="4" class="ltx_Math" display="inline" id="p2.m136">
<mn>
4
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="3" class="ltx_Math" display="inline" id="p2.m137">
<mn>
3
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2.4" class="ltx_Math" display="inline" id="p2.m138">
<mn>
2.4
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2" class="ltx_Math" display="inline" id="p2.m139">
<mn>
2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{714285}" class="ltx_Math" display="inline" id="p2.m140">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
714285
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.5" class="ltx_Math" display="inline" id="p2.m141">
<mn>
1.5
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{3}" class="ltx_Math" display="inline" id="p2.m142">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
3
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.2" class="ltx_Math" display="inline" id="p2.m143">
<mn>
1.2
</mn>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1.\overline{09}" class="ltx_Math" display="inline" id="p2.m144">
<mrow>
<mn>
1
</mn>
<mo>
.
</mo>
<mover accent="true">
<mn>
09
</mn>
<mo>
¯
</mo>
</mover>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="1" class="ltx_Math" display="inline" id="p2.m145">
<mn>
1
</mn>
</math>
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
The longest northwest to southeast diagonal obviously contains ones.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/tableofdivisionupto12">
table of division up to 12
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
TableOfDivisionUpTo12
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:36:16
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:36:16
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Data Structure
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A06
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 12E99
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A05
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
<footer class="ltx_page_footer">
<div class="ltx_page_logo">
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|
0 | TableOfMultiplicationUpTo12 | http://planetmath.org/TableOfMultiplicationUpTo12 | <!DOCTYPE html>
<html>
<head>
<title>
table of multiplication up to 12
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
table of multiplication up to 12
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Because of the commutative property of
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
multiplication
</a>
, it does not matter if the row or the column gives the first operand.
</p>
</div>
<div class="ltx_para" id="p2">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
<math alttext="\times" class="ltx_Math" display="inline" id="p2.m1">
<mo>
×
</mo>
</math>
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
1
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
2
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
3
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
4
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
5
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
7
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
8
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
9
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
10
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
11
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
12
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
1
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
1
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
2
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
3
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
4
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
5
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
7
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
8
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
9
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
10
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
11
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
12
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
2
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
2
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
4
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
8
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
10
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
12
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
14
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
16
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
18
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
20
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
22
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
24
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
3
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
3
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
9
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
12
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
15
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
18
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
21
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
24
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
27
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
30
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
33
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
36
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
4
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
4
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
8
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
12
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
16
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
20
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
24
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
28
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
32
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
36
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
40
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
44
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
48
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
5
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
5
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
10
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
15
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
20
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
25
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
30
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
35
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
40
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
45
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
50
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
55
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
60
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
6
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
12
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
18
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
24
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
30
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
36
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
42
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
48
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
54
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
60
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
66
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
72
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
7
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
7
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
14
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
21
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
28
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
35
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
42
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
49
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
56
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
63
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
70
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
77
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
84
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
8
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
8
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
16
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
24
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
32
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
40
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
48
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
56
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
64
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
72
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
80
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
88
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
96
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
9
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
9
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
18
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
27
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
36
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
45
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
54
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
63
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
72
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
81
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
90
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
99
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
108
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
10
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
10
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
20
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
30
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
40
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
50
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
60
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
70
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
80
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
90
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
100
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
110
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
120
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
11
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
11
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
22
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
33
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
44
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
55
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
66
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
77
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
88
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
99
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
110
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
121
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
132
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_l ltx_border_r">
12
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
12
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
24
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
36
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
48
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
60
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
72
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
84
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
96
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
108
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
120
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
132
</td>
<td class="ltx_td ltx_align_right ltx_border_r">
144
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Obviously the longest northwest to southeast diagonal contains numbers of the form
<math alttext="n^{2}" class="ltx_Math" display="inline" id="p3.m1">
<msup>
<mi>
n
</mi>
<mn>
2
</mn>
</msup>
</math>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/tableofmultiplicationupto12">
table of multiplication up to 12
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
TableOfMultiplicationUpTo12
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:36:14
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:36:14
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Data Structure
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A06
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A05
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 11B25
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
<footer class="ltx_page_footer">
<div class="ltx_page_logo">
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</html>
|
0 | ThereforeSign | http://planetmath.org/ThereforeSign | <!DOCTYPE html>
<html>
<head>
<title>
therefore sign
</title>
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
therefore sign
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The
<span class="ltx_text ltx_font_italic">
therefore sign
</span>
“
<math alttext="\therefore" class="ltx_Math" display="inline" id="p1.m1">
<mo>
∴
</mo>
</math>
” is used especially in handwritten mathematical text as a shorthand of the or
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
relations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="S_{1}\quad\therefore\;S_{2}" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<msub>
<mi>
S
</mi>
<mn>
1
</mn>
</msub>
<mo mathvariant="italic" separator="true">
</mo>
<mo rspace="5.3pt">
∴
</mo>
<msub>
<mi>
S
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
It expresses that
<math alttext="S_{2}" class="ltx_Math" display="inline" id="p1.m2">
<msub>
<mi>
S
</mi>
<mn>
2
</mn>
</msub>
</math>
has been inferred from
<math alttext="S_{1}" class="ltx_Math" display="inline" id="p1.m3">
<msub>
<mi>
S
</mi>
<mn>
1
</mn>
</msub>
</math>
or from
<math alttext="S_{1}" class="ltx_Math" display="inline" id="p1.m4">
<msub>
<mi>
S
</mi>
<mn>
1
</mn>
</msub>
</math>
and some preceding facts. The sign is rather a punctuation mark than a symbol of
<a class="nnexus_concept" href="http://planetmath.org/logicalimplication">
logical implication
</a>
. Grammatically, it could be characterised a conclusive coordinating . The usage of the symbol is not mathematically
<a class="nnexus_concept" href="http://planetmath.org/welldefined">
well-defined
</a>
, and it often means ‘we can conclude in
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
context
</a>
’ or ‘we can conclude from statements already shown or assumed to be true’.
<br class="ltx_break"/>
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
For example, in determining an angle of a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
right triangle
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RightTriangle.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/triangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, one may write
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\sin\alpha=\frac{1}{2}\quad\therefore\;\alpha=30^{\circ}" class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mi>
sin
</mi>
<mi>
α
</mi>
<mo>
=
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo mathvariant="italic" separator="true">
</mo>
<mo rspace="5.3pt">
∴
</mo>
<mi>
α
</mi>
<mo>
=
</mo>
<msup>
<mn>
30
</mn>
<mo>
∘
</mo>
</msup>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Here, “
<math alttext="\therefore" class="ltx_Math" display="inline" id="p2.m1">
<mo>
∴
</mo>
</math>
” does not a proper
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
implication
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Implication.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/implication">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
“
<math alttext="\Rightarrow" class="ltx_Math" display="inline" id="p2.m2">
<mo>
⇒
</mo>
</math>
”, since the exact implication here would be
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\sin\alpha=\frac{1}{2}\;\,\Leftarrow\;\,\alpha=30^{\circ}." class="ltx_Math" display="block" id="S0.Ex3.m1">
<mrow>
<mrow>
<mrow>
<mi>
sin
</mi>
<mo>
</mo>
<mi>
α
</mi>
</mrow>
<mo>
=
</mo>
<mpadded width="+4.4pt">
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
</mpadded>
<mo rspace="6.9pt">
⇐
</mo>
<mi>
α
</mi>
<mo>
=
</mo>
<msup>
<mn>
30
</mn>
<mo>
∘
</mo>
</msup>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
To obtain a strict implication, we would need to introduce some of the context.
For
<a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">
instance
</a>
, we know that, since
<math alttext="\alpha" class="ltx_Math" display="inline" id="p2.m3">
<mi>
α
</mi>
</math>
is an angle of a right triangle,
<math alttext="0^{\circ}\leq\alpha\leq 90^{\circ}" class="ltx_Math" display="inline" id="p2.m4">
<mrow>
<msup>
<mn>
0
</mn>
<mo>
∘
</mo>
</msup>
<mo>
≤
</mo>
<mi>
α
</mi>
<mo>
≤
</mo>
<msup>
<mn>
90
</mn>
<mo>
∘
</mo>
</msup>
</mrow>
</math>
,
so what we wrote could be interpreted as the implication
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\sin\alpha=\frac{1}{2}\;\land\;0^{\circ}\leq\alpha\leq 90^{\circ}\;\,%
\Rightarrow\;\,\alpha=30^{\circ}." class="ltx_Math" display="block" id="S0.Ex4.m1">
<mrow>
<mrow>
<mrow>
<mi>
sin
</mi>
<mo>
</mo>
<mi>
α
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mpadded width="+2.8pt">
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
</mpadded>
<mo>
∧
</mo>
<msup>
<mn>
0
</mn>
<mo>
∘
</mo>
</msup>
</mrow>
<mo>
≤
</mo>
<mi>
α
</mi>
<mo>
≤
</mo>
<mpadded width="+4.4pt">
<msup>
<mn>
90
</mn>
<mo>
∘
</mo>
</msup>
</mpadded>
<mo rspace="6.9pt">
⇒
</mo>
<mi>
α
</mi>
<mo>
=
</mo>
<msup>
<mn>
30
</mn>
<mo>
∘
</mo>
</msup>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
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pahio (2872)
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</th>
<td class="ltx_td ltx_align_left ltx_border_r">
8
</td>
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pahio (2872)
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msc 00A06
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msc 00A05
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RingsOfRationalNumbers
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LogarithmicScale
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0 | NesbittsInequality | http://planetmath.org/NesbittsInequality | <!DOCTYPE html>
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<title>
Nesbitt’s inequality
</title>
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<h1 class="ltx_title ltx_title_document">
Nesbitt’s inequality
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Nesbitt’s
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
inequality
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Inequality.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inequalitiesforrealnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
says, that for positive real
<math alttext="a" class="ltx_Math" display="inline" id="p1.m1">
<mi>
a
</mi>
</math>
,
<math alttext="b" class="ltx_Math" display="inline" id="p1.m2">
<mi>
b
</mi>
</math>
and
<math alttext="c" class="ltx_Math" display="inline" id="p1.m3">
<mi>
c
</mi>
</math>
we have:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}." class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mrow>
<mfrac>
<mi>
a
</mi>
<mrow>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mi>
b
</mi>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mi>
c
</mi>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
</mfrac>
</mrow>
<mo>
≥
</mo>
<mfrac>
<mn>
3
</mn>
<mn>
2
</mn>
</mfrac>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
This is a special case of Shapiro’s inequality.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
Nesbitt’s inequality
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
NesbittsInequality
</td>
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<tr class="ltx_tr">
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Date of creation
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<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:36:59
</td>
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Last modified on
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<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:36:59
</td>
</tr>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathwizard (128)
</td>
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mathwizard (128)
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Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
</tr>
<tr class="ltx_tr">
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Author
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<td class="ltx_td ltx_align_left ltx_border_r">
mathwizard (128)
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Theorem
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<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
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<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A07
</td>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
ShapiroInequality
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0 | ProofOfNesbittsInequality | http://planetmath.org/ProofOfNesbittsInequality | <!DOCTYPE html>
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<title>
proof of Nesbitt’s inequality
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
proof of Nesbitt’s inequality
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Starting from Nesbitt’s
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
inequality
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Inequality.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inequalitiesforrealnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mfrac>
<mi>
a
</mi>
<mrow>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mi>
b
</mi>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mi>
c
</mi>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
</mfrac>
</mrow>
<mo>
≥
</mo>
<mfrac>
<mn>
3
</mn>
<mn>
2
</mn>
</mfrac>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
we transform the left hand side:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}-3\geq\frac{3}{2}." class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mrow>
<mrow>
<mrow>
<mfrac>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
<mrow>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
</mfrac>
</mrow>
<mo>
-
</mo>
<mn>
3
</mn>
</mrow>
<mo>
≥
</mo>
<mfrac>
<mn>
3
</mn>
<mn>
2
</mn>
</mfrac>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Now this can be transformed into:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="((a+b)+(a+c)+(b+c))\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)\geq 9." class="ltx_Math" display="block" id="S0.Ex3.m1">
<mrow>
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
+
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
+
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo>
(
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<mrow>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
</mrow>
<mo>
)
</mo>
</mrow>
</mrow>
<mo>
≥
</mo>
<mn>
9
</mn>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Division by 3 and the right yields:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{(a+b)+(a+c)+(b+c)}{3}\geq\frac{3}{\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b%
+c}}." class="ltx_Math" display="block" id="S0.Ex4.m1">
<mrow>
<mrow>
<mfrac>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
+
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
+
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mn>
3
</mn>
</mfrac>
<mo>
≥
</mo>
<mfrac>
<mn>
3
</mn>
<mrow>
<mfrac>
<mn>
1
</mn>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<mrow>
<mi>
b
</mi>
<mo>
+
</mo>
<mi>
c
</mi>
</mrow>
</mfrac>
</mrow>
</mfrac>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Now on the left we have the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
arithmetic mean
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ArithmeticMean.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/arithmeticmean">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and on the right the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
harmonic mean
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/HarmonicMean.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/harmonicmean">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, so this inequality is true.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
proof of Nesbitt’s inequality
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
ProofOfNesbittsInequality
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:37:01
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 12:37:01
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathwizard (128)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathwizard (128)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathwizard (128)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Proof
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A07
</td>
</tr>
</tbody>
</table>
</div>
</article>
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|
0 | CabtaxiNumber | http://planetmath.org/CabtaxiNumber | <!DOCTYPE html>
<html>
<head>
<title>
cabtaxi number
</title>
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<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
cabtaxi number
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
cabtaxi number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CabtaxiNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cabtaxinumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
for a given
<math alttext="n" class="ltx_Math" display="inline" id="p1.m1">
<mi>
n
</mi>
</math>
is the smallest positive number which can be written as
<math alttext="a^{3}+b^{3}" class="ltx_Math" display="inline" id="p1.m2">
<mrow>
<msup>
<mi>
a
</mi>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mi>
b
</mi>
<mn>
3
</mn>
</msup>
</mrow>
</math>
in
<math alttext="n" class="ltx_Math" display="inline" id="p1.m3">
<mi>
n
</mi>
</math>
different ways, with either
<math alttext="a" class="ltx_Math" display="inline" id="p1.m4">
<mi>
a
</mi>
</math>
or
<math alttext="b" class="ltx_Math" display="inline" id="p1.m5">
<mi>
b
</mi>
</math>
allowed to be negative integers. For example, 91 is the 2nd cabtaxi number since it can be expressed
<math alttext="(-5)^{3}+6^{3}=3^{3}+4^{3}=91" class="ltx_Math" display="inline" id="p1.m6">
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
5
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mn>
6
</mn>
<mn>
3
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mn>
3
</mn>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mn>
4
</mn>
<mn>
3
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<mn>
91
</mn>
</mrow>
</math>
. The known cabtaxi numbers are 1, 91, 728, 2741256, 6017193, 1412774811, 11302198488, 137513849003496, 424910390480793000, listed in A047696 of Sloane’s OEIS. Adding the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
restriction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/subalgebraofanalgebraicsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/subfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="a\geq b>0" class="ltx_Math" display="inline" id="p1.m7">
<mrow>
<mi>
a
</mi>
<mo>
≥
</mo>
<mi>
b
</mi>
<mo>
>
</mo>
<mn>
0
</mn>
</mrow>
</math>
gives the
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
for the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
taxicab numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TaxicabNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/taxicabnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
cabtaxi number
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CabtaxiNumber
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:56:08
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:56:08
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
4
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
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Entry type
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<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A08
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0 | IndexOfImportantIrrationalConstants | http://planetmath.org/IndexOfImportantIrrationalConstants | <!DOCTYPE html>
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<title>
index of important irrational constants
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
index of important irrational constants
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The following table lists some of the most important
<a class="nnexus_concept" href="http://planetmath.org/irrational">
irrational
</a>
constants in mathematics.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Of course importance is sometimes debatable. Hardly anyone disputes the importance of
<math alttext="\pi" class="ltx_Math" display="inline" id="p2.m1">
<mi>
π
</mi>
</math>
or
<math alttext="e" class="ltx_Math" display="inline" id="p2.m2">
<mi>
e
</mi>
</math>
(in fact, these are the only two constants in the OEIS to have the keyword “core” attached to them), but for other constants it is not quite clear cut. In general, if a given constant has a name (especially a name hyphenating two famous mathematicians’ last names) I consider it important.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Irrationality is not always clear cut either, e.g., it might be a mistake to exclude the
<a class="nnexus_concept" href="http://planetmath.org/eulersconstant">
Euler-Mascheroni constant
</a>
<math alttext="\gamma" class="ltx_Math" display="inline" id="p3.m1">
<mi>
γ
</mi>
</math>
from this list.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
The constants are given to 20
<a class="nnexus_concept" href="http://planetmath.org/decimalplace">
decimal places
</a>
.
</p>
</div>
<div class="ltx_para" id="p5">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.1149420448532962007
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Kepler-BouwkampConstant.html">
Kepler-Bouwkamp constant
</a>
or polygon-inscribing constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.1234567891011121314
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Champernowne’s constant
<math alttext="C_{10}" class="ltx_Math" display="inline" id="p5.m1">
<msub>
<mi>
C
</mi>
<mn>
10
</mn>
</msub>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.2078795763507619085
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="i^{i}" class="ltx_Math" display="inline" id="p5.m2">
<msup>
<mi>
i
</mi>
<mi>
i
</mi>
</msup>
</math>
(has no
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.9#E2">
imaginary part
</a>
) or
<math alttext="e^{\frac{-\pi}{2}}" class="ltx_Math" display="inline" id="p5.m3">
<msup>
<mi>
e
</mi>
<mfrac>
<mrow>
<mo>
-
</mo>
<mi>
π
</mi>
</mrow>
<mn>
2
</mn>
</mfrac>
</msup>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.2357111317192329313
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CopelandErdsConstant
</span>
Copeland-Erdős constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.2614972128476427837
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Meissel-Mertens constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.3275822918721811159
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Lévy’s constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.4146825098511116602
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
The
<a class="nnexus_concept" href="http://planetmath.org/primeconstant">
prime constant
</a>
<math alttext="\rho" class="ltx_Math" display="inline" id="p5.m4">
<mi>
ρ
</mi>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.5926327182016361971
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Lehmer’s constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.6079271018540266286
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="{6\over{\pi^{2}}}" class="ltx_Math" display="inline" id="p5.m5">
<mfrac>
<mn>
6
</mn>
<msup>
<mi>
π
</mi>
<mn>
2
</mn>
</msup>
</mfrac>
</math>
, the probability that a random
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
is
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
squarefree
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Squarefree.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/squarefreenumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.6434105462883380261
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Cahen’s constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.7642236535892206629
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Landau-RamanujanConstant.html">
Landau-Ramanujan constant
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.8346268416740731862
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Gauss’s constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.8862269254527580136
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="\Gamma(\frac{3}{2})=\frac{1}{2}\sqrt{\pi}" class="ltx_Math" display="inline" id="p5.m6">
<mrow>
<mrow>
<mi mathvariant="normal">
Γ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mfrac>
<mn>
3
</mn>
<mn>
2
</mn>
</mfrac>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo>
</mo>
<msqrt>
<mi>
π
</mi>
</msqrt>
</mrow>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
0.9159655941772190150
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Catalan’s constant
<math alttext="K" class="ltx_Math" display="inline" id="p5.m7">
<mi>
K
</mi>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.2020569031595942853
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Apéry’s constant
<math alttext="\zeta(3)" class="ltx_Math" display="inline" id="p5.m8">
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.2254167024651776451
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="\Gamma(\frac{3}{4})" class="ltx_Math" display="inline" id="p5.m9">
<mrow>
<mi mathvariant="normal">
Γ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mfrac>
<mn>
3
</mn>
<mn>
4
</mn>
</mfrac>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.3063778838630806904
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mills’ constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.3247179572447460260
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
The
<a class="nnexus_concept" href="http://planetmath.org/plasticconstant">
plastic constant
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.4142135623730950488
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Square root
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SquareRoot.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/squareroot">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of two
<math alttext="\sqrt{2}" class="ltx_Math" display="inline" id="p5.m10">
<msqrt>
<mn>
2
</mn>
</msqrt>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.4513692348833810502
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Ramanujan-SoldnerConstant.html">
Ramanujan-Soldner constant
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.6066951524152917637
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Erdős-Borwein constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.6180339887498948482
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
The
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
golden ratio
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GoldenRatio.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/goldenratio">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="\phi" class="ltx_Math" display="inline" id="p5.m11">
<mi>
ϕ
</mi>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.6449340668482264364
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="\zeta(2)=\frac{\pi^{2}}{6}" class="ltx_Math" display="inline" id="p5.m12">
<mrow>
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mfrac>
<msup>
<mi>
π
</mi>
<mn>
2
</mn>
</msup>
<mn>
6
</mn>
</mfrac>
</mrow>
</math>
, the solution to the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Basel problem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BaselProblem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/baselproblem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.7320508075688772935
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Square root of three
<math alttext="\sqrt{3}" class="ltx_Math" display="inline" id="p5.m13">
<msqrt>
<mn>
3
</mn>
</msqrt>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.7579327566180045327
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Vijayaraghavan’s infinite
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NestedRadical.html">
nested radical
</a>
<math alttext="\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5+\ldots}}}}}" class="ltx_Math" display="inline" id="p5.m14">
<msqrt>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<msqrt>
<mrow>
<mn>
2
</mn>
<mo>
+
</mo>
<msqrt>
<mrow>
<mn>
3
</mn>
<mo>
+
</mo>
<msqrt>
<mrow>
<mn>
4
</mn>
<mo>
+
</mo>
<msqrt>
<mrow>
<mn>
5
</mn>
<mo>
+
</mo>
<mi mathvariant="normal">
…
</mi>
</mrow>
</msqrt>
</mrow>
</msqrt>
</mrow>
</msqrt>
</mrow>
</msqrt>
</mrow>
</msqrt>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
1.7724538509055160273
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="\Gamma(\frac{1}{2})=\sqrt{\pi}" class="ltx_Math" display="inline" id="p5.m15">
<mrow>
<mrow>
<mi mathvariant="normal">
Γ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<msqrt>
<mi>
π
</mi>
</msqrt>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
2.2360679774997896964
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Square root of five
<math alttext="\sqrt{5}" class="ltx_Math" display="inline" id="p5.m16">
<msqrt>
<mn>
5
</mn>
</msqrt>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
2.6651441426902251887
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="2^{\sqrt{2}}" class="ltx_Math" display="inline" id="p5.m17">
<msup>
<mn>
2
</mn>
<msqrt>
<mn>
2
</mn>
</msqrt>
</msup>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
2.6854520010653064453
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Khinchin’s constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
2.4142135623730950488
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
The
<a class="nnexus_concept" href="http://planetmath.org/silverratio">
silver ratio
</a>
<math alttext="\delta_{S}" class="ltx_Math" display="inline" id="p5.m18">
<msub>
<mi>
δ
</mi>
<mi>
S
</mi>
</msub>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
2.5849817595792532170
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Sierpiński’s constant
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
2.7182818284590452354
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
The
<a class="nnexus_concept" href="http://planetmath.org/naturallogbase">
natural log base
</a>
<math alttext="e" class="ltx_Math" display="inline" id="p5.m19">
<mi>
e
</mi>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
3.1415926535897932385
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
The ratio of a circle’s radius to its
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
circumference
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Circumference.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/circle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="\pi" class="ltx_Math" display="inline" id="p5.m20">
<mi>
π
</mi>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
3.6256099082219083119
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="\Gamma(\frac{1}{4})" class="ltx_Math" display="inline" id="p5.m21">
<mrow>
<mi mathvariant="normal">
Γ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mfrac>
<mn>
1
</mn>
<mn>
4
</mn>
</mfrac>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
4.1327313541224929385
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="\sqrt{2e\pi}" class="ltx_Math" display="inline" id="p5.m22">
<msqrt>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
</msqrt>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
4.6692116609102990671
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Feigenbaum’s constant
<math alttext="\delta" class="ltx_Math" display="inline" id="p5.m23">
<mi>
δ
</mi>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
7.3890560989306502272
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="e^{2}" class="ltx_Math" display="inline" id="p5.m24">
<msup>
<mi>
e
</mi>
<mn>
2
</mn>
</msup>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
14.1347251417346937904
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
The imaginary part of the first nontrivial zero of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Riemann zeta function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/25.1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://dlmf.nist.gov/25.2#E1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RiemannZetaFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/riemannzetafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(the real part is
<math alttext="\frac{1}{2}" class="ltx_Math" display="inline" id="p5.m25">
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
</math>
)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
15.1542622414792641898
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="e^{e}" class="ltx_Math" display="inline" id="p5.m26">
<msup>
<mi>
e
</mi>
<mi>
e
</mi>
</msup>
</math>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r">
36.4621596072079117710
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<math alttext="\pi^{\pi}" class="ltx_Math" display="inline" id="p5.m27">
<msup>
<mi>
π
</mi>
<mi>
π
</mi>
</msup>
</math>
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
In looking these up in the OEIS, you can simply type them with a
<a class="nnexus_concept" href="http://planetmath.org/decimalpoint">
decimal point
</a>
and no commas between the digits. If you get no results, try chopping off a couple of the least significant digits.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Alan Jeffrey,
<span class="ltx_text ltx_font_italic">
Handbook of Mathematical Formulas and
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.4#iv">
Integrals
</a>
</span>
, 3rd Edition. New York: Elsevier Academic Press (2004): 223,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Section.html">
Section
</a>
11.1.4 Special values of
<math alttext="\Gamma(x)" class="ltx_Math" display="inline" id="bib.bib1.m1">
<mrow>
<mi mathvariant="normal">
Γ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p7">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/indexofimportantirrationalconstants">
index of important irrational constants
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
IndexOfImportantIrrationalConstants
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:03:10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:03:10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
17
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Topic
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A08
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind | http://planetmath.org/LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind | <!DOCTYPE html>
<html>
<head>
<title>
large integers that are or might be the smallest of their kind
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
large integers that are or might be the smallest of their kind
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
For the purpose of this feature, the arbitrary cutoff is
<math alttext="10^{7}" class="ltx_Math" display="inline" id="p2.m1">
<msup>
<mn>
10
</mn>
<mn>
7
</mn>
</msup>
</math>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
19099919
</span>
is the smallest prime to start a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Cunningham chain
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CunninghamChain.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cunninghamchain">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of length 8.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
85864769
</span>
is the smallest prime to start a Cunningham chain of length 9.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
545587687
</span>
is the smallest class 13+ prime in the
<a class="nnexus_concept" href="http://planetmath.org/erdhosselfridgeclassificationofprimes">
Erdos-Selfridge classification of primes
</a>
.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
635318657
</span>
is the smallest number that can be expressed as a sum of two
<a class="nnexus_concept" href="http://planetmath.org/fourthpower">
fourth powers
</a>
in two different ways.
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
823766851
</span>
is the smallest prime with
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
primitive root
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimitiveRoot.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/primitiveroot">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
48.
</p>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
906150257
</span>
is the smallest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
counterexample
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Counterexample.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/counterexample">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to Pólya’s
<a class="nnexus_concept" href="http://planetmath.org/openquestion">
conjecture
</a>
.
</p>
</div>
<div class="ltx_para" id="p9">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
1023456789
</span>
is the smallest
<a class="nnexus_concept" href="http://planetmath.org/pandigitalnumber">
pandigital number
</a>
in base 10.
</p>
</div>
<div class="ltx_para" id="p10">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
1704961513
</span>
is the smallest class 14+ prime in the Erdős-Selfridge classification of primes.
</p>
</div>
<div class="ltx_para" id="p11">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
10123457689
</span>
is the smallest pandigital prime in base 10.
</p>
</div>
<div class="ltx_para" id="p12">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
26089808579
</span>
is the smallest prime to start a Cunningham chain of length 10.
</p>
</div>
<div class="ltx_para" id="p13">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
665043081119
</span>
is the smallest prime to start a Cunningham chain of length 11.
</p>
</div>
<div class="ltx_para" id="p14">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
554688278429
</span>
is the smallest prime to start a Cunningham chain of length 12.
</p>
</div>
<div class="ltx_para" id="p15">
<p class="ltx_p">
<math alttext="10^{13}+1" class="ltx_Math" display="inline" id="p15.m1">
<mrow>
<msup>
<mn>
10
</mn>
<mn>
13
</mn>
</msup>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
is, as of 2005, the smallest candidate for a counterexample to the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Mertens conjecture
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MertensConjecture.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mertensconjecture">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(though the smallest counterexample could turn out to be as large as
<math alttext="3.21\times 10^{64}" class="ltx_Math" display="inline" id="p15.m2">
<mrow>
<mn>
3.21
</mn>
<mo>
×
</mo>
<msup>
<mn>
10
</mn>
<mn>
64
</mn>
</msup>
</mrow>
</math>
).
</p>
</div>
<div class="ltx_para" id="p16">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
4090932431513069
</span>
is the smallest prime to start a Cunningham chain of length 13.
</p>
</div>
<div class="ltx_para" id="p17">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
95405042230542329
</span>
is the smallest prime to start a Cunningham chain of length 14.
</p>
</div>
<div class="ltx_para" id="p18">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
810433818265726529159
</span>
is the smallest prime known to start a Cunningham chain of length 16, but there could be a smaller such prime.
</p>
</div>
<div class="ltx_para" id="p19">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
439351292910452432574786963588089477522344721
</span>
is the smallest prime in Paul Hoffman’s erroneous version of Wilf’s
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
primefree sequence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimefreeSequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/primefreesequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in which
<math alttext="a_{1}=3794765361567513" class="ltx_Math" display="inline" id="p19.m1">
<mrow>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<mn>
3794765361567513
</mn>
</mrow>
</math>
,
<math alttext="a_{2}=20615674205555510" class="ltx_Math" display="inline" id="p19.m2">
<mrow>
<msub>
<mi>
a
</mi>
<mn>
2
</mn>
</msub>
<mo>
=
</mo>
<mn>
20615674205555510
</mn>
</mrow>
</math>
and
<math alttext="a_{n}=a_{n-2}+a_{n-1}" class="ltx_Math" display="inline" id="p19.m3">
<mrow>
<msub>
<mi>
a
</mi>
<mi>
n
</mi>
</msub>
<mo>
=
</mo>
<mrow>
<msub>
<mi>
a
</mi>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
</msub>
<mo>
+
</mo>
<msub>
<mi>
a
</mi>
<mrow>
<mi>
n
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msub>
</mrow>
</mrow>
</math>
for
<math alttext="n>2" class="ltx_Math" display="inline" id="p19.m4">
<mrow>
<mi>
n
</mi>
<mo>
>
</mo>
<mn>
2
</mn>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p20">
<p class="ltx_p">
If an odd perfect number exists, it is at least
<math alttext="10^{300}+1" class="ltx_Math" display="inline" id="p20.m1">
<mrow>
<msup>
<mn>
10
</mn>
<mn>
300
</mn>
</msup>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
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Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
large
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
that are or might be the smallest of their kind
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
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</a>
name
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<td class="ltx_td ltx_align_left ltx_border_r">
LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind
</td>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:04:14
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:04:14
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mravinci (12996)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mravinci (12996)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
15
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mravinci (12996)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
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</td>
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Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A08
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
SmallIntegersThatAreOrMightBeTheLargestOfTheirKind
</td>
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small integers that are or might be the largest of their kind
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
small integers that are or might be the largest of their kind
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
1
</span>
is the largest
<a class="nnexus_concept" href="http://planetmath.org/sumproductnumber">
sum-product number
</a>
in binary, and the largest to be a sum-product number in any standard positional base. 1 is the largest
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
whose
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Zeckendorf representation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ZeckendorfRepresentation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/zeckendorfstheorem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
has no significant zeroes. Also, it is the largest (and the only) integer to be labelled non-prime without ever being a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
probable prime
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ProbablePrime.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/probableprime">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
with unknown factorization.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
2
</span>
is the largest (and the only)
<a class="nnexus_concept" href="http://mathworld.wolfram.com/EvenPrime.html">
even prime
</a>
number. Also, it might be the largest
<math alttext="n" class="ltx_Math" display="inline" id="p2.m1">
<mi>
n
</mi>
</math>
such that no
<math alttext="n\times n" class="ltx_Math" display="inline" id="p2.m2">
<mrow>
<mi>
n
</mi>
<mo>
×
</mo>
<mi>
n
</mi>
</mrow>
</math>
<a class="nnexus_concept" href="http://planetmath.org/magicsquare">
magic square
</a>
consisting of consecutive primes can be constructed. The total of such magic squares is only known up to
<math alttext="n=6" class="ltx_Math" display="inline" id="p2.m3">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mn>
6
</mn>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
3
</span>
is the largest and only prime
<a class="nnexus_concept" href="http://planetmath.org/perfecttotientnumber">
perfect totient number
</a>
.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
4
</span>
is the largest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
composite number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CompositeNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/compositenumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
such that the union of the set of its
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
totatives
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Totative.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/totative">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and the set of its
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
divisors
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Divisor.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divisibility">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divisortheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complete
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/ordersinanumberfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/completebinarytree">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/completegraph">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
range of integers from 1 to itself. All larger numbers satisfying this
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/RelationshipBetweenTotativesAndDivisors
</span>
relationship between their totatives and divisors are prime. In another commonality with primes, 4 might be the largest composite
<math alttext="n" class="ltx_Math" display="inline" id="p4.m1">
<mi>
n
</mi>
</math>
such that
<math alttext="\phi(n)\sigma_{0}(n)+2" class="ltx_Math" display="inline" id="p4.m2">
<mrow>
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<msub>
<mi>
σ
</mi>
<mn>
0
</mn>
</msub>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
</math>
is a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
multiple
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Multiple.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generalassociativity">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<math alttext="n" class="ltx_Math" display="inline" id="p4.m3">
<mi>
n
</mi>
</math>
, with
<math alttext="\phi(n)" class="ltx_Math" display="inline" id="p4.m4">
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
being Euler’s
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TotientFunction.html">
totient function
</a>
and
<math alttext="\sigma_{0}(n)" class="ltx_Math" display="inline" id="p4.m5">
<mrow>
<msub>
<mi>
σ
</mi>
<mn>
0
</mn>
</msub>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
being a count of the divisors. (For a prime
<math alttext="p" class="ltx_Math" display="inline" id="p4.m6">
<mi>
p
</mi>
</math>
,
<math alttext="\phi(p)\sigma_{0}(p)+2=2p" class="ltx_Math" display="inline" id="p4.m7">
<mrow>
<mrow>
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<msub>
<mi>
σ
</mi>
<mn>
0
</mn>
</msub>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
p
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
p
</mi>
</mrow>
</mrow>
</math>
).
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
5
</span>
might be the largest
<a class="nnexus_concept" href="http://planetmath.org/untouchablenumber">
untouchable number
</a>
to be odd and prime. If a larger, odd though composite untouchable number were to be found, 5 would retain the distinction of being the largest such prime. Finding a larger prime untouchable number would strip 5 of both distinctions.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
6
</span>
is the largest integer to be both a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
factorial
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Factorial.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/factorial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and a
<a class="nnexus_concept" href="http://planetmath.org/primorial">
primorial
</a>
, and it’s the largest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
squarefree
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Squarefree.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/squarefreenumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
factorial. 6 is the largest
<math alttext="n" class="ltx_Math" display="inline" id="p6.m1">
<mi>
n
</mi>
</math>
for which the inequality
<math alttext="\phi(n)>\sqrt{n}" class="ltx_Math" display="inline" id="p6.m2">
<mrow>
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
>
</mo>
<msqrt>
<mi>
n
</mi>
</msqrt>
</mrow>
</math>
is false. Also, it is a
<a class="nnexus_concept" href="http://planetmath.org/harshadnumber">
Harshad number
</a>
regardless of the base in any standard positional base notation. 6 is 110 in binary, 20 in ternary, 12 in base 4, 11 in base 5 and 10 in its own base, and then 6 in all bases afterwards. It is not palindromic in binary, ternary or quaternary, and is thus the largest composite
<a class="nnexus_concept" href="http://planetmath.org/strictlynonpalindromicnumber">
strictly non-palindromic number
</a>
.
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
8
</span>
is the largest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Fibonacci number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FibonacciNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fibonaccisequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
that is also a cube.
</p>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
9
</span>
is the largest composite center of a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
prime quadruplet
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimeQuadruplet.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/primequadruplet">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
that is not a multiple of 15.
</p>
</div>
<div class="ltx_para" id="p9">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
19
</span>
is the largest prime Roman numeral
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
palindromic number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PalindromicNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/palindromicnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
(XIX). If one allows overlines, one would have to ignore the
<a class="nnexus_concept" href="http://planetmath.org/quantumsymmetry">
symmetry
</a>
of the overlines in order to permit for a larger prime
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
palindrome
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Palindrome.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/palindrome">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p10">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
23
</span>
is the largest integer to have the same
<a class="nnexus_concept" href="http://planetmath.org/grouprepresentation">
representation
</a>
in both
<a class="nnexus_concept" href="http://planetmath.org/factorialbase">
factorial base
</a>
and primorial base (specifically, 321).
</p>
</div>
<div class="ltx_para" id="p11">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
24
</span>
is the largest
<math alttext="n" class="ltx_Math" display="inline" id="p11.m1">
<mi>
n
</mi>
</math>
such that
<math alttext="m|n" class="ltx_Math" display="inline" id="p11.m2">
<mrow>
<mi>
m
</mi>
<mo stretchy="false">
|
</mo>
<mi>
n
</mi>
</mrow>
</math>
for all
<math alttext="0<m<\sqrt{n}" class="ltx_Math" display="inline" id="p11.m3">
<mrow>
<mn>
0
</mn>
<mo>
<
</mo>
<mi>
m
</mi>
<mo>
<
</mo>
<msqrt>
<mi>
n
</mi>
</msqrt>
</mrow>
</math>
. Also, it is the largest integer to satisfy the equality
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\sum_{i=1}^{n}i^{2}=m^{2}," class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mrow>
<munderover>
<mo largeop="true" movablelimits="false" symmetric="true">
∑
</mo>
<mrow>
<mi>
i
</mi>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
<mi>
n
</mi>
</munderover>
<msup>
<mi>
i
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<msup>
<mi>
m
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
where
<math alttext="m" class="ltx_Math" display="inline" id="p11.m4">
<mi>
m
</mi>
</math>
is an integer. [Tattersall, 2005]
</p>
</div>
<div class="ltx_para" id="p12">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
26
</span>
is the largest (and only) integer sandwiched between a square and a cube.
</p>
</div>
<div class="ltx_para" id="p13">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
30
</span>
is the largest integer such that none of its totatives are composite,
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/EveryPositiveIntegerGreaterThan30HasAtLeastOneCompositeTotative
</span>
all greater integers have at least one composite totative. The count of those totatives happens to be equal to the count of its divisors, 30 is the largest integer for which this is true.
</p>
</div>
<div class="ltx_para" id="p14">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
41
</span>
is the largest
<math alttext="n" class="ltx_Math" display="inline" id="p14.m1">
<mi>
n
</mi>
</math>
such that the
<a class="nnexus_concept" href="http://planetmath.org/polynomialring">
polynomial
</a>
<math alttext="m^{2}-m+n" class="ltx_Math" display="inline" id="p14.m2">
<mrow>
<mrow>
<msup>
<mi>
m
</mi>
<mn>
2
</mn>
</msup>
<mo>
-
</mo>
<mi>
m
</mi>
</mrow>
<mo>
+
</mo>
<mi>
n
</mi>
</mrow>
</math>
yields primes for any
<a class="nnexus_concept" href="http://planetmath.org/positive">
positive
</a>
<math alttext="m<n" class="ltx_Math" display="inline" id="p14.m3">
<mrow>
<mi>
m
</mi>
<mo>
<
</mo>
<mi>
n
</mi>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p15">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
46
</span>
is the largest
<a class="nnexus_concept" href="http://planetmath.org/evennumber">
even integer
</a>
for which there is no pair of abudant numbers that add up to it. (See the empirical proof that every sufficiently large even integer can be expressed as the sum of a pair of
<a class="nnexus_concept" href="http://planetmath.org/abundantnumber">
abundant numbers
</a>
).
</p>
</div>
<div class="ltx_para" id="p16">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
55
</span>
is the largest Fibonacci number that is also a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
triangular number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TriangularNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/triangularnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p17">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
60
</span>
is thought to be the largest integer that does not admit to a representation under Chen’s
<a class="nnexus_concept" href="http://planetmath.org/lemma">
theorem
</a>
(as a sum of two distinct primes or a sum of a prime and a
<a class="nnexus_concept" href="http://planetmath.org/semiprime">
semiprime
</a>
; see A100952 in Sloane’s OEIS).60 is also the largest
<math alttext="n" class="ltx_Math" display="inline" id="p17.m1">
<mi>
n
</mi>
</math>
such that
<math alttext="\pi(n)<\phi(n)" class="ltx_Math" display="inline" id="p17.m2">
<mrow>
<mrow>
<mi>
π
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
<
</mo>
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
is false (with
<math alttext="\pi(x)" class="ltx_Math" display="inline" id="p17.m3">
<mrow>
<mi>
π
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
being the
<a class="nnexus_concept" href="http://planetmath.org/primecountingfunction">
prime counting function
</a>
).
</p>
</div>
<div class="ltx_para" id="p18">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
61
</span>
might be the largest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
prime number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimeNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/prime">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="p_{x}" class="ltx_Math" display="inline" id="p18.m1">
<msub>
<mi>
p
</mi>
<mi>
x
</mi>
</msub>
</math>
(where
<math alttext="x" class="ltx_Math" display="inline" id="p18.m2">
<mi>
x
</mi>
</math>
, the index of
<math alttext="p" class="ltx_Math" display="inline" id="p18.m3">
<mi>
p
</mi>
</math>
in an ordered list of the primes in ascending order,
<math alttext="x=\pi(x)" class="ltx_Math" display="inline" id="p18.m4">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mi>
π
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
) such that
<math alttext="p_{x}|p_{x+1}p_{x+2}+1" class="ltx_Math" display="inline" id="p18.m5">
<mrow>
<msub>
<mi>
p
</mi>
<mi>
x
</mi>
</msub>
<mo stretchy="false">
|
</mo>
<msub>
<mi>
p
</mi>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msub>
<msub>
<mi>
p
</mi>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
</msub>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p19">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
71
</span>
is the largest
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SupersingularPrime.html">
supersingular prime
</a>
.
</p>
</div>
<div class="ltx_para" id="p20">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
90
</span>
is the largest
<math alttext="n" class="ltx_Math" display="inline" id="p20.m1">
<mi>
n
</mi>
</math>
such that
<math alttext="\phi(n)=\pi(n)" class="ltx_Math" display="inline" id="p20.m2">
<mrow>
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mi>
π
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
</math>
, where
<math alttext="\phi(x)" class="ltx_Math" display="inline" id="p20.m3">
<mrow>
<mi>
ϕ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is Euler’s totient function and
<math alttext="\pi(x)" class="ltx_Math" display="inline" id="p20.m4">
<mrow>
<mi>
π
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is the prime-counting function.
</p>
</div>
<div class="ltx_para" id="p21">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
127
</span>
might be the largest prime
<math alttext="p" class="ltx_Math" display="inline" id="p21.m1">
<mi>
p
</mi>
</math>
satisfying the three conditions of the new Mersenne conjecture [Ribenboim, 2004]. Also, it might be the largest prime
<math alttext="p" class="ltx_Math" display="inline" id="p21.m2">
<mi>
p
</mi>
</math>
such that
<math alttext="2^{p}-1" class="ltx_Math" display="inline" id="p21.m3">
<mrow>
<msup>
<mn>
2
</mn>
<mi>
p
</mi>
</msup>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</math>
is a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Chen prime
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ChenPrime.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/chenprime">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p22">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
144
</span>
is the largest Fibonacci number that is also a square. In base 10 it is the largest sum-product number, a fact that is amazing when you consider that in order to prove it so David Wilson had to test sum-product number
<span class="ltx_text ltx_font_italic">
candidates
</span>
as large as
<math alttext="10^{84}" class="ltx_Math" display="inline" id="p22.m1">
<msup>
<mn>
10
</mn>
<mn>
84
</mn>
</msup>
</math>
.
</p>
</div>
<div class="ltx_para" id="p23">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
163
</span>
is the largest Heegner
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
discriminant
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/discriminantinalgebraicnumberfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/discriminant1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p24">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
216
</span>
might be the largest integer which is not the sum of a prime number and a triangular number. (Sun, 2008)
</p>
</div>
<div class="ltx_para" id="p25">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
454
</span>
is the largest integer such that its shortest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
partition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Partition.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integerpartition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
into cubes requires eight of them.
</p>
</div>
<div class="ltx_para" id="p26">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
563
</span>
might be the largest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Wilson prime
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/WilsonPrime.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/wilsonprime">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p27">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
786
</span>
might be the largest integer for which
<math alttext="{}_{2n}\!C_{n}" class="ltx_Math" display="inline" id="p27.m1">
<mmultiscripts>
<mi>
C
</mi>
<mi>
n
</mi>
<none>
</none>
<mprescripts>
</mprescripts>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
<none>
</none>
</mmultiscripts>
</math>
is not
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Divisible.html">
divisible
</a>
by the square of an
<a class="nnexus_concept" href="http://mathworld.wolfram.com/OddPrime.html">
odd prime
</a>
.
</p>
</div>
<div class="ltx_para" id="p28">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
1493
</span>
might be the largest
<a class="nnexus_concept" href="http://planetmath.org/sternprime">
Stern prime
</a>
, since there is no way to put it in the form
<math alttext="p+2b^{2}" class="ltx_Math" display="inline" id="p28.m1">
<mrow>
<mi>
p
</mi>
<mo>
+
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msup>
<mi>
b
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mrow>
</math>
, where
<math alttext="p" class="ltx_Math" display="inline" id="p28.m2">
<mi>
p
</mi>
</math>
is a different prime and
<math alttext="b>0" class="ltx_Math" display="inline" id="p28.m3">
<mrow>
<mi>
b
</mi>
<mo>
>
</mo>
<mn>
0
</mn>
</mrow>
</math>
. The first
<a class="nnexus_concept" href="http://planetmath.org/decimalplace">
hundred
</a>
thousand primes have been checked and all greater than 1493 can be put into the given form.
</p>
</div>
<div class="ltx_para" id="p29">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
1806
</span>
is the largest
<math alttext="n" class="ltx_Math" display="inline" id="p29.m1">
<mi>
n
</mi>
</math>
such that
<math alttext="m^{n+1}=m\mod n" class="ltx_Math" display="inline" id="p29.m2">
<mrow>
<msup>
<mi>
m
</mi>
<mrow>
<mi>
n
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</msup>
<mo>
=
</mo>
<mrow>
<mi>
m
</mi>
<mo lspace="2.5pt" rspace="2.5pt">
mod
</mo>
<mi>
n
</mi>
</mrow>
</mrow>
</math>
for any
<math alttext="m" class="ltx_Math" display="inline" id="p29.m3">
<mi>
m
</mi>
</math>
.
</p>
</div>
<div class="ltx_para" id="p30">
<p class="ltx_p">
For the purpose of this feature, the arbitrary cutoff is
<math alttext="10^{4}" class="ltx_Math" display="inline" id="p30.m1">
<msup>
<mn>
10
</mn>
<mn>
4
</mn>
</msup>
</math>
.
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
R. K. Guy,
<span class="ltx_text ltx_font_italic">
Unsolved Problems in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Number Theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NumberTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/numbertheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
, B37. New York: Springer-Verlag (2004)
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
P. Ribenboim,
<span class="ltx_text ltx_font_italic">
The Little Book of Bigger Primes
</span>
, p. 83. New York: Springer-Verlag (2004)
</span>
</li>
<li class="ltx_bibitem" id="bib.bib3">
<span class="ltx_bibtag ltx_role_refnum">
3
</span>
<span class="ltx_bibblock">
J. J. Tattersall,
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ElementaryNumberTheory.html">
Elementary number theory
</a>
in nine chapters
</span>
, p. 58. Cambridge: Cambridge University Press (2005)
</span>
</li>
<li class="ltx_bibitem" id="bib.bib4">
<span class="ltx_bibtag ltx_role_refnum">
4
</span>
<span class="ltx_bibblock">
Zhi-Wei Sun, “On Sums of Primes and Triangular Numbers” ArXiv preprint (2008): 2
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p31">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
small integers that are or might be the largest of their kind
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
SmallIntegersThatAreOrMightBeTheLargestOfTheirKind
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:53:04
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:53:04
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mravinci (12996)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mravinci (12996)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
22
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Mravinci (12996)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Feature
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A08
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
EveryPositiveIntegerGreaterThan30HasAtLeastOneCompositeTotative
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | TaxicabNumbers | http://planetmath.org/TaxicabNumbers | <!DOCTYPE html>
<html>
<head>
<title>
taxicab numbers
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
taxicab numbers
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The number
<math alttext="1729" class="ltx_Math" display="inline" id="p1.m1">
<mn>
1729
</mn>
</math>
has a reputation of its own. The reason is the famous exchange between
<span class="ltx_text ltx_font_typewriter">
http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Hardy.html
</span>
G. H. Hardy, a famous British mathematician (1877-1947), and
<span class="ltx_text ltx_font_typewriter">
http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Ramanujan.html
</span>
Srinivasa Ramanujan , one of India’s greatest mathematical geniuses (1887-1920):
</p>
</div>
<div class="ltx_para" id="p2">
<blockquote class="ltx_quote">
<p class="ltx_p">
In 1917, during one visit to Ramanujan in a hospital (he was ill for much of his last three years), Hardy mentioned that the number of the taxi cab
that had brought him was
<math alttext="1729" class="ltx_Math" display="inline" id="p2.m1">
<mn>
1729
</mn>
</math>
, which, as numbers go, Hardy thought
was “rather a dull number”. At this, Ramanujan perked up, and said
“No, it is a very interesting number; it is the smallest number
expressible as a sum of two cubes in two different ways.”
</p>
</blockquote>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Indeed:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="1729=1+12^{3}=9^{3}+10^{3}." class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mn>
1729
</mn>
<mo>
=
</mo>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<msup>
<mn>
12
</mn>
<mn>
3
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mn>
9
</mn>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mn>
10
</mn>
<mn>
3
</mn>
</msup>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Moreover, there are other reasons why
<math alttext="1729" class="ltx_Math" display="inline" id="p3.m1">
<mn>
1729
</mn>
</math>
is far from dull.
<math alttext="1729" class="ltx_Math" display="inline" id="p3.m2">
<mn>
1729
</mn>
</math>
is the third
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Carmichael number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CarmichaelNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fermatcompositenesstest">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. Even more strange, beginning at the
<math alttext="1729" class="ltx_Math" display="inline" id="p3.m3">
<mn>
1729
</mn>
</math>
th decimal digit of the transcental number
<math alttext="e" class="ltx_Math" display="inline" id="p3.m4">
<mi>
e
</mi>
</math>
, the next ten
successive digits of
<math alttext="e" class="ltx_Math" display="inline" id="p3.m5">
<mi>
e
</mi>
</math>
are 0719425863. This is the first appearance
of all ten digits in a row without repititions.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
More generally, the smallest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
natural number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NaturalNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/naturalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
which can be expressed as the sum of
<math alttext="n" class="ltx_Math" display="inline" id="p4.m1">
<mi>
n
</mi>
</math>
<a class="nnexus_concept" href="http://planetmath.org/positive">
positive
</a>
cubes is called the
<math alttext="n" class="ltx_Math" display="inline" id="p4.m2">
<mi>
n
</mi>
</math>
th
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
taxicab number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TaxicabNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/taxicabnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. The first taxicab numbers are:
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="2=1^{3}+1^{3},\ 1729=1^{3}+12^{3}=9^{3}+10^{3},\ 87539319=167^{3}+436^{3}=228^%
{3}+423^{3}=255^{3}+414^{3}" class="ltx_Math" display="block" id="S0.Ex2.m1">
<mrow>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
=
</mo>
<mrow>
<msup>
<mn>
1
</mn>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mn>
1
</mn>
<mn>
3
</mn>
</msup>
</mrow>
</mrow>
<mo>
,
</mo>
<mrow>
<mn>
1729
</mn>
<mo>
=
</mo>
<mrow>
<msup>
<mn>
1
</mn>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mn>
12
</mn>
<mn>
3
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mn>
9
</mn>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mn>
10
</mn>
<mn>
3
</mn>
</msup>
</mrow>
</mrow>
</mrow>
<mo>
,
</mo>
<mrow>
<mn>
87539319
</mn>
<mo>
=
</mo>
<mrow>
<msup>
<mn>
167
</mn>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mn>
436
</mn>
<mn>
3
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mn>
228
</mn>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mn>
423
</mn>
<mn>
3
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mn>
255
</mn>
<mn>
3
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mn>
414
</mn>
<mn>
3
</mn>
</msup>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
followed by
<math alttext="6963472309248" class="ltx_Math" display="inline" id="p4.m3">
<mn>
6963472309248
</mn>
</math>
(found by E. Rosenstiel, J.A. Dardis, and C.R. Rosenstiel in 1991) and
<math alttext="48988659276962496" class="ltx_Math" display="inline" id="p4.m4">
<mn>
48988659276962496
</mn>
</math>
(found by David Wilson on November 21st, 1997).
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
taxicab numbers
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
TaxicabNumbers
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:43:00
</td>
</tr>
<tr class="ltx_tr">
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Last modified on
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<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:43:00
</td>
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Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
alozano (2414)
</td>
</tr>
<tr class="ltx_tr">
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Last modified by
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<td class="ltx_td ltx_align_left ltx_border_r">
alozano (2414)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
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alozano (2414)
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Classification
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msc 00A08
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top ten coolest numbers
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
top ten coolest numbers
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
This is an attempt to give a count-down of the
<a class="nnexus_concept" href="http://planetmath.org/toptencoolestnumbers">
top ten coolest
numbers
</a>
. Let’s first admit that this is a highly subjective
ordering–one person’s 14.38 is another’s
<math alttext="\frac{\pi^{2}}{6}" class="ltx_Math" display="inline" id="p1.m1">
<mfrac>
<msup>
<mi>
π
</mi>
<mn>
2
</mn>
</msup>
<mn>
6
</mn>
</mfrac>
</math>
. The
astute (or probably simply “awake”) reader will notice, for example,
a definite bias toward numbers interesting to a number theorist in the
below list. (On the other hand, who better to gauge the coolness of
numbers than a number-theorist…) But who knows? Maybe I can be
convinced that I’ve left something out, or that my ordering should be
switched in some cases. But let’s first set down some ground rules.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
What’s in the list?
</span>
What makes a number cool? I think a
word that sums up the key
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
characteristic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/characteristicsubgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/characteristic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of cool numbers is
“canonicity”. Numbers that appear in this list should be somehow
fundamental to the nature of mathematics. They could represent a
fundamental fact or
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of mathematics, be the first
<a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">
instance
</a>
of
an amazing class of numbers, be omnipresent in modern mathematics, or
simply have an eerily long list of interesting
<a class="nnexus_concept" href="http://planetmath.org/property">
properties
</a>
. Perhaps a
more appropriate question to ask is the following:
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
What’s
<em class="ltx_emph ltx_font_italic">
not
</em>
in the list?
</span>
There are some really awesome
numbers that I didn’t include in the list. I’ll go through several
examples to get a feel for what sorts of numbers don’t fit the
characteristics mentioned above.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Shocking as it may seem, I first disqualify the
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
constants
</a>
appearing in
Euler’s
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
formula
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Formula.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/firstorderlanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="e^{i\pi}+1=0" class="ltx_Math" display="inline" id="p4.m1">
<mrow>
<mrow>
<msup>
<mi>
e
</mi>
<mrow>
<mi>
i
</mi>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
</msup>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
. This was a tough decision. Perhaps
these five (
<math alttext="e" class="ltx_Math" display="inline" id="p4.m2">
<mi>
e
</mi>
</math>
,
<math alttext="i" class="ltx_Math" display="inline" id="p4.m3">
<mi>
i
</mi>
</math>
,
<math alttext="\pi" class="ltx_Math" display="inline" id="p4.m4">
<mi>
π
</mi>
</math>
, 1, and 0) belong at the top of the list, or
perhaps they’re just too fundamentally important to be considered
exceptionally
<em class="ltx_emph ltx_font_italic">
cool
</em>
. Or maybe they’re just so cliché’d that
we’ll get a significantly more interesting list by excluding them.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
Also disqualified are numbers whose
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Primary.html">
primary
</a>
significance is cultural,
rather than mathematical: despite being the answer to life, the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
universe
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/universe">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universeofdiscourse">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, and everything, 42 is a comparatively uninteresting number. Similarly not included in the list were
876-5309, 666, and the first illegal prime number. Similarly disqualified were constants of nature like Newton’s
<math alttext="g" class="ltx_Math" display="inline" id="p5.m1">
<mi>
g
</mi>
</math>
and
<math alttext="G" class="ltx_Math" display="inline" id="p5.m2">
<mi>
G
</mi>
</math>
, the fine structure constant, Avogadro’s number, etc.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
Finally, I disqualified number that were highly non-canonical in
construction. For example, the
<a class="nnexus_concept" href="http://planetmath.org/primeconstant">
prime constant
</a>
and Champoleon’s
constant are both mathematically interesting, but only because they
were, at least in an admittedly vague sense, constructed to be as
such. Also along these lines are numbers like G63 and Skewe’s
constant, which while mathematically interesting because of roles
they’ve played in proofs, are not inherently interesting in and of
themselves.
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
That said, I felt free to ignore any of these disqualifications when I
felt like it. I hope you enjoy the following list, and I welcome
feedback.
<br class="ltx_break"/>
</p>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
Honorable Mentions
<br class="ltx_break"/>
</span>
</p>
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
65,537 - This number is arguably the number with the most
potential. It’s currently the largest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Fermat prime
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FermatPrime.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fermatnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
known. If it
turns out to be
<em class="ltx_emph ltx_font_italic">
the
</em>
largest Fermat prime, it might earn itself
a place on the list, by virtue of thus also being the largest odd
value of
<math alttext="n" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mi>
n
</mi>
</math>
for which an
<math alttext="n" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mi>
n
</mi>
</math>
-gon is
<a class="nnexus_concept" href="http://planetmath.org/constructiblenumbers">
constructible
</a>
using only a rule
and compass.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
Conway’s constant - The construction of the number can be found
here
<a class="ltx_ref ltx_url ltx_font_typewriter" href="http://mathworld.wolfram.com/ConwaysConstant.html" title="">
http://mathworld.wolfram.com/ConwaysConstant.html
</a>
. Though
this number has some remarkable properties (not the least of which is
being unexpectedly
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Algebraics.html">
algebraic
</a>
), it’s completely non-canonical
construction kept it from overtaking any of our list’s current
members.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
1728 and 1729 - This pair just didn’t have quite enough going
for them to make it. 1728 is an important
<math alttext="j" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mi>
j
</mi>
</math>
-
<a class="nnexus_concept" href="http://planetmath.org/invariant">
invariant
</a>
of
<a class="nnexus_concept" href="http://planetmath.org/ellipticcurve">
elliptic
curves
</a>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
modular forms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ModularForm.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/modularform">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, and is a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
perfect
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/perfectgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/perfectfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
cube. 1729 happens to be
the third
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Carmichael number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CarmichaelNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fermatcompositenesstest">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, but the primary motivation for
including 1729 is because of the mathematical folklore associated it
to being the first
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
taxicab number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TaxicabNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/taxicabnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
, making it more interesting
(math-)historically than mathematically.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
28 - Aside from being a
<a class="nnexus_concept" href="http://planetmath.org/perfectnumber">
perfect number
</a>
, a fairly interesting
fact in and of itself, the number 28 has some extra interesting
“aliquot” properties that propels it beyond other perfect numbers.
Specifically, the largest known
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
collection
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Collection.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/collection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of sociable numbers has
<a class="nnexus_concept" href="http://planetmath.org/cardinality">
cardinality
</a>
28, and though this might seem a silly feat in and of
itself, the fact that sociable numbers and perfect numbers are
so closely related may reveal something slightly more profound about
28 than it just being perfect.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
26 - being the only number between a square and a cube is pretty cool; as well as that to Actuaries, this number has relavance to life expectancy - in than it is a turning point. (this will change over time as is just a tenuous arguement to
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Support.html">
support
</a>
giving 26 a mention!).
</p>
</div>
</li>
</ul>
<p class="ltx_p">
And now, on to the top 10:
</p>
</div>
<div class="ltx_para" id="p9">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#10) The
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Golden Ratio
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GoldenRatio.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/goldenratio">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<math alttext="\phi" class="ltx_Math" display="inline" id="p9.m1">
<mi>
ϕ
</mi>
</math>
<br class="ltx_break"/>
</span>
This was a tough
one. Yes, it’s cool that it
<a class="nnexus_concept" href="http://planetmath.org/satisfactionrelation">
satisfies
</a>
the property that its
<a class="nnexus_concept" href="http://planetmath.org/inversenumber">
reciprocal
</a>
is one less than it, but this merely
<a class="nnexus_concept" href="http://planetmath.org/preservationandreflection">
reflects
</a>
that it’s a
root of the wholly generic polynomial
<math alttext="x^{2}-x-1=0" class="ltx_Math" display="inline" id="p9.m2">
<mrow>
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
-
</mo>
<mi>
x
</mi>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo>
=
</mo>
<mn>
0
</mn>
</mrow>
</math>
. Yes, it’s cool
that it may have an aesthetic quality revered by the Greeks, but this
is void from consideration for being non-mathematical. Only slightly
less
<a class="nnexus_concept" href="http://planetmath.org/canonical">
canonical
</a>
is that it gives the limiting ratio of subsequent
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Fibonacci numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/26.11#p4">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FibonacciNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fibonaccisequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. Redeeming it, however, is that this generalizes to
<em class="ltx_emph ltx_font_italic">
all
</em>
“Fibonacci-like”
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
sequences
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Sequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, and is the
<a class="nnexus_concept" href="http://planetmath.org/equation">
solution
</a>
to two
sort of canonical
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
:
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex1">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\ddots}}}}" class="ltx_Math" display="inline" id="S0.Ex1.m1">
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mfrac>
<mn>
1
</mn>
<mi mathvariant="normal">
⋱
</mi>
</mfrac>
</mrow>
</mfrac>
</mrow>
</mfrac>
</mrow>
</mfrac>
</mstyle>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
and
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex2">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}" class="ltx_Math" display="inline" id="S0.Ex2.m1">
<msqrt>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<msqrt>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<msqrt>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mi mathvariant="normal">
⋯
</mi>
</mrow>
</msqrt>
</mrow>
</msqrt>
</mrow>
</msqrt>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Also, this number plays an important role in the hstory of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebraic
number theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicNumberTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraicnumbertheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. The field it generates is the first known example of a
field in which unique factorization fails. Trying to come to grips with
this fact led to the invention of ideal theory, class nubers, etc.
</p>
</div>
<div class="ltx_para" id="p10">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#9) 691
<br class="ltx_break"/>
</span>
The
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
prime number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimeNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/prime">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
691 made it on here for
a couple of reasons: First, it’s prime, but more importantly, it’s the
first example of an
<em class="ltx_emph ltx_font_italic">
irregular
</em>
prime, a class of primes of
immense importance in algebraic number theory. (A word of caution:
it’s not the
<em class="ltx_emph ltx_font_italic">
smallest
</em>
<a class="nnexus_concept" href="http://planetmath.org/regularprime">
irregular prime
</a>
, but it’s the one that
corresponds to the earliest
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Bernoulli number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/24.1#P1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://dlmf.nist.gov/24.2#i">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BernoulliNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bernoullinumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bernoullipolynomialsandnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<math alttext="B_{12}" class="ltx_Math" display="inline" id="p10.m1">
<msub>
<mi>
B
</mi>
<mn>
12
</mn>
</msub>
</math>
, so 691 is only
“first” in that sense). It also shows up as a
<a class="nnexus_concept" href="http://planetmath.org/polynomial">
coefficient
</a>
of every
non-constant term in the
<math alttext="q" class="ltx_Math" display="inline" id="p10.m2">
<mi>
q
</mi>
</math>
-expansion of the modular form
<math alttext="E_{12}(z)" class="ltx_Math" display="inline" id="p10.m3">
<mrow>
<msub>
<mi>
E
</mi>
<mn>
12
</mn>
</msub>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
z
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
. Further testimony to the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
arithmetic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/arithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/arithmeticalhierarchy">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
significance is its
seemingly magical appearance in the algebraic
<math alttext="K" class="ltx_Math" display="inline" id="p10.m4">
<mi>
K
</mi>
</math>
-theory: It’s known
that
<math alttext="K_{22}(\mathbb{Z})" class="ltx_Math" display="inline" id="p10.m5">
<mrow>
<msub>
<mi>
K
</mi>
<mn>
22
</mn>
</msub>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
ℤ
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
surjects onto 691.
</p>
</div>
<div class="ltx_para" id="p11">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#8) 78,557
<br class="ltx_break"/>
</span>
The number 78,557 is here to represent
an amazing class of numbers called
<em class="ltx_emph ltx_font_italic">
Sierpinski
</em>
numbers, defined
to be numbers
<math alttext="k" class="ltx_Math" display="inline" id="p11.m1">
<mi>
k
</mi>
</math>
such that
<math alttext="k2^{n}+1" class="ltx_Math" display="inline" id="p11.m2">
<mrow>
<mrow>
<mi>
k
</mi>
<mo>
</mo>
<msup>
<mn>
2
</mn>
<mi>
n
</mi>
</msup>
</mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
is
<a class="nnexus_concept" href="http://planetmath.org/compositenumber">
composite
</a>
for
<em class="ltx_emph ltx_font_italic">
every
</em>
<math alttext="n\geq 1" class="ltx_Math" display="inline" id="p11.m3">
<mrow>
<mi>
n
</mi>
<mo>
≥
</mo>
<mn>
1
</mn>
</mrow>
</math>
. That such numbers exist is flabbergasting…we know from
Dirichlet’s theorem that primes occur infinitely often in non-trivial
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ArithmeticSequence.html">
arithmetic sequences
</a>
. Though the sequence formed by
<math alttext="78557\cdot 2^{n}+1" class="ltx_Math" display="inline" id="p11.m4">
<mrow>
<mrow>
<mn>
78557
</mn>
<mo>
⋅
</mo>
<msup>
<mn>
2
</mn>
<mi>
n
</mi>
</msup>
</mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
isn’t arithmetic, it certainly doesn’t behave multiplicatively
either, and there’s no apparent reason why there shouldn’t be a large
(or
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinite
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinite.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extendedrealnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/infinite">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) number of primes in
<em class="ltx_emph ltx_font_italic">
every
</em>
such sequence. This
notwithstanding, Sierpinski’s
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CompositeNumber.html">
composite number
</a>
theorem proves there
are in fact
<em class="ltx_emph ltx_font_italic">
infinitely
</em>
many odd such numbers
<math alttext="k" class="ltx_Math" display="inline" id="p11.m5">
<mi>
k
</mi>
</math>
. As a small
disclaimer, though it’s proven that 78,557 is indeed a
<a class="nnexus_concept" href="http://planetmath.org/sierpinskinumber">
Sierpinski
number
</a>
, it is not quite yet known that it is the smallest. There are
17
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PositiveInteger.html">
positive integers
</a>
smaller than 78,557 not yet known to be
non-Sierpinski.
</p>
</div>
<div class="ltx_para" id="p12">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#7)
<math alttext="\frac{\pi^{2}}{6}" class="ltx_Math" display="inline" id="p12.m1">
<mfrac>
<msup>
<mi>
π
</mi>
<mn mathvariant="normal">
2
</mn>
</msup>
<mn mathvariant="normal">
6
</mn>
</mfrac>
</math>
<br class="ltx_break"/>
</span>
Perhaps the first striking
this about this number is that it is the sum of the reciprocals of the
positive integer squares:
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex3">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^{2}}+\cdots=\frac{%
\pi^{2}}{6}." class="ltx_Math" display="inline" id="S0.Ex3.m1">
<mrow>
<mrow>
<mrow>
<mn>
1
</mn>
<mo>
+
</mo>
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<mn>
4
</mn>
</mfrac>
</mstyle>
<mo>
+
</mo>
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<mn>
9
</mn>
</mfrac>
</mstyle>
<mo>
+
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
+
</mo>
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<msup>
<mi>
n
</mi>
<mn>
2
</mn>
</msup>
</mfrac>
</mstyle>
<mo>
+
</mo>
<mi mathvariant="normal">
⋯
</mi>
</mrow>
<mo>
=
</mo>
<mstyle displaystyle="true">
<mfrac>
<msup>
<mi>
π
</mi>
<mn>
2
</mn>
</msup>
<mn>
6
</mn>
</mfrac>
</mstyle>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Though the choice of
<math alttext="2" class="ltx_Math" display="inline" id="p12.m2">
<mn>
2
</mn>
</math>
here for the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
exponent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Exponent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/exponent">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is somewhat
non-canonical (i.e. we’ve just noted that
<math alttext="\zeta(2)=\frac{\pi^{2}}{6}" class="ltx_Math" display="inline" id="p12.m3">
<mrow>
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
2
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mfrac>
<msup>
<mi>
π
</mi>
<mn>
2
</mn>
</msup>
<mn>
6
</mn>
</mfrac>
</mrow>
</math>
,
where
<math alttext="\zeta" class="ltx_Math" display="inline" id="p12.m4">
<mi>
ζ
</mi>
</math>
stands for the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Riemann zeta function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/25.1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://dlmf.nist.gov/25.2#E1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RiemannZetaFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/riemannzetafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
), and that this is
largely interesting for math-historical reasons (it was the first sum
of this type that Euler computed), we can at least include it here to
represent the amazing array of numbers of the form
<math alttext="\zeta(n)" class="ltx_Math" display="inline" id="p12.m5">
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
for
<math alttext="n" class="ltx_Math" display="inline" id="p12.m6">
<mi>
n
</mi>
</math>
a positive integer. This class of numbers incorporates two amazing
and seemingly disparate collections, depending on whether
<math alttext="n" class="ltx_Math" display="inline" id="p12.m7">
<mi>
n
</mi>
</math>
is even
(in which case
<math alttext="\zeta(n)" class="ltx_Math" display="inline" id="p12.m8">
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is known to be a
<a class="nnexus_concept" href="http://planetmath.org/rationalnumber">
rational
</a>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
multiple
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Multiple.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generalassociativity">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divisibility">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<math alttext="\pi^{n}" class="ltx_Math" display="inline" id="p12.m9">
<msup>
<mi>
π
</mi>
<mi>
n
</mi>
</msup>
</math>
) or odd (in which case extremely little is known, even for
<math alttext="\zeta(3)" class="ltx_Math" display="inline" id="p12.m10">
<mrow>
<mi>
ζ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
3
</mn>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p13">
<p class="ltx_p">
Further, there’s something slightly more canonical about the fact that
its reciprocal,
<math alttext="\frac{6}{\pi^{2}}" class="ltx_Math" display="inline" id="p13.m1">
<mfrac>
<mn>
6
</mn>
<msup>
<mi>
π
</mi>
<mn>
2
</mn>
</msup>
</mfrac>
</math>
, gives the “probability” (in a
suitably-defined sense) that two randomly chosen positive integers are
relatively prime.
</p>
</div>
<div class="ltx_para" id="p14">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#6) Feigenbaum’s constant
<br class="ltx_break"/>
</span>
- This is the entry on
this the list with which I have the least familiarity. The one thing
going for it is that it seems to be highly canonical, representing
the limiting ratio of distance between bifurcation
<a class="nnexus_concept" href="http://planetmath.org/interval">
intervals
</a>
for a
fairly large class of one-dimensional maps. In other words, all maps
that fall in to this category will bifurcate at the same rate, giving
us a glimpse of order in the realm of chaotic systems.
</p>
</div>
<div class="ltx_para" id="p15">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#5) 2
<br class="ltx_break"/>
</span>
This number caused quite a bit of
controversy in discussions leading up to the construction of this list.
The question here is canonicality. The first
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
argument
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/argument">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of “It’s the
only
<a class="nnexus_concept" href="http://mathworld.wolfram.com/EvenPrime.html">
even prime
</a>
” is merely a re-wording of “It’s the only prime
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Divisible.html">
divisible
</a>
by 2,” which could uniquely characterizes
<em class="ltx_emph ltx_font_italic">
any
</em>
prime
(e.g. 5 is the only prime divisible by 5, etc.). Of debatable
canonicality is the immensely prevalent notion of “working in
binary.” To a
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computer
</a>
scientist, this may seem extremely canonical,
but to a mathematician, it may simply be an (not quite) arbitrary
choice of a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
finite field
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FiniteField.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/finitefield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
over which to work.
</p>
</div>
<div class="ltx_para" id="p16">
<p class="ltx_p">
Yet 2 has some remarkable features even ignoring aspects relating to
its
<a class="nnexus_concept" href="http://planetmath.org/primality">
primality
</a>
. For instance, the somewhat canonical
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FieldofReals.html">
field of real
</a>
numbers
<math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="p16.m1">
<mi>
ℝ
</mi>
</math>
has index 2 in its
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebraic
closure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicClosure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraicallyclosed">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="\mathbb{C}" class="ltx_Math" display="inline" id="p16.m2">
<mi>
ℂ
</mi>
</math>
. The factor
<math alttext="2\pi i" class="ltx_Math" display="inline" id="p16.m3">
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
π
</mi>
<mo>
</mo>
<mi>
i
</mi>
</mrow>
</math>
is prevalent enough in
complex and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FourierAnalysis.html">
Fourier analysis
</a>
that I’ve heard people lament that
<math alttext="\pi" class="ltx_Math" display="inline" id="p16.m4">
<mi>
π
</mi>
</math>
should have been defined to be twice its current value. It’s also the
<em class="ltx_emph ltx_font_italic">
only
</em>
prime number
<math alttext="p" class="ltx_Math" display="inline" id="p16.m5">
<mi>
p
</mi>
</math>
such that
<math alttext="x^{p}+y^{p}=z^{p}" class="ltx_Math" display="inline" id="p16.m6">
<mrow>
<mrow>
<msup>
<mi>
x
</mi>
<mi>
p
</mi>
</msup>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mi>
p
</mi>
</msup>
</mrow>
<mo>
=
</mo>
<msup>
<mi>
z
</mi>
<mi>
p
</mi>
</msup>
</mrow>
</math>
has any rational solutions.
</p>
</div>
<div class="ltx_para" id="p17">
<p class="ltx_p">
Finally, if nothing else, it is certainly the first prime, and could
at least be included for being the first representative of such an
amazing class of numbers.
</p>
</div>
<div class="ltx_para" id="p18">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#4) 808017424794512875886459904961710757005754368
<math alttext="\times 10^{9}" class="ltx_Math" display="inline" id="p18.m1">
<mrow>
<mi>
</mi>
<mo mathvariant="normal">
×
</mo>
<msup>
<mn mathvariant="normal">
10
</mn>
<mn mathvariant="normal">
9
</mn>
</msup>
</mrow>
</math>
<br class="ltx_break"/>
</span>
</p>
</div>
<div class="ltx_para" id="p19">
<p class="ltx_p">
The above
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
is the size of the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MonsterGroup.html">
monster group
</a>
, the largest
of the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SporadicGroup.html">
sporadic groups
</a>
. This gives it a relatively high degree of
canonicality. It’s unclear (at least to me) why there should be
<em class="ltx_emph ltx_font_italic">
any
</em>
sporadic groups, or why, given that they exist, there
should only be finitely many. Since there
<em class="ltx_emph ltx_font_italic">
is
</em>
, however, there
must be something fairly special about the largest possible one.
</p>
</div>
<div class="ltx_para" id="p20">
<p class="ltx_p">
Also contributing to this number’s rank on this list is the remarkable
properties of the monster group itself, which has been realized
(actually, was constructed as) a group of rotations in
196,883-dimensional space, representing in some sense a
<em class="ltx_emph ltx_font_italic">
limit
</em>
to the amount of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
symmetry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/symmetricrelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumsymmetry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
such a space can possess.
</p>
</div>
<div class="ltx_para" id="p21">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#3)
<a class="nnexus_concept" href="http://planetmath.org/eulersconstant">
Euler-Mascheroni Constant
</a>
,
<math alttext="\gamma" class="ltx_Math" display="inline" id="p21.m1">
<mi>
γ
</mi>
</math>
<br class="ltx_break"/>
</span>
One of
the most amazing facts from
<a class="nnexus_concept" href="http://planetmath.org/elementaryrecursivefunction">
elementary
</a>
calculus is that the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
harmonic
series
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/HarmonicSeries.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/harmonicseries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
diverges
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/unlimitedregistermachine">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/convergentsequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, but that if you put an exponent on the
<a class="nnexus_concept" href="http://planetmath.org/fraction">
denominators
</a>
even just a
<em class="ltx_emph ltx_font_italic">
hair
</em>
above 1, the result is a
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ConvergentSequence.html">
convergent sequence
</a>
.
A refined statement says that the
<a class="nnexus_concept" href="http://planetmath.org/sumofseries">
partial sums
</a>
of the harmonic series
grow like
<math alttext="\ln(n)" class="ltx_Math" display="inline" id="p21.m2">
<mrow>
<mi>
ln
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
, and a further refinement says that the error of
this approximation approaches our constant:
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex4">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle\lim\limits_{n\rightarrow\infty}1+\frac{1}{2}+\frac{1}{3}+\cdots+%
\frac{1}{n}-\ln(n)=\gamma." class="ltx_Math" display="inline" id="S0.Ex4.m1">
<mrow>
<mrow>
<mrow>
<mrow>
<mrow>
<munder>
<mo movablelimits="false">
lim
</mo>
<mrow>
<mi>
n
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</munder>
<mo>
</mo>
<mn>
1
</mn>
</mrow>
<mo>
+
</mo>
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<mn>
2
</mn>
</mfrac>
</mstyle>
<mo>
+
</mo>
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<mn>
3
</mn>
</mfrac>
</mstyle>
<mo>
+
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
+
</mo>
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<mi>
n
</mi>
</mfrac>
</mstyle>
</mrow>
<mo>
-
</mo>
<mrow>
<mi>
ln
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mrow>
<mo>
=
</mo>
<mi>
γ
</mi>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
This seems to represent something fundamental about the harmonic series, and
thus of the integers themselves.
</p>
</div>
<div class="ltx_para" id="p22">
<p class="ltx_p">
Finally, perhaps due to importance inherited from the crucially
important harmonic series, the Euler-Mascheroni constant appears
magically all over mathematics.
</p>
</div>
<div class="ltx_para" id="p23">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#2) Khinchin’s constant,
<math alttext="K\approx 2.685452..." class="ltx_Math" display="inline" id="p23.m1">
<mrow>
<mi>
K
</mi>
<mo mathvariant="normal">
≈
</mo>
<mrow>
<mn mathvariant="normal">
2.685452
</mn>
<mo mathvariant="bold">
</mo>
<mi mathvariant="normal">
…
</mi>
</mrow>
</mrow>
</math>
<br class="ltx_break"/>
</span>
For a
<a class="nnexus_concept" href="http://planetmath.org/realnumber">
real number
</a>
<math alttext="x" class="ltx_Math" display="inline" id="p23.m2">
<mi>
x
</mi>
</math>
, we define a
<a class="nnexus_concept" href="http://planetmath.org/geometricmean">
geometric mean
</a>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex5">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle f(x)=\lim\limits_{n\rightarrow\infty}(a_{1}\cdots a_{n})^{1/n}," class="ltx_Math" display="inline" id="S0.Ex5.m1">
<mrow>
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<munder>
<mo movablelimits="false">
lim
</mo>
<mrow>
<mi>
n
</mi>
<mo>
→
</mo>
<mi mathvariant="normal">
∞
</mi>
</mrow>
</munder>
<mo>
</mo>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
</mo>
<msub>
<mi>
a
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mrow>
<mn>
1
</mn>
<mo>
/
</mo>
<mi>
n
</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
where the
<math alttext="a_{i}" class="ltx_Math" display="inline" id="p23.m3">
<msub>
<mi>
a
</mi>
<mi>
i
</mi>
</msub>
</math>
are the terms of the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
simple continued fraction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SimpleContinuedFraction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuedfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
expansion of
<math alttext="x" class="ltx_Math" display="inline" id="p23.m4">
<mi>
x
</mi>
</math>
. By nothing short of a miracle of mathematics, this
function of
<math alttext="x" class="ltx_Math" display="inline" id="p23.m5">
<mi>
x
</mi>
</math>
is almost everywhere (i.e. everywhere except for a set
of measure 0)
<em class="ltx_emph ltx_font_italic">
independent of
</em>
<math alttext="x" class="ltx_Math" display="inline" id="p23.m6">
<mi>
x
</mi>
</math>
!!! In other words, except for
a “small” number of exceptions, this function
<math alttext="f(x)" class="ltx_Math" display="inline" id="p23.m7">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
always outputs
the same value, which is called Khinchin’s constant and is denoted by
<math alttext="K" class="ltx_Math" display="inline" id="p23.m8">
<mi>
K
</mi>
</math>
. It’s hard to impress upon a casual reader just how astounding
this is, but consider the following:
<em class="ltx_emph ltx_font_italic">
Any
</em>
infinite collection of
non-negative integers
<math alttext="a_{0},a_{1},\ldots" class="ltx_Math" display="inline" id="p23.m9">
<mrow>
<msub>
<mi>
a
</mi>
<mn>
0
</mn>
</msub>
<mo>
,
</mo>
<msub>
<mi>
a
</mi>
<mn>
1
</mn>
</msub>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
</mrow>
</math>
forms a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
continued fraction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.12#i">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ContinuedFraction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
</sup>
,
and indeed each continued fraction gives an infinite collection of
that form. That the partial geometric means of these sequences is
<em class="ltx_emph ltx_font_italic">
almost everywhere constant
</em>
tells us a great deal about the
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.16#SS1.p5">
distribution
</a>
of sequences showing up as continued fraction sequences,
in turn revealing something very fundamental about the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
real numbers.
</p>
</div>
<div class="ltx_para" id="p24">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold" style="font-size:144%;">
#1) 163
<br class="ltx_break"/>
</span>
Well, we’ve come down to it, this
author’s humble opinion of the coolest number in
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Existence.html">
existence
</a>
. Though an
unlikely candidate, I hope to show you that 163 satisfies so many
eerily related properties as to earn this title.
</p>
</div>
<div class="ltx_para" id="p25">
<p class="ltx_p">
I’ll begin with something that most number theorists already know
about this number – it is the largest value of
<math alttext="d" class="ltx_Math" display="inline" id="p25.m1">
<mi>
d
</mi>
</math>
such that the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
number field
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NumberField.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/numberfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="\mathbb{Q}(\sqrt{-d})" class="ltx_Math" display="inline" id="p25.m2">
<mrow>
<mi>
ℚ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msqrt>
<mrow>
<mo>
-
</mo>
<mi>
d
</mi>
</mrow>
</msqrt>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
has
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
class number
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ClassNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/idealclassesformanabeliangroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/idealclass">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
1, meaning that its
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RingofIntegers.html">
ring
of integers
</a>
is a unique factorization domain. The issue of
factorization in
<a class="nnexus_concept" href="http://mathworld.wolfram.com/QuadraticField.html">
quadratic fields
</a>
, and of number fields in general, is
one of the principal driving forces of algebraic number theory, and to
be able to pinpoint the end of perfect factorization in the quadratic
case like this seems at least arguably fundamental.
</p>
</div>
<div class="ltx_para" id="p26">
<p class="ltx_p">
But even if you don’t care about factorization in number fields, the
above fact has some amazing repercussions to more basic
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
number theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NumberTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/numbertheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
The two following facts in particular jump out:
</p>
<ul class="ltx_itemize" id="I2">
<li class="ltx_item" id="I2.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i1.p1">
<p class="ltx_p">
<math alttext="e^{\pi\sqrt{163}}" class="ltx_Math" display="inline" id="I2.i1.p1.m1">
<msup>
<mi>
e
</mi>
<mrow>
<mi>
π
</mi>
<mo>
</mo>
<msqrt>
<mn>
163
</mn>
</msqrt>
</mrow>
</msup>
</math>
is within
<math alttext="10^{-12}" class="ltx_Math" display="inline" id="I2.i1.p1.m2">
<msup>
<mn>
10
</mn>
<mrow>
<mo>
-
</mo>
<mn>
12
</mn>
</mrow>
</msup>
</math>
of an integer.
</p>
</div>
</li>
<li class="ltx_item" id="I2.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i2.p1">
<p class="ltx_p">
The
<a class="nnexus_concept" href="http://planetmath.org/polynomialring">
polynomial
</a>
<math alttext="f(x)=x^{2}+x+41" class="ltx_Math" display="inline" id="I2.i2.p1.m1">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mi>
x
</mi>
<mo>
+
</mo>
<mn>
41
</mn>
</mrow>
</mrow>
</math>
has the property that for integers
<math alttext="1\leq x\leq 41" class="ltx_Math" display="inline" id="I2.i2.p1.m2">
<mrow>
<mn>
1
</mn>
<mo>
≤
</mo>
<mi>
x
</mi>
<mo>
≤
</mo>
<mn>
41
</mn>
</mrow>
</math>
,
<math alttext="f(x)" class="ltx_Math" display="inline" id="I2.i2.p1.m3">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is prime.
</p>
</div>
</li>
</ul>
<p class="ltx_p">
Both of these are tied intimately (the former using deep properties of
the
<math alttext="j" class="ltx_Math" display="inline" id="p26.m1">
<mi>
j
</mi>
</math>
-function, the latter using relatively simple arguments
concerning the splitting of primes in number fields) to the above
<a class="nnexus_concept" href="http://planetmath.org/imaginaryquadraticfield">
quadratic imaginary number field
</a>
having class number 1. Further,
since
<math alttext="\mathbb{Q}(\sqrt{-d})" class="ltx_Math" display="inline" id="p26.m2">
<mrow>
<mi>
ℚ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msqrt>
<mrow>
<mo>
-
</mo>
<mi>
d
</mi>
</mrow>
</msqrt>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is the
<em class="ltx_emph ltx_font_italic">
last
</em>
such field, the two listed
properties are in some sense the best possible.
</p>
</div>
<div class="ltx_para" id="p27">
<p class="ltx_p">
Most striking to me, however, is the amazing frequency with which 163
shows up in a wide
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
variety
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Variety.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equationalclass">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/varietyofgroups">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of class number problems. In
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to
being the last value of
<math alttext="d" class="ltx_Math" display="inline" id="p27.m1">
<mi>
d
</mi>
</math>
such that
<math alttext="\mathbb{Q}(\sqrt{-d})" class="ltx_Math" display="inline" id="p27.m2">
<mrow>
<mi>
ℚ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msqrt>
<mrow>
<mo>
-
</mo>
<mi>
d
</mi>
</mrow>
</msqrt>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
has class number
1, it is the
<em class="ltx_emph ltx_font_italic">
first
</em>
value of
<math alttext="p" class="ltx_Math" display="inline" id="p27.m3">
<mi>
p
</mi>
</math>
such that
<math alttext="\mathbb{Q}(\zeta_{p}+\zeta_{p}^{-1})" class="ltx_Math" display="inline" id="p27.m4">
<mrow>
<mi>
ℚ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msub>
<mi>
ζ
</mi>
<mi>
p
</mi>
</msub>
<mo>
+
</mo>
<msubsup>
<mi>
ζ
</mi>
<mi>
p
</mi>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msubsup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
(the
<a class="nnexus_concept" href="http://planetmath.org/maximalsubgroup">
maximal
</a>
real
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
subfield
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Subfield.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/subfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of the
<math alttext="p" class="ltx_Math" display="inline" id="p27.m5">
<mi>
p
</mi>
</math>
-th
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
cyclotomic field
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CyclotomicField.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cyclotomicfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) has class number
<em class="ltx_emph ltx_font_italic">
greater
</em>
than 1. That 163
appears as the last instance of a quadratic field having unique
factorization, and the first instance of a real cyclotomic field
<em class="ltx_emph ltx_font_italic">
not
</em>
having unique factorization, seems too remarkable to be
coincidental. This is (maybe) further substantiated by a couple of
other factoids
</p>
<ul class="ltx_itemize" id="I3">
<li class="ltx_item" id="I3.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i1.p1">
<p class="ltx_p">
Hasse asked for an example of a prime and an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
extension
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/substructure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extension">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extensionofafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extensionfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
such that
the prime splits completely into
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Divisor.html">
divisors
</a>
which
<em class="ltx_emph ltx_font_italic">
do not
</em>
lie in a
<a class="nnexus_concept" href="http://planetmath.org/cyclicgroup">
cyclic subgroup
</a>
of the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ClassGroup.html">
class group
</a>
. The first such example is any prime
less than 163 which splits completely in the cubic field generated by
the polynomial
<math alttext="x^{3}=11x^{2}+14x+1" class="ltx_Math" display="inline" id="I3.i1.p1.m1">
<mrow>
<msup>
<mi>
x
</mi>
<mn>
3
</mn>
</msup>
<mo>
=
</mo>
<mrow>
<mrow>
<mn>
11
</mn>
<mo>
</mo>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
+
</mo>
<mrow>
<mn>
14
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
</math>
. This field has
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
discriminant
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/discriminant">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/discriminantinalgebraicnumberfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/discriminant1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="163^{2}" class="ltx_Math" display="inline" id="I3.i1.p1.m2">
<msup>
<mn>
163
</mn>
<mn>
2
</mn>
</msup>
</math>
. (See Shanks’
<em class="ltx_emph ltx_font_italic">
The Simplest Cubic Fields
</em>
).
</p>
</div>
</li>
<li class="ltx_item" id="I3.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i2.p1">
<p class="ltx_p">
The maximal
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
conductor
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/rayclassfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dirichletcharacter">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
if an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
imaginary
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/imaginaries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/imaginary">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://planetmath.org/abeliannumberfield">
abelian number field
</a>
of
class number 1 corresponds to the field
<math alttext="\mathbb{Q}(\sqrt{-67},\sqrt{-163})" class="ltx_Math" display="inline" id="I3.i2.p1.m1">
<mrow>
<mi>
ℚ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msqrt>
<mrow>
<mo>
-
</mo>
<mn>
67
</mn>
</mrow>
</msqrt>
<mo>
,
</mo>
<msqrt>
<mrow>
<mo>
-
</mo>
<mn>
163
</mn>
</mrow>
</msqrt>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
,
which has conductor
<math alttext="10921=67*163" class="ltx_Math" display="inline" id="I3.i2.p1.m2">
<mrow>
<mn>
10921
</mn>
<mo>
=
</mo>
<mrow>
<mn>
67
</mn>
<mo>
*
</mo>
<mn>
163
</mn>
</mrow>
</mrow>
</math>
.
</p>
</div>
</li>
</ul>
<p class="ltx_p">
It is unclear whether or not these additional arithmetical properties
reflect deeper properties of the
<math alttext="j" class="ltx_Math" display="inline" id="p27.m6">
<mi>
j
</mi>
</math>
-function or other modular forms,
and remains a wide open field of study.
</p>
</div>
<div class="ltx_para" id="p28">
<p class="ltx_p">
Originally posted on
<span class="ltx_text ltx_font_typewriter">
http://math.arizona.edu/ mcleman
</span>
Cam’s homepage
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
top ten coolest numbers
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
TopTenCoolestNumbers
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:38:03
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:38:03
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
17
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Feature
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A08
</td>
</tr>
</tbody>
</table>
</div>
</article>
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</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A mathematical object is said to be
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
canonical
</a>
</em>
if it arises in a natural way without introducing any additional objects.
</p>
</div>
<section class="ltx_subsubsection" id="S0.SS0.SSSx1">
<h2 class="ltx_title ltx_title_subsubsection">
Examples
</h2>
<div class="ltx_para" id="S0.SS0.SSSx1.p1">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
Suppose
<math alttext="A\times B" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mrow>
<mi>
A
</mi>
<mo>
×
</mo>
<mi>
B
</mi>
</mrow>
</math>
is the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Cartesian product
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CartesianProduct.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cartesianproduct">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of sets
<math alttext="A,B" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mrow>
<mi>
A
</mi>
<mo>
,
</mo>
<mi>
B
</mi>
</mrow>
</math>
.
Then
<math alttext="A\times B" class="ltx_Math" display="inline" id="I1.i1.p1.m3">
<mrow>
<mi>
A
</mi>
<mo>
×
</mo>
<mi>
B
</mi>
</mrow>
</math>
has two
<math alttext="A\times B\to A" class="ltx_Math" display="inline" id="I1.i1.p1.m4">
<mrow>
<mrow>
<mi>
A
</mi>
<mo>
×
</mo>
<mi>
B
</mi>
</mrow>
<mo>
→
</mo>
<mi>
A
</mi>
</mrow>
</math>
and
<math alttext="A\times B\to B" class="ltx_Math" display="inline" id="I1.i1.p1.m5">
<mrow>
<mrow>
<mi>
A
</mi>
<mo>
×
</mo>
<mi>
B
</mi>
</mrow>
<mo>
→
</mo>
<mi>
B
</mi>
</mrow>
</math>
defined in a natural way. Of course, if
we assume more
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<math alttext="A,B" class="ltx_Math" display="inline" id="I1.i1.p1.m6">
<mrow>
<mi>
A
</mi>
<mo>
,
</mo>
<mi>
B
</mi>
</mrow>
</math>
there are also other
<a class="nnexus_concept" href="http://planetmath.org/involutoryring">
projections
</a>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CanonicalProjection
</span>
<a class="nnexus_concept" href="http://planetmath.org/canonicalprojection">
canonical projection
</a>
(in
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GroupTheory.html">
group theory
</a>
)
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx2">
<h2 class="ltx_title ltx_title_subsubsection">
Notes
</h2>
<div class="ltx_para" id="S0.SS0.SSSx2.p1">
<p class="ltx_p">
For a discussion of the theological use of canonical, see
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
]
</cite>
.
</p>
</div>
</section>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Wikipedia, article on
<span class="ltx_text ltx_font_typewriter">
http://en.wikipedia.org/wiki/Canonical
</span>
canonical.
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p2">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
canonical
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Canonical
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:44:32
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:44:32
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
6
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
CanonicalFormOfElementOfNumberField
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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|
0 | CompletingTheSquare | http://planetmath.org/CompletingTheSquare | <!DOCTYPE html>
<html>
<head>
<title>
completing the square
</title>
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
completing the square
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Let us consider the
<a class="nnexus_concept" href="http://planetmath.org/expression">
expression
</a>
<math alttext="x^{2}+xy" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
</math>
, where
<math alttext="x" class="ltx_Math" display="inline" id="p1.m2">
<mi>
x
</mi>
</math>
and
<math alttext="y" class="ltx_Math" display="inline" id="p1.m3">
<mi>
y
</mi>
</math>
are real (or
<a class="nnexus_concept" href="http://planetmath.org/complex">
complex
</a>
) numbers.
Using the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
formula
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Formula.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/firstorderlanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="(x+y)^{2}=x^{2}+2xy+y^{2}" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<mi>
y
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
x
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
we can write
</p>
<table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A0.EGx1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex4">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle x^{2}+xy" class="ltx_Math" display="inline" id="S0.Ex2.m1">
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S0.Ex2.m2">
<mo>
=
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle x^{2}+xy+0" class="ltx_Math" display="inline" id="S0.Ex2.m3">
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
<mo>
+
</mo>
<mn>
0
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S0.Ex3.m2">
<mo>
=
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle x^{2}+xy+\frac{y^{2}}{4}-\frac{y^{2}}{4}" class="ltx_Math" display="inline" id="S0.Ex3.m3">
<mrow>
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
<mo>
+
</mo>
<mstyle displaystyle="true">
<mfrac>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
<mn>
4
</mn>
</mfrac>
</mstyle>
</mrow>
<mo>
-
</mo>
<mstyle displaystyle="true">
<mfrac>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
<mn>
4
</mn>
</mfrac>
</mstyle>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S0.Ex4.m2">
<mo>
=
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle\left(x+\frac{y}{2}\right)^{2}-\frac{y^{2}}{4}." class="ltx_Math" display="inline" id="S0.Ex4.m3">
<mrow>
<mrow>
<msup>
<mrow>
<mo>
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<mstyle displaystyle="true">
<mfrac>
<mi>
y
</mi>
<mn>
2
</mn>
</mfrac>
</mstyle>
</mrow>
<mo>
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
<mo>
-
</mo>
<mstyle displaystyle="true">
<mfrac>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
<mn>
4
</mn>
</mfrac>
</mstyle>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
This manipulation is called
<em class="ltx_emph ltx_font_italic">
completing the square
</em>
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
]
</cite>
in
<math alttext="x^{2}+xy" class="ltx_Math" display="inline" id="p1.m4">
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
</math>
, or completing the square
<math alttext="x^{2}" class="ltx_Math" display="inline" id="p1.m5">
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</math>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Replacing
<math alttext="y" class="ltx_Math" display="inline" id="p2.m1">
<mi>
y
</mi>
</math>
by
<math alttext="-y" class="ltx_Math" display="inline" id="p2.m2">
<mrow>
<mo>
-
</mo>
<mi>
y
</mi>
</mrow>
</math>
, we also have
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="x^{2}-xy=\left(x-\frac{y}{2}\right)^{2}-\frac{y^{2}}{4}." class="ltx_Math" display="block" id="S0.Ex5.m1">
<mrow>
<mrow>
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
-
</mo>
<mrow>
<mi>
x
</mi>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mrow>
<mo>
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
-
</mo>
<mfrac>
<mi>
y
</mi>
<mn>
2
</mn>
</mfrac>
</mrow>
<mo>
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
<mo>
-
</mo>
<mfrac>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
<mn>
4
</mn>
</mfrac>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Here are some applications of this method:
</p>
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/DerivationOfQuadraticFormula
</span>
<a class="nnexus_concept" href="http://planetmath.org/derivationsinnaturaldeduction">
Derivation
</a>
of the solution formula to the
<a class="nnexus_concept" href="http://planetmath.org/quadraticequationinmathbbc">
quadratic equation
</a>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
Putting the general equation of a circle, ellipse, or hyperbola into standard form, e.g. the circle
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A0.EGx2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex6">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle x^{2}+y^{2}+2x+4y=5\Rightarrow(x+1)^{2}+(y+2)^{2}=10," class="ltx_Math" display="inline" id="S0.Ex6.m1">
<mrow>
<mrow>
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mi>
y
</mi>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<mrow>
<mn>
4
</mn>
<mo>
</mo>
<mi>
y
</mi>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
5
</mn>
<mo>
⇒
</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
x
</mi>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mi>
y
</mi>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
=
</mo>
<mn>
10
</mn>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
from which it is frequently easier to read off important information (the center, radius, etc.)
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
Completing the square can also be used to find the extremal value
of a quadratic polynomial
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib2" title="">
2
</a>
]
</cite>
without
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Calculus.html">
calculus
</a>
.
Let us illustrate this for the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
polynomial
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/polynomial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomialring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="p(x)=4x^{2}+8x+9" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mrow>
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<mn>
4
</mn>
<mo>
</mo>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
+
</mo>
<mrow>
<mn>
8
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<mn>
9
</mn>
</mrow>
</mrow>
</math>
.
Completing the square yields
</p>
<table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A0.EGx3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex9">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle p(x)" class="ltx_Math" display="inline" id="S0.Ex7.m1">
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S0.Ex7.m2">
<mo>
=
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle(2x+2)^{2}-4+9" class="ltx_Math" display="inline" id="S0.Ex7.m3">
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
<mo>
-
</mo>
<mn>
4
</mn>
</mrow>
<mo>
+
</mo>
<mn>
9
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S0.Ex8.m2">
<mo>
=
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle(2x+2)^{2}+5" class="ltx_Math" display="inline" id="S0.Ex8.m3">
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
<mo>
+
</mo>
<mn>
5
</mn>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
</td>
<td class="ltx_td ltx_align_center ltx_eqn_cell">
<math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S0.Ex9.m2">
<mo>
≥
</mo>
</math>
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math alttext="\displaystyle 5," class="ltx_Math" display="inline" id="S0.Ex9.m3">
<mrow>
<mn>
5
</mn>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
since
<math alttext="(2x+2)^{2}\geq 0" class="ltx_Math" display="inline" id="I1.i3.p1.m2">
<mrow>
<msup>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<mn>
2
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mn>
2
</mn>
</msup>
<mo>
≥
</mo>
<mn>
0
</mn>
</mrow>
</math>
. Here, equality holds if and
only if
<math alttext="x=-1" class="ltx_Math" display="inline" id="I1.i3.p1.m3">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
</math>
.
Thus
<math alttext="p(x)\geq 5" class="ltx_Math" display="inline" id="I1.i3.p1.m4">
<mrow>
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
≥
</mo>
<mn>
5
</mn>
</mrow>
</math>
for all
<math alttext="x\in\mathbb{R}" class="ltx_Math" display="inline" id="I1.i3.p1.m5">
<mrow>
<mi>
x
</mi>
<mo>
∈
</mo>
<mi>
ℝ
</mi>
</mrow>
</math>
, and
<math alttext="p(x)=5" class="ltx_Math" display="inline" id="I1.i3.p1.m6">
<mrow>
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
5
</mn>
</mrow>
</math>
if and only if
<math alttext="x=-1" class="ltx_Math" display="inline" id="I1.i3.p1.m7">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
</math>
.
It follows that
<math alttext="p(x)" class="ltx_Math" display="inline" id="I1.i3.p1.m8">
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
has a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
global minimum
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GlobalMinimum.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extremum">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
at
<math alttext="x=-1" class="ltx_Math" display="inline" id="I1.i3.p1.m9">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</mrow>
</math>
, where
<math alttext="p(-1)=5" class="ltx_Math" display="inline" id="I1.i3.p1.m10">
<mrow>
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mn>
5
</mn>
</mrow>
</math>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
Completing the square can also be used as an
<a class="nnexus_concept" href="http://planetmath.org/integrationtechniques">
integration technique
</a>
to
<a class="nnexus_concept" href="http://planetmath.org/integralsign">
integrate
</a>
, for example the
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
<math alttext="\displaystyle\frac{1}{4x^{2}+8x+9}" class="ltx_Math" display="inline" id="I1.i4.p1.m1">
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<mrow>
<mrow>
<mn>
4
</mn>
<mo>
</mo>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo>
+
</mo>
<mrow>
<mn>
8
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<mn>
9
</mn>
</mrow>
</mfrac>
</mstyle>
</math>
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
]
</cite>
.
</p>
</div>
</li>
</ul>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
R. Adams,
<em class="ltx_emph ltx_font_italic">
Calculus, a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complete
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/ordersinanumberfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/soundcomplete">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/kripkesemantics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/maximallyconsistent">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
course
</em>
,
Addison-Wesley Publishers Ltd, 3rd ed.
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
<em class="ltx_emph ltx_font_italic">
Matematiklexikon
</em>
(in Swedish),
J. Thompson, T. Martinsson, Wahlström & Widstrand, 1991.
</span>
</li>
</ul>
</section>
<div class="ltx_para" id="p4">
<p class="ltx_p">
(Anyone has an English reference?)
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
completing the square
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompletingTheSquare
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:36:27
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 13:36:27
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
14
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
mathcam (2727)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Algorithm
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
SquareOfSum
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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<div class="ltx_page_logo">
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0 | Counterexample | http://planetmath.org/Counterexample | <!DOCTYPE html>
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<title>
counterexample
</title>
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counterexample
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
counterexample
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Counterexample.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
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<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</em>
is an example which is used to prove that a statement is false.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
For instance, to disprove that the statement “For all
<math alttext="n" class="ltx_Math" display="inline" id="p2.m1">
<mi>
n
</mi>
</math>
,
<math alttext="2^{2^{n}}+1" class="ltx_Math" display="inline" id="p2.m2">
<mrow>
<msup>
<mn>
2
</mn>
<msup>
<mn>
2
</mn>
<mi>
n
</mi>
</msup>
</msup>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</math>
is prime.”, one could exhibit
<math alttext="5" class="ltx_Math" display="inline" id="p2.m3">
<mn>
5
</mn>
</math>
as a counterexample; since
<math alttext="2^{2^{5}}+1=641\times 6700417" class="ltx_Math" display="inline" id="p2.m4">
<mrow>
<mrow>
<msup>
<mn>
2
</mn>
<msup>
<mn>
2
</mn>
<mn>
5
</mn>
</msup>
</msup>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
641
</mn>
<mo>
×
</mo>
<mn>
6700417
</mn>
</mrow>
</mrow>
</math>
, the statement is false.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
counterexample
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Counterexample
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:43:00
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:43:00
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A20
</td>
</tr>
</tbody>
</table>
</div>
</article>
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Generated on Tue Feb 6 22:19:49 2018 by
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0 | DigitalObject | http://planetmath.org/DigitalObject | <!DOCTYPE html>
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<title>
digital object
</title>
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<div class="ltx_page_main">
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
digital object
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/digitalobject">
digital object
</a>
</em>
in a digital library is the textual or multimedia data and the metadata.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Formally, a digital object
<math alttext="DO" class="ltx_Math" display="inline" id="p2.m1">
<mrow>
<mi>
D
</mi>
<mo>
</mo>
<mi>
O
</mi>
</mrow>
</math>
is a quadruple
<math alttext="(h,SM,ST,SS)" class="ltx_Math" display="inline" id="p2.m2">
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
h
</mi>
<mo>
,
</mo>
<mrow>
<mi>
S
</mi>
<mo>
</mo>
<mi>
M
</mi>
</mrow>
<mo>
,
</mo>
<mrow>
<mi>
S
</mi>
<mo>
</mo>
<mi>
T
</mi>
</mrow>
<mo>
,
</mo>
<mrow>
<mi>
S
</mi>
<mo>
</mo>
<mi>
S
</mi>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
where
</p>
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<math alttext="h\in H" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mrow>
<mi>
h
</mi>
<mo>
∈
</mo>
<mi>
H
</mi>
</mrow>
</math>
, where
<math alttext="H" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mi>
H
</mi>
</math>
is a set of universally unique handles (labels);
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
<math alttext="SM=\{sm_{1},sm_{2},...,sm_{n}\}" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mrow>
<mrow>
<mi>
S
</mi>
<mo>
</mo>
<mi>
M
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
s
</mi>
<mo>
</mo>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mo>
,
</mo>
<mrow>
<mi>
s
</mi>
<mo>
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
,
</mo>
<mrow>
<mi>
s
</mi>
<mo>
</mo>
<msub>
<mi>
m
</mi>
<mi>
n
</mi>
</msub>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
</math>
is a set of streams;
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
<math alttext="ST=\{st_{1},st_{2},...,st_{m}\}" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mrow>
<mrow>
<mi>
S
</mi>
<mo>
</mo>
<mi>
T
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
s
</mi>
<mo>
</mo>
<msub>
<mi>
t
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mo>
,
</mo>
<mrow>
<mi>
s
</mi>
<mo>
</mo>
<msub>
<mi>
t
</mi>
<mn>
2
</mn>
</msub>
</mrow>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
,
</mo>
<mrow>
<mi>
s
</mi>
<mo>
</mo>
<msub>
<mi>
t
</mi>
<mi>
m
</mi>
</msub>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
</math>
is a set of structural metadata
<a class="nnexus_concept" href="http://planetmath.org/comprehensionaxiom">
specifications
</a>
; and
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
<math alttext="SS=\{stsm_{1},stsm_{2},...,stsm_{p}\}" class="ltx_Math" display="inline" id="I1.i4.p1.m1">
<mrow>
<mrow>
<mi>
S
</mi>
<mo>
</mo>
<mi>
S
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
{
</mo>
<mrow>
<mi>
s
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mo>
,
</mo>
<mrow>
<mi>
s
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
,
</mo>
<mrow>
<mi>
s
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
s
</mi>
<mo>
</mo>
<msub>
<mi>
m
</mi>
<mi>
p
</mi>
</msub>
</mrow>
<mo stretchy="false">
}
</mo>
</mrow>
</mrow>
</math>
is a set of Structured Streams functions defined from the streams in
<math alttext="SM" class="ltx_Math" display="inline" id="I1.i4.p1.m2">
<mrow>
<mi>
S
</mi>
<mo>
</mo>
<mi>
M
</mi>
</mrow>
</math>
and from the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structures
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in
<math alttext="ST" class="ltx_Math" display="inline" id="I1.i4.p1.m3">
<mrow>
<mi>
S
</mi>
<mo>
</mo>
<mi>
T
</mi>
</mrow>
</math>
.
</p>
</div>
</li>
</ol>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
digital object
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
DigitalObject
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:30:38
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:30:38
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
11
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
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Definition
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msc 00A20
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0 | FamousOpenQuestionsInMathematics | http://planetmath.org/FamousOpenQuestionsInMathematics | <!DOCTYPE html>
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famous open questions in mathematics
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<h1 class="ltx_title ltx_title_document">
famous open questions in mathematics
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Despite the fact that at least a generation of mathematicians has tried to solve these problems, they still remain open:
</p>
</div>
<div class="ltx_para" id="p2">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Goldbach conjecture
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GoldbachConjecture.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/goldbachsconjecture">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Twin prime conjecture
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TwinPrimeConjecture.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/twinprimeconjecture">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
Poincaré
<a class="nnexus_concept" href="http://planetmath.org/openquestion">
conjecture
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Riemann Hypothesis
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RiemannHypothesis.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/riemannzetafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/birchandswinnertondyerconjecture">
Birch and Swinnerton-Dyer conjecture
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I1.i6.p1">
<p class="ltx_p">
P vs NP problem
</p>
</div>
</li>
<li class="ltx_item" id="I1.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I1.i7.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/HodgeConjecture.html">
Hodge conjecture
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I1.i8.p1">
<p class="ltx_p">
Solution to the Navier-Stokes equations
</p>
</div>
</li>
<li class="ltx_item" id="I1.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I1.i9.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Collatz problem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CollatzProblem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/collatzproblem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
10.
</span>
<div class="ltx_para" id="I1.i10.p1">
<p class="ltx_p">
Schanuel’s conjecture
</p>
</div>
</li>
<li class="ltx_item" id="I1.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
11.
</span>
<div class="ltx_para" id="I1.i11.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Beal conjecture
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BealsConjecture.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bealconjecture">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
12.
</span>
<div class="ltx_para" id="I1.i12.p1">
<p class="ltx_p">
Irrationality of Euler’s
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
constant
</a>
<math alttext="\gamma" class="ltx_Math" display="inline" id="I1.i12.p1.m1">
<mi>
γ
</mi>
</math>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
13.
</span>
<div class="ltx_para" id="I1.i13.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Existence.html">
Existence
</a>
of an odd perfect number
</p>
</div>
</li>
<li class="ltx_item" id="I1.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
14.
</span>
<div class="ltx_para" id="I1.i14.p1">
<p class="ltx_p">
Koethe conjecture
</p>
</div>
</li>
<li class="ltx_item" id="I1.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
15.
</span>
<div class="ltx_para" id="I1.i15.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/invariantsubspaceproblem">
Invariant subspace problem
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
16.
</span>
<div class="ltx_para" id="I1.i16.p1">
<p class="ltx_p">
Lehmer’s conjecture
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/openproblems">
Open problems
</a>
3 to 8, together with the quest for a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
mathematical foundation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/bibliographyforaxiomaticsandmathematicsfoundationsincategories">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/analyticsandformallogicsinmetamathematics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
explaining the mass gap
<a class="nnexus_concept" href="http://planetmath.org/property">
property
</a>
in Yang-Mills theory, constitute a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
collection
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Collection.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/collection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of problems known as the
<a class="nnexus_concept" href="http://planetmath.org/millenniumproblems">
Millennium Problems
</a>
. Please see
<span class="ltx_text ltx_font_typewriter">
http://www.claymath.org/millennium/
</span>
http://www.claymath.org/millennium/ for more detail.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Note:
</span>
A mathematical foundation for the partial solution of the the mass gap property in Yang-Mills theory has been published by G. Cleaver and K. Tanaka (2000): “Ratio of Quark Masses in Duality Theories.”,
<span class="ltx_text ltx_font_typewriter">
http://arxiv.org/abs/hep-th/0002089v1
</span>
on line, and can be accessed as a
<span class="ltx_text ltx_font_typewriter">
http://arxiv.org/PS_cache/hep-th/pdf/0002/0002089v1.pdf
</span>
PDF file .
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<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/famousopenquestionsinmathematics">
famous open questions in mathematics
</a>
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</tr>
<tr class="ltx_tr">
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name
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FamousOpenQuestionsInMathematics
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2013-03-22 14:42:52
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2013-03-22 14:42:52
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Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
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rspuzio (6075)
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15
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rspuzio (6075)
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msc 00A20
</td>
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Related topic
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TwoGeneratorProperty
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MillenniumProblems
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0 | HighSchoolMathematics | http://planetmath.org/HighSchoolMathematics | <!DOCTYPE html>
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high school mathematics
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high school mathematics
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<br class="ltx_break"/>
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The aim of this meta entry is to index entries suitable for
high school students.
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<h2 class="ltx_title ltx_title_subsubsection">
Basics
</h2>
<div class="ltx_para" id="S0.SS0.SSSx1.p1">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
set, union,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
intersection
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r26">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Intersection.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/intersection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
natural numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NaturalNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/naturalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
rational numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/egyptianfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/liberabaci">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/rationalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/realnumber">
real numbers
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/associativityofmultiplication">
associativity of multiplication
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/productofnegativenumbers">
product of negative numbers
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
equation,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
inequality
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Inequality.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inequalitiesforrealnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I1.i6.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/proportionequation">
proportion equation
</a>
, proportionality of numbers
</p>
</div>
</li>
<li class="ltx_item" id="I1.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I1.i7.p1">
<p class="ltx_p">
per cent
</p>
</div>
</li>
<li class="ltx_item" id="I1.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I1.i8.p1">
<p class="ltx_p">
mathematical induction
</p>
</div>
</li>
<li class="ltx_item" id="I1.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I1.i9.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
proof by contradiction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ProofbyContradiction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradiction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/deductiontheoremholdsforclassicalpropositionallogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
10.
</span>
<div class="ltx_para" id="I1.i10.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
converse
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Converse.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/converse">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
11.
</span>
<div class="ltx_para" id="I1.i11.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/contrapositive">
contrapositive
</a>
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx2">
<h2 class="ltx_title ltx_title_subsubsection">
Algebra
</h2>
<div class="ltx_para" id="S0.SS0.SSSx2.p1">
<ol class="ltx_enumerate" id="I2">
<li class="ltx_item" id="I2.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I2.i1.p1">
<p class="ltx_p">
opposite number,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
difference
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/difference">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/setdifference">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I2.i2.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/inversenumber">
inverse number
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/division">
division
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I2.i3.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
multiple
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Multiple.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divisibility">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
product
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/product">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I2.i4.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/topicentryonrationalnumbers">
entries on rational numbers
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I2.i5.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/irrational">
irrational numbers
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I2.i6.p1">
<p class="ltx_p">
factorization of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I2.i7.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/linearequation">
linear equation
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I2.i8.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/squareofsum">
square of sum
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I2.i9.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/differenceofsquares">
difference of squares
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
10.
</span>
<div class="ltx_para" id="I2.i10.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/groupingmethodforfactoringpolynomials">
grouping method for factoring polynomials
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
11.
</span>
<div class="ltx_para" id="I2.i11.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Factoring.html">
factoring
</a>
a sum or difference of two cubes
</p>
</div>
</li>
<li class="ltx_item" id="I2.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
12.
</span>
<div class="ltx_para" id="I2.i12.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/zeroruleofproduct">
zero rule of product
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
13.
</span>
<div class="ltx_para" id="I2.i13.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://
<a class="nnexus_concept" href="http://planetmath.org/planetmath">
planetmath
</a>
.org/ConjugationMnemonic
</span>
conjugation
</p>
</div>
</li>
<li class="ltx_item" id="I2.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
14.
</span>
<div class="ltx_para" id="I2.i14.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/evenevenoddrule">
even-even-odd rule
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
15.
</span>
<div class="ltx_para" id="I2.i15.p1">
<p class="ltx_p">
completing the square
</p>
</div>
</li>
<li class="ltx_item" id="I2.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
16.
</span>
<div class="ltx_para" id="I2.i16.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/squarerootsofrationals">
square roots of rationals
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
17.
</span>
<div class="ltx_para" id="I2.i17.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/quadraticformula">
quadratic formula
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
18.
</span>
<div class="ltx_para" id="I2.i18.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/quadraticinequality">
quadratic inequality
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i19" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
19.
</span>
<div class="ltx_para" id="I2.i19.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/strangeroot">
strange root
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i20" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
20.
</span>
<div class="ltx_para" id="I2.i20.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/inequalitywithabsolutevalues">
inequality with absolute values
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i21" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
21.
</span>
<div class="ltx_para" id="I2.i21.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/absolutevalueinequalities">
absolute value inequalities
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i22" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
22.
</span>
<div class="ltx_para" id="I2.i22.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/longdivision">
long division
</a>
of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
polynomials
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/polynomial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polynomialring">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx3">
<h2 class="ltx_title ltx_title_subsubsection">
Geometry
</h2>
<div class="ltx_para" id="S0.SS0.SSSx3.p1">
<ol class="ltx_enumerate" id="I3">
<li class="ltx_item" id="I3.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I3.i1.p1">
<p class="ltx_p">
basic geometric figures:
</p>
</div>
<div class="ltx_para" id="I3.i1.p2">
<ul class="ltx_itemize" id="I3.I1">
<li class="ltx_item" id="I3.I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i1.p1">
<p class="ltx_p">
points
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i2.p1">
<p class="ltx_p">
lines
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i3.p1">
<p class="ltx_p">
planes
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i4.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
line segments
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LineSegment.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/linesegment">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i5.p1">
<p class="ltx_p">
rays
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i6.p1">
<p class="ltx_p">
angles
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i7.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
triangles
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Triangle.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/triangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i8.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
parallelograms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Parallelogram.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/parallelogram">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i9.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
rectangles
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Rectangle.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/specialelementsinarelationalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/rectangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i10.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
trapezoids
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Trapezoid.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/trapezoid">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i11.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
polygons
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Polygon.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polygon">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polygon1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i12.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
regular polygons
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RegularPolygon.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/regularpolygon">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i13.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/baseandheightoftriangle">
base and height of triangle
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i14.p1">
<p class="ltx_p">
circles
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i15.p1">
<p class="ltx_p">
parts of a ball
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i16.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/cylinder">
cylinder
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I1.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I1.i17.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/coneinmathbbr3">
solid cone
</a>
</p>
</div>
</li>
</ul>
</div>
</li>
<li class="ltx_item" id="I3.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I3.i2.p1">
<p class="ltx_p">
basic geometric
<a class="nnexus_concept" href="http://planetmath.org/property">
properties
</a>
:
</p>
</div>
<div class="ltx_para" id="I3.i2.p2">
<ul class="ltx_itemize" id="I3.I2">
<li class="ltx_item" id="I3.I2.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i1.p1">
<p class="ltx_p">
intersections
</p>
</div>
</li>
<li class="ltx_item" id="I3.I2.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/Betweenness
</span>
between
</p>
</div>
</li>
<li class="ltx_item" id="I3.I2.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/endpoint">
endpoints
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I2.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i4.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Midpoint
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Midpoint.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/proofofuniquenessofcenterofacircle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/midpoint">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/midpoint1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
midpoints
</p>
</div>
</li>
<li class="ltx_item" id="I3.I2.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i5.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
parallelism
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/mutualpositionsofvectors">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/parallellismineuclideanplane">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/parallelismoftwoplanes">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I2.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i6.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/perpendicularityineuclideanplane">
perpendicularity
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I2.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i7.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Congruence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Congruence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence12">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mayanmath">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruenceonapartialalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
congruence
</p>
</div>
</li>
<li class="ltx_item" id="I3.I2.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i8.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
similarity
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Similarity.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/similarityingeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I2.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i9.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
similar triangles
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SimilarTriangles.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/similarityoftriangles">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I2.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I2.i10.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/tangentofcircle">
tangent of circle
</a>
</p>
</div>
</li>
</ul>
</div>
</li>
<li class="ltx_item" id="I3.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I3.i3.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
acute angles
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AcuteAngle.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/angle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/convexangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, convex angles, radian
</p>
</div>
</li>
<li class="ltx_item" id="I3.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I3.i4.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complementary angles
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplementaryAngles.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complementaryangles">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
supplementary angles
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SupplementaryAngles.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/supplementaryangles">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/explementary">
explementary angles
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I3.i5.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/anglebetweentwolines">
angle between two lines
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/angleofviewofalinesegment">
angle of view
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I3.i6.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/projectionofpoint">
projection of point
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I3.i7.p1">
<p class="ltx_p">
locus
</p>
</div>
</li>
<li class="ltx_item" id="I3.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I3.i8.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
normal line
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NormalLine.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/normalline">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
angle bisector
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AngleBisector.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/anglebisector">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/anglebisectoraslocus">
angle bisector as locus
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/centernormalandcenternormalplaneasloci">
center normal as locus
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I3.i9.p1">
<p class="ltx_p">
measurements (lengths, areas, and volumes) of basic geometric figures
</p>
</div>
</li>
<li class="ltx_item" id="I3.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
10.
</span>
<div class="ltx_para" id="I3.i10.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/compassandstraightedgeconstruction">
compass and straightedge constructions
</a>
:
</p>
</div>
<div class="ltx_para" id="I3.i10.p2">
<ul class="ltx_itemize" id="I3.I3">
<li class="ltx_item" id="I3.I3.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/Midpoint
</span>
midpoint
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i2.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
perpendicular bisector
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PerpendicularBisector.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/perpendicularbisector">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i3.p1">
<p class="ltx_p">
dropping the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
perpendicular
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Perpendicular.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dihedralangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
from a point to a line
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i4.p1">
<p class="ltx_p">
erecting the perpendicular to a line at a point
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i5.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfAngleBisector
</span>
angle bisector
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i6.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfRegularTriangle
</span>
<a class="nnexus_concept" href="http://planetmath.org/regulartriangle">
regular triangle
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i7.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfDuplicatingAnAngle
</span>
duplicating an angle
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i8.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfCenterOfGivenCircle
</span>
center of a circle
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i9.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/constructionoftangent">
construction of tangent
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i10.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Circumcenter
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Circumcenter.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/circumcircle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
circle
<a class="nnexus_concept" href="http://planetmath.org/incidencegeometry">
passing through
</a>
three noncollinear points
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i11.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfParallelLine
</span>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ParallelLines.html">
parallel line
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i12.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfSquare
</span>
square
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i13.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/NSectionOfLineSegmentWithCompassAndStraightedge
<math alttext="n" class="ltx_Math" display="inline" id="I3.I3.i13.p1.m1">
<mi mathvariant="normal">
n
</mi>
</math>
</span>
-
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
section
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Section.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operationsonrelations">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of line segment
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i14.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/circlewithgivencenterandgivenradius">
circle with given center and given radius
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i15.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfSimilarTriangles
</span>
similar triangles
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i16.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfGeometricMean
</span>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
geometric mean
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GeometricMean.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/geometricmean">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i17.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/ConstructionOfCentralProportion
</span>
central proportional
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i18.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfInversePoint
</span>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
inverse point
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/InversePoints.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inversionofplane">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
with respect to a circle
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i19" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i19.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstructionOfRegularPentagon
</span>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RegularPentagon.html">
regular pentagon
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I3.i20" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I3.i20.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/ConstructionOfRegular2nGonFromRegularNGon
</span>
construction of
<a class="nnexus_concept" href="http://planetmath.org/cofinality">
regular
</a>
<math alttext="2n" class="ltx_Math" display="inline" id="I3.I3.i20.p1.m1">
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
n
</mi>
</mrow>
</math>
-gon from regular
<math alttext="n" class="ltx_Math" display="inline" id="I3.I3.i20.p1.m2">
<mi>
n
</mi>
</math>
-gon
</p>
</div>
</li>
</ul>
</div>
</li>
<li class="ltx_item" id="I3.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
11.
</span>
<div class="ltx_para" id="I3.i11.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/trisectionofangle">
trisection of angle
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
12.
</span>
<div class="ltx_para" id="I3.i12.p1">
<p class="ltx_p">
axiomatic proofs in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
geometry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Geometry.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/geometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/moscowmathematicalpapyrus">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
:
</p>
</div>
<div class="ltx_para" id="I3.i12.p2">
<ul class="ltx_itemize" id="I3.I4">
<li class="ltx_item" id="I3.I4.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i1.p1">
<p class="ltx_p">
angles of an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
isosceles triangle
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/IsoscelesTriangle.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/isoscelestriangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i2.p1">
<p class="ltx_p">
determining from angles that a triangle is isosceles
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/isoscelestriangletheorem">
isosceles triangle theorem
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i4.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/converseofisoscelestriangletheorem">
converse of isosceles triangle theorem
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i5.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/parallelogramtheorems">
parallelogram theorems
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i6.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/regularpolygonandcircles">
regular polygon and circles
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i7.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Pythagorean theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PythagoreanTheorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/pythagoreantheorem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generalizedpythagoreantheorem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and its various proofs:
</p>
</div>
<div class="ltx_para" id="I3.I4.i7.p2">
<ul class="ltx_itemize" id="I3.I4.I1">
<li class="ltx_item" id="I3.I4.I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
*
</span>
<div class="ltx_para" id="I3.I4.I1.i1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/ProofOfPythagoreasTheorem
</span>
using four
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
congruent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Congruent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/geometriccongruence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
triangles and a square
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
*
</span>
<div class="ltx_para" id="I3.I4.I1.i2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/ProofOfPythagoreanTheorem
</span>
dropping
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
altitude
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Altitude.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/prismatoid">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to form three similar triangles
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
*
</span>
<div class="ltx_para" id="I3.I4.I1.i3.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/GarfieldsProofOfPythagoreanTheorem
</span>
Garfield’s proof
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
*
</span>
<div class="ltx_para" id="I3.I4.I1.i4.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/ProofOfPythagoreanTheorem2
</span>
two dissections of a square with side
<math alttext="a+b" class="ltx_Math" display="inline" id="I3.I4.I1.i4.p1.m1">
<mrow>
<mi>
a
</mi>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
</math>
</p>
</div>
</li>
</ul>
</div>
</li>
<li class="ltx_item" id="I3.I4.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i8.p1">
<p class="ltx_p">
construct the center of a given circle
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i9.p1">
<p class="ltx_p">
Thales’
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and its
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/ProofOfThalesTheorem
</span>
proof
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i10.p1">
<p class="ltx_p">
opposing angles in a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
cyclic quadrilateral
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CyclicQuadrilateral.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cyclicquadrilateral">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
are supplementary
</p>
</div>
</li>
<li class="ltx_item" id="I3.I4.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
<span class="ltx_text ltx_font_bold">
–
</span>
</span>
<div class="ltx_para" id="I3.I4.i11.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/trianglemidsegmenttheorem">
mid-segment theorem
</a>
</p>
</div>
</li>
</ul>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx4">
<h2 class="ltx_title ltx_title_subsubsection">
Analytic geometry
</h2>
<div class="ltx_para" id="S0.SS0.SSSx4.p1">
<ol class="ltx_enumerate" id="I4">
<li class="ltx_item" id="I4.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I4.i1.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
analytic geometry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AnalyticGeometry.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/analyticgeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I4.i2.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Cartesian coordinates
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CartesianCoordinates.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cartesiancoordinates">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I4.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/coordinatesofmidpoint">
coordinates of midpoint
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I4.i4.p1">
<p class="ltx_p">
slope
</p>
</div>
</li>
<li class="ltx_item" id="I4.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I4.i5.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
tangent line
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TangentLine.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/tangentline">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I4.i6.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/conditionoforthogonality">
condition of orthogonality
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I4.i7.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/AngleBetweenTwoLines
</span>
angle between two lines
</p>
</div>
</li>
<li class="ltx_item" id="I4.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I4.i8.p1">
<p class="ltx_p">
conics (
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
ellipse
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/ellipse">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/conicsection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/hyperbola">
hyperbola
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/parabola">
parabola
</a>
)
</p>
</div>
</li>
<li class="ltx_item" id="I4.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I4.i9.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/polarcoordinates">
polar coordinates
</a>
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx5">
<h2 class="ltx_title ltx_title_subsubsection">
Vectors and Matrices
</h2>
<div class="ltx_para" id="S0.SS0.SSSx5.p1">
<ol class="ltx_enumerate" id="I5">
<li class="ltx_item" id="I5.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I5.i1.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/parallelogramprinciple">
sum of vectors
</a>
(i.e. parallelogram principle),
<a class="nnexus_concept" href="http://planetmath.org/differenceofvectors">
difference of vectors
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I5.i2.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/euclideanvector">
Euclidean vectors
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I5.i3.p1">
<p class="ltx_p">
mutual positions of vectors
</p>
</div>
</li>
<li class="ltx_item" id="I5.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I5.i4.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
scalar product
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ScalarProduct.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dotproduct">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I5.i5.p1">
<p class="ltx_p">
matrices,
<a class="nnexus_concept" href="http://planetmath.org/addition">
addition
</a>
and
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
multiplication
</a>
of matrices
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx6">
<h2 class="ltx_title ltx_title_subsubsection">
Trigonometry
</h2>
<div class="ltx_para" id="S0.SS0.SSSx6.p1">
<ol class="ltx_enumerate" id="I6">
<li class="ltx_item" id="I6.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I6.i1.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/RightTriangle.html">
right triangle
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I6.i2.p1">
<p class="ltx_p">
regular triangle
</p>
</div>
</li>
<li class="ltx_item" id="I6.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I6.i3.p1">
<p class="ltx_p">
isosceles triangle
</p>
</div>
</li>
<li class="ltx_item" id="I6.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I6.i4.p1">
<p class="ltx_p">
altitudes
</p>
</div>
</li>
<li class="ltx_item" id="I6.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I6.i5.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Bisector.html">
bisectors
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I6.i6.p1">
<p class="ltx_p">
ASA, SSS, SAS, SSA (
<a class="nnexus_concept" href="http://planetmath.org/trianglesolving">
triangle solving
</a>
)
</p>
</div>
</li>
<li class="ltx_item" id="I6.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I6.i7.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/exacttrigonometrytables">
exact trigonometry tables
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I6.i8.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
sohcahtoa
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SOHCAHTOA.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sohcahtoa">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I6.i9.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/determiningsignsoftrigonometricfunctions">
determining signs of trigonometric functions
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
10.
</span>
<div class="ltx_para" id="I6.i10.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/additionandsubtractionformulasforsineandcosine">
addition formulas for sine and cosine
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
11.
</span>
<div class="ltx_para" id="I6.i11.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/additionandsubtractionformulasfortangent">
addition formula for tangent
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
12.
</span>
<div class="ltx_para" id="I6.i12.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/goniometricformulas">
goniometric formulae
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
13.
</span>
<div class="ltx_para" id="I6.i13.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/trigonometricequations">
trigonometric equation
</a>
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx7">
<h2 class="ltx_title ltx_title_subsubsection">
Functions
</h2>
<div class="ltx_para" id="S0.SS0.SSSx7.p1">
<ol class="ltx_enumerate" id="I7">
<li class="ltx_item" id="I7.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I7.i1.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/definition">
definitions
</a>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I7.i2.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
argument
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Argument.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/argument">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I7.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/polynomialfunction">
polynomial functions
</a>
(including
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
linear functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LinearFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/linearfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
)
</p>
</div>
</li>
<li class="ltx_item" id="I7.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I7.i4.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/rationalfunction">
rational functions
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I7.i5.p1">
<p class="ltx_p">
functions involving
</p>
</div>
</li>
<li class="ltx_item" id="I7.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I7.i6.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
exponential functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/4.2#iii">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://dlmf.nist.gov/4.2#E19">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ExponentialFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complexexponentialfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I7.i7.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Briggsian logarithms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BriggsianLogarithm.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/briggsianlogarithms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I7.i8.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
trigonometric functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/4#PT3">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TrigonometricFunctions.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/definitionsintrigonometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I7.i9.p1">
<p class="ltx_p">
limit of real number sequence
</p>
</div>
</li>
<li class="ltx_item" id="I7.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
10.
</span>
<div class="ltx_para" id="I7.i10.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GeometricSequence.html">
geometric sequence
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
11.
</span>
<div class="ltx_para" id="I7.i11.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
sequences
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Sequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and series
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx8">
<h2 class="ltx_title ltx_title_subsubsection">
Differential calculus
</h2>
<div class="ltx_para" id="S0.SS0.SSSx8.p1">
<ol class="ltx_enumerate" id="I8">
<li class="ltx_item" id="I8.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I8.i1.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
concept
</a>
of a limit
</p>
</div>
</li>
<li class="ltx_item" id="I8.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I8.i2.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/limitrulesoffunctions">
limit rules of functions
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/improperlimits">
improper limit
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I8.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I8.i3.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
continuous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Continuous.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/classesofordinalsandenumeratingfunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I8.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I8.i4.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0
</span>
limit of sine divided by angle at 0
</p>
</div>
</li>
<li class="ltx_item" id="I8.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I8.i5.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/intermediatevaluetheorem">
intermediate value theorem
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I8.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I8.i6.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
derivative
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Derivative.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fixedpointsofnormalfunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivative">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I8.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I8.i7.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/derivativesofsineandcosine">
derivatives of sine and cosine
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I8.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I8.i8.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/derivativeofinversefunction">
derivative of inverse function
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I8.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I8.i9.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/relatedrates">
related rates
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I8.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
10.
</span>
<div class="ltx_para" id="I8.i10.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Minimum.html">
minimum
</a>
and
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Maximum.html">
maximum
</a>
of functions (
<a class="nnexus_concept" href="http://planetmath.org/extremum">
extrema
</a>
)
</p>
</div>
</li>
<li class="ltx_item" id="I8.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
11.
</span>
<div class="ltx_para" id="I8.i11.p1">
<p class="ltx_p">
least and greatest value of function
</p>
</div>
</li>
<li class="ltx_item" id="I8.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
12.
</span>
<div class="ltx_para" id="I8.i12.p1">
<p class="ltx_p">
mean value theorem
</p>
</div>
</li>
<li class="ltx_item" id="I8.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
13.
</span>
<div class="ltx_para" id="I8.i13.p1">
<p class="ltx_p">
Rolle’s theorem
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx9">
<h2 class="ltx_title ltx_title_subsubsection">
Integral calculus
</h2>
<div class="ltx_para" id="S0.SS0.SSSx9.p1">
<ol class="ltx_enumerate" id="I9">
<li class="ltx_item" id="I9.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I9.i1.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/riemannsum">
Riemann sum
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I9.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I9.i2.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
integral
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.4#iv">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lebesgueintegral">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integral">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I9.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I9.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/lefthandrule">
left hand rule
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I9.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I9.i4.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/righthandrule">
right hand rule
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I9.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I9.i5.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/midpointrule">
midpoint rule
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I9.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I9.i6.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompositeTrapezoidalRule
</span>
trapezoid rule
</p>
</div>
</li>
<li class="ltx_item" id="I9.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I9.i7.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental theorem of calculus
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremofcalculus">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentaltheoremsofcalculusforlebesgueintegration">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I9.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I9.i8.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/integrationtechniques">
integration techniques
</a>
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx10">
<h2 class="ltx_title ltx_title_subsubsection">
Complex numbers
</h2>
<div class="ltx_para" id="S0.SS0.SSSx10.p1">
<ol class="ltx_enumerate" id="I10">
<li class="ltx_item" id="I10.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I10.i1.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complex numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complexnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I10.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I10.i2.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexFunction.html">
complex function
</a>
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx11">
<h2 class="ltx_title ltx_title_subsubsection">
Applications (word problems)
</h2>
<div class="ltx_para" id="S0.SS0.SSSx11.p1">
<ol class="ltx_enumerate" id="I11">
<li class="ltx_item" id="I11.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I11.i1.p1">
<p class="ltx_p">
graphing of equations and inequalities
</p>
</div>
</li>
<li class="ltx_item" id="I11.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I11.i2.p1">
<p class="ltx_p">
counting and basic probability
</p>
</div>
</li>
<li class="ltx_item" id="I11.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I11.i3.p1">
<p class="ltx_p">
length, area, volume
</p>
</div>
</li>
<li class="ltx_item" id="I11.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I11.i4.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Distance.html">
distance
</a>
, rate, speed,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Velocity.html">
velocity
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I11.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I11.i5.p1">
<p class="ltx_p">
money,
<a class="nnexus_concept" href="http://planetmath.org/interest">
interest
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/compoundinterest">
compound interest
</a>
</p>
</div>
</li>
</ol>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/highschoolmathematics">
high school mathematics
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
HighSchoolMathematics
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:16:09
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:16:09
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
93
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Topic
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
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IndexOfEntriesOnCompassAndStraightedgeConstructions
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0 | IndexOfTables | http://planetmath.org/IndexOfTables | <!DOCTYPE html>
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<title>
index of tables
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
index of tables
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Since early in the history of mathematics, tables have served to aid in various computational tasks. Even today, with the ready availability of powerful
<a class="nnexus_concept" href="http://planetmath.org/calculator">
calculators
</a>
, tables still serve an important pedagogical purpose. Tables are found in the main text of mathematical papers and books, to illustrate the
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
concepts
</a>
in question, and as the back matter of books.
</p>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
Elementary tables
</h2>
<div class="ltx_para" id="S1.p1">
<p class="ltx_p">
The most
<a class="nnexus_concept" href="http://planetmath.org/elementaryrecursivefunction">
elementary
</a>
tables in mathematics are those for the four basic
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
:
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
subtraction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/subtraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/additionandsubtractionformulasforsineandcosine">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concept" href="http://planetmath.org/multiplication">
multiplication
</a>
and
<a class="nnexus_concept" href="http://planetmath.org/division">
division
</a>
. In the old days, schoolchildren had to learn these up to 12. Nowadays teachers stop at 10.
</p>
</div>
<div class="ltx_para" id="S1.p2">
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/tableofadditionupto12">
Table of addition up to 12
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/tableofsubtractionupto12">
Table of subtraction up to 12
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/tableofmultiplicationupto12">
Table of multiplication up to 12
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/tableofdivisionupto12">
Table of division up to 12
</a>
</p>
</div>
</li>
</ul>
</div>
<div class="ltx_para" id="S1.p3">
<p class="ltx_p">
Before the age of
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computers
</a>
, tables of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Logarithm.html">
logarithms
</a>
were of paramount importance to most scientists, as well as tables of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
square roots
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SquareRoot.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/squareroot">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
</section>
<section class="ltx_section" id="S2">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
2
</span>
Logical tables
</h2>
<div class="ltx_para" id="S2.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Truth tables
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TruthTable.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/truthtable">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
can be made for some fairly complicated
<a class="nnexus_concept" href="http://planetmath.org/expression">
expressions
</a>
, but the ones of most use to computer programming students are those for the basic expressions, a truth table for
<a class="nnexus_concept" href="http://planetmath.org/conjunction">
logical AND
</a>
, a truth table for
<a class="nnexus_concept" href="http://planetmath.org/disjunction">
logical OR
</a>
, a truth table for
<a class="nnexus_concept" href="http://planetmath.org/exclusiveor">
logical XOR
</a>
, etc. These can be expressed as binary operations by simply replacing TRUEs with 1s and FALSEs with 0s.
</p>
</div>
</section>
<section class="ltx_section" id="S3">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
3
</span>
Combinatorics tables
</h2>
<div class="ltx_para" id="S3.p1">
<p class="ltx_p">
Some tables have garnered
<a class="nnexus_concept" href="http://planetmath.org/interest">
interest
</a>
beyond just looking up values to become classics. This is perhaps truest of Pascal’s
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
triangle
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Triangle.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/triangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, which has many interesting
<a class="nnexus_concept" href="http://planetmath.org/property">
properties
</a>
besides being a convenient way to look up
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
binomial coefficients
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.2#E1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://dlmf.nist.gov/26.3#SS1.p1">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BinomialCoefficient.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/binomialcoefficient">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
without having to compute large
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
factorials
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Factorial.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/factorial">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
</section>
<section class="ltx_section" id="S4">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
4
</span>
Number theory tables
</h2>
<div class="ltx_para" id="S4.p1">
<p class="ltx_p">
Books about
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
prime numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimeNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/prime">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
can be reliably counted upon to list the first
<a class="nnexus_concept" href="http://planetmath.org/decimalplace">
thousand
</a>
<a class="nnexus_concept" href="http://planetmath.org/positive">
positive
</a>
prime numbers, or perhaps the first ten thousand, but certainly the first hundred. A book about prime numbers, or a book about
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
number theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NumberTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/numbertheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in general, might have as an appendix a table of
<a class="nnexus_concept" href="http://planetmath.org/integerfactorization">
integer factorizations
</a>
for
<math alttext="0<n<1001" class="ltx_Math" display="inline" id="S4.p1.m1">
<mrow>
<mn>
0
</mn>
<mo>
<
</mo>
<mi>
n
</mi>
<mo>
<
</mo>
<mn>
1001
</mn>
</mrow>
</math>
. A
<a class="nnexus_concept" href="http://planetmath.org/tableofmersenneprimes">
table of Mersenne primes
</a>
generally gives the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
exponent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Exponent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generalpower">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/exponent">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
rather than writing out the prime in base 10 since these numbers get very large very quickly. A book specifically on
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Mersenne primes
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MersennePrime.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mersennenumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
might also have a
<a class="nnexus_concept" href="http://planetmath.org/tableoffactorsofsmallmersennenumbers">
table of factors of small Mersenne numbers
</a>
. The
<a class="nnexus_concept" href="http://planetmath.org/fermatnumbers">
Fermat numbers
</a>
grow large even more quickly, and thus a table of factors of Fermat numbers is even less likely to write out Fermat numbers.
</p>
</div>
<div class="ltx_para" id="S4.p2">
<p class="ltx_p">
Similarly to a book about prime numbers, a book on the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Fibonacci numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/26.11#p4">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FibonacciNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fibonaccisequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
might have a
<a class="nnexus_concept" href="http://planetmath.org/listoffibonaccinumbers">
list of Fibonacci numbers
</a>
(the appendices of Koshy’s book on the subject are actually tables of the factorizations of the Fibonacci and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Lucas numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LucasNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lucasnumbers">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
).
</p>
</div>
<div class="ltx_para" id="S4.p3">
<p class="ltx_p">
For
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
with a very small range of possible output values, such as the Möbius function
<math alttext="-2<\mu(n)<2" class="ltx_Math" display="inline" id="S4.p3.m1">
<mrow>
<mrow>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
<mo>
<
</mo>
<mrow>
<mi>
μ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
<
</mo>
<mn>
2
</mn>
</mrow>
</math>
or the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Liouville function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/27.2#E13">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LiouvilleFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/liouvillefunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="|\lambda(n)|=1" class="ltx_Math" display="inline" id="S4.p3.m2">
<mrow>
<mrow>
<mo stretchy="false">
|
</mo>
<mrow>
<mi>
λ
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
n
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo stretchy="false">
|
</mo>
</mrow>
<mo>
=
</mo>
<mn>
1
</mn>
</mrow>
</math>
, it makes sense for a table to pair them up with their
<a class="nnexus_concept" href="http://planetmath.org/matching">
matching
</a>
<a class="nnexus_concept" href="http://planetmath.org/summatoryfunctionofarithmeticfunction">
summatory functions
</a>
. Thus we have a table of values of the Möbius function and the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Mertens function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MertensFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mertensfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and a table of values of the Liouville function and its summatory function.
</p>
</div>
<div class="ltx_para" id="S4.p4">
<p class="ltx_p">
There are also tables using
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
primitive roots
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimitiveRoot.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/primitiveroot">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and index to solve
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
congruences
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Congruence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence12">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruenceonapartialalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
relation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/presentationofagroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
to specific roots and indices. A pair of tables concern
<a class="nnexus_concept" href="http://planetmath.org/equation">
solutions
</a>
of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Diophantine equations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DiophantineEquation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/diophantineequation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
: table of
<a class="nnexus_concept" href="http://planetmath.org/integercontraharmonicmeans">
integer contraharmonic means
</a>
and table of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
<a class="nnexus_concept" href="http://planetmath.org/harmonicmean">
harmonic means
</a>
.
</p>
</div>
<div class="ltx_para" id="S4.p5">
<p class="ltx_p">
In the study of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GlobalField.html">
global fields
</a>
, a
<a class="nnexus_concept" href="http://planetmath.org/tableofsomefundamentalunits">
table of some fundamental units
</a>
and a table of
<a class="nnexus_concept" href="http://planetmath.org/classnumbersofimaginaryquadraticfields">
class numbers of imaginary quadratic fields
</a>
is a useful aid to the student.
</p>
</div>
<div class="ltx_para" id="S4.p6">
<p class="ltx_p">
Moving on to the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complex plane
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexPlane.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complex">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, just a few
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complex multiplication
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ComplexMultiplication.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/complexmultiplication">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
tables would be
<a class="nnexus_concept" href="http://planetmath.org/necessaryandsufficient">
sufficient
</a>
, as it would be easier for the student to learn the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
formula
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Formula.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/firstorderlanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
for complex multiplication rather than to try to memorize some multidimensional table which would be too large even if limited to a small range.
</p>
</div>
</section>
<section class="ltx_section" id="S5">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
5
</span>
Geometry
</h2>
<div class="ltx_para" id="S5.p1">
<p class="ltx_p">
To illustrate the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Pythagorean theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PythagoreanTheorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/pythagoreantheorem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generalizedpythagoreantheorem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, older books might have an appendix of the first
<a class="nnexus_concept" href="http://planetmath.org/pythagoreantriplet">
primitive Pythagorean triplets
</a>
. Tables of values of the sine, cosine and
<a class="nnexus_concept" href="http://dlmf.nist.gov/4.14#E4">
tangent functions
</a>
would not be out of place even in a basic
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
geometry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Geometry.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/geometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
book.
</p>
</div>
</section>
<section class="ltx_section" id="S6">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
6
</span>
Reference tables
</h2>
<div class="ltx_para" id="S6.p1">
<p class="ltx_p">
For reference purposes, there are tables where one can look up the specific value of a
<a class="nnexus_concept" href="http://planetmath.org/realnumber">
real number
</a>
or a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
sequence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Sequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of numbers. For example, an
<a class="nnexus_concept" href="http://planetmath.org/indexofimportantirrationalconstants">
index of important irrational constants
</a>
such as Borwein’s dictionary of real numbers. For sequences of integers, there are the books by Neil Sloane, the
<span class="ltx_text ltx_font_italic">
Handbook of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/IntegerSequence.html">
Integer Sequences
</a>
</span>
and the
<span class="ltx_text ltx_font_italic">
Encyclopedia of Integer Sequences
</span>
, both
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Predecessor.html">
predecessors
</a>
to the
<a class="nnexus_concept" href="http://planetmath.org/onlineencyclopediaofintegersequences">
Online Encyclopedia of Integer Sequences
</a>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/indexoftables">
index of tables
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
IndexOfTables
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:07:31
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:07:31
</td>
</tr>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
</tr>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
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<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
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Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
8
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
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<tr class="ltx_tr">
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Entry type
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<td class="ltx_td ltx_align_left ltx_border_r">
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<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A20
</td>
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</tbody>
</table>
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0 | MathematicsVocabulary | http://planetmath.org/MathematicsVocabulary | <!DOCTYPE html>
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<title>
mathematics vocabulary
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
mathematics vocabulary
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_subsubsection" id="S0.SS0.SSSx1">
<h2 class="ltx_title ltx_title_subsubsection">
A-B
</h2>
<div class="ltx_para" id="S0.SS0.SSSx1.p1">
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
a posteriori
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
a priori
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
abstract
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
abuse
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
acute
</p>
</div>
</li>
<li class="ltx_item" id="I1.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i6.p1">
<p class="ltx_p">
ad hoc
</p>
</div>
</li>
<li class="ltx_item" id="I1.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i7.p1">
<p class="ltx_p">
add,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
addition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/addition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i8.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
adjacent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/adjacent">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/adjacent1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i9.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
adjoint
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/adjointfunctor">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dualhomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/adjointendomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i10.p1">
<p class="ltx_p">
affine
</p>
</div>
</li>
<li class="ltx_item" id="I1.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i11.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/alphabet">
alphabet
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i12.p1">
<p class="ltx_p">
analogous,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
analogy
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Analogy.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/similarityandanalogoussystemsdynamicadjointnessandtopologicalequivalence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i13.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
analysis
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Analysis.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicentryonanalysis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
analytic
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/logicism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/analytic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i14.p1">
<p class="ltx_p">
angle
</p>
</div>
</li>
<li class="ltx_item" id="I1.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i15.p1">
<p class="ltx_p">
ansatz
</p>
</div>
</li>
<li class="ltx_item" id="I1.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i16.p1">
<p class="ltx_p">
apex
</p>
</div>
</li>
<li class="ltx_item" id="I1.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i17.p1">
<p class="ltx_p">
applied
</p>
</div>
</li>
<li class="ltx_item" id="I1.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i18.p1">
<p class="ltx_p">
approximate
</p>
</div>
</li>
<li class="ltx_item" id="I1.i19" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i19.p1">
<p class="ltx_p">
arc
</p>
</div>
</li>
<li class="ltx_item" id="I1.i20" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i20.p1">
<p class="ltx_p">
area
</p>
</div>
</li>
<li class="ltx_item" id="I1.i21" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i21.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
argument
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Argument.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/argument">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i22" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i22.p1">
<p class="ltx_p">
assume,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
assumption
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/deduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivationsinnaturaldeduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i23" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i23.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/asymptote">
asymptote
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i24" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i24.p1">
<p class="ltx_p">
autonomous
</p>
</div>
</li>
<li class="ltx_item" id="I1.i25" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i25.p1">
<p class="ltx_p">
axiom, axiomatic
</p>
</div>
</li>
<li class="ltx_item" id="I1.i26" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i26.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Azimuth.html">
azimuth
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i27" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i27.p1">
<p class="ltx_p">
basis
</p>
</div>
</li>
<li class="ltx_item" id="I1.i28" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i28.p1">
<p class="ltx_p">
boundary
</p>
</div>
</li>
<li class="ltx_item" id="I1.i29" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i29.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/upperbound">
bounded
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i30" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i30.p1">
<p class="ltx_p">
branch
</p>
</div>
</li>
<li class="ltx_item" id="I1.i31" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i31.p1">
<p class="ltx_p">
bundle (as in tangent bundle)
</p>
</div>
</li>
</ul>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx2">
<h2 class="ltx_title ltx_title_subsubsection">
C
</h2>
<div class="ltx_para" id="S0.SS0.SSSx2.p1">
<ul class="ltx_itemize" id="I2">
<li class="ltx_item" id="I2.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i1.p1">
<p class="ltx_p">
calculate
</p>
</div>
</li>
<li class="ltx_item" id="I2.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i2.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Calculus.html">
calculus
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
canonical
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i4.p1">
<p class="ltx_p">
Cartesian
</p>
</div>
</li>
<li class="ltx_item" id="I2.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i5.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
category
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Category.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/category">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i6.p1">
<p class="ltx_p">
cell
</p>
</div>
</li>
<li class="ltx_item" id="I2.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i7.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
character
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/characterofafinitegroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/character">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i8.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
characterization
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Characterization.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/characterisation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i9.p1">
<p class="ltx_p">
chord
</p>
</div>
</li>
<li class="ltx_item" id="I2.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i10.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/circle">
circular
</a>
(as in
<a class="nnexus_concept" href="http://planetmath.org/circularreasoning">
circular argument
</a>
and
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CircularReasoning
</span>
circular reasoning)
</p>
</div>
</li>
<li class="ltx_item" id="I2.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i11.p1">
<p class="ltx_p">
circulation
</p>
</div>
</li>
<li class="ltx_item" id="I2.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i12.p1">
<p class="ltx_p">
class
</p>
</div>
</li>
<li class="ltx_item" id="I2.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i13.p1">
<p class="ltx_p">
classic
</p>
</div>
</li>
<li class="ltx_item" id="I2.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i14.p1">
<p class="ltx_p">
clear, clearly
</p>
</div>
</li>
<li class="ltx_item" id="I2.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i15.p1">
<p class="ltx_p">
closed,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Closure.html">
closure
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i16.p1">
<p class="ltx_p">
collapse
</p>
</div>
</li>
<li class="ltx_item" id="I2.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i17.p1">
<p class="ltx_p">
compact
</p>
</div>
</li>
<li class="ltx_item" id="I2.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i18.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/compassandstraightedgeconstruction">
compass
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i19" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i19.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
complete
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/ordersinanumberfield">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/soundcomplete">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/supplementalaxiomsforanabeliancategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/kripkesemantics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/maximallyconsistent">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i20" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i20.p1">
<p class="ltx_p">
complex
</p>
</div>
</li>
<li class="ltx_item" id="I2.i21" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i21.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
concave
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/dihedralangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/convexfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, concavity
</p>
</div>
</li>
<li class="ltx_item" id="I2.i22" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i22.p1">
<p class="ltx_p">
concrete
</p>
</div>
</li>
<li class="ltx_item" id="I2.i23" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i23.p1">
<p class="ltx_p">
condition
</p>
</div>
</li>
<li class="ltx_item" id="I2.i24" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i24.p1">
<p class="ltx_p">
cone
</p>
</div>
</li>
<li class="ltx_item" id="I2.i25" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i25.p1">
<p class="ltx_p">
conformal
</p>
</div>
</li>
<li class="ltx_item" id="I2.i26" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i26.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
congruence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Congruence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruencerelationonanalgebraicsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence12">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruenceonapartialalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/congruence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
congruent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Congruent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/geometriccongruence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i27" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i27.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/openquestion">
conjecture
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i28" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i28.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
conjugate
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/algebraicconjugates">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/similarmatrix">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/conjugacyclass">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
conjugation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Conjugation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/innerautomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i29" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i29.p1">
<p class="ltx_p">
connected
</p>
</div>
</li>
<li class="ltx_item" id="I2.i30" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i30.p1">
<p class="ltx_p">
constrain, constraint
</p>
</div>
</li>
<li class="ltx_item" id="I2.i31" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i31.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/concretecategory">
construct
</a>
, constructible,
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CompassAndStraightedgeConstruction
</span>
construction, constructive
</p>
</div>
</li>
<li class="ltx_item" id="I2.i32" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i32.p1">
<p class="ltx_p">
contact
</p>
</div>
</li>
<li class="ltx_item" id="I2.i33" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i33.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
continuous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Continuous.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/classesofordinalsandenumeratingfunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i34" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i34.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
continuum
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Continuum.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuumhypothesis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i35" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i35.p1">
<p class="ltx_p">
contract,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
contraction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/contraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sequent">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i36" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i36.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Contractible.html">
contractible
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i37" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i37.p1">
<p class="ltx_p">
convex, convexity
</p>
</div>
</li>
<li class="ltx_item" id="I2.i38" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i38.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
coordinate
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Coordinates.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/coordinatevector">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/frame">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i39" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i39.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/lemma">
corollary
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i40" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i40.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
countable
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Countable.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/countable">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i41" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i41.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
counterexample
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Counterexample.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/counterexample">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i42" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i42.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
contradiction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Contradiction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradiction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradictorystatement">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i43" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i43.p1">
<p class="ltx_p">
convergence,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
convergent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Convergent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/convergentstoacontinuedfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/convergentseries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i44" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i44.p1">
<p class="ltx_p">
cover,
<a class="nnexus_concept" href="http://planetmath.org/site">
covering
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I2.i45" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i45.p1">
<p class="ltx_p">
criteria, criterion
</p>
</div>
</li>
<li class="ltx_item" id="I2.i46" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i46.p1">
<p class="ltx_p">
cross (as in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
cross product
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CrossProduct.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/crossproduct">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and cross section)
</p>
</div>
</li>
<li class="ltx_item" id="I2.i47" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i47.p1">
<p class="ltx_p">
crystalline
</p>
</div>
</li>
<li class="ltx_item" id="I2.i48" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i48.p1">
<p class="ltx_p">
curve
</p>
</div>
</li>
<li class="ltx_item" id="I2.i49" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i49.p1">
<p class="ltx_p">
curvilinear
</p>
</div>
</li>
<li class="ltx_item" id="I2.i50" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I2.i50.p1">
<p class="ltx_p">
cut
</p>
</div>
</li>
</ul>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx3">
<h2 class="ltx_title ltx_title_subsubsection">
D-F
</h2>
<div class="ltx_para" id="S0.SS0.SSSx3.p1">
<ul class="ltx_itemize" id="I3">
<li class="ltx_item" id="I3.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i1.p1">
<p class="ltx_p">
deduce,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Deduction.html">
deduction
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i2.p1">
<p class="ltx_p">
dependence, dependent
</p>
</div>
</li>
<li class="ltx_item" id="I3.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
, define
</p>
</div>
</li>
<li class="ltx_item" id="I3.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i4.p1">
<p class="ltx_p">
deformable, deformation
</p>
</div>
</li>
<li class="ltx_item" id="I3.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i5.p1">
<p class="ltx_p">
degree
</p>
</div>
</li>
<li class="ltx_item" id="I3.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i6.p1">
<p class="ltx_p">
dense
</p>
</div>
</li>
<li class="ltx_item" id="I3.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i7.p1">
<p class="ltx_p">
derivation
</p>
</div>
</li>
<li class="ltx_item" id="I3.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i8.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
derivative
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/fixedpointsofnormalfunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivative">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i9.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
difference
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/difference">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/setdifference">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i10.p1">
<p class="ltx_p">
different
</p>
</div>
</li>
<li class="ltx_item" id="I3.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i11.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/higherorderderivatives">
differentiate
</a>
,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Differentiation.html">
differentiation
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i12.p1">
<p class="ltx_p">
dilate,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
dilation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Dilation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/affinetransformation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i13.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
dilemma
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Dilemma.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/paradox">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i14.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
divergence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Divergence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divergence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i15.p1">
<p class="ltx_p">
divide,
<a class="nnexus_concept" href="http://planetmath.org/division">
division
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i16.p1">
<p class="ltx_p">
divides,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Divisible.html">
divisible
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
divisor
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Divisor.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divisibility">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/divisortheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i17.p1">
<p class="ltx_p">
domain
</p>
</div>
</li>
<li class="ltx_item" id="I3.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i18.p1">
<p class="ltx_p">
dual
</p>
</div>
</li>
<li class="ltx_item" id="I3.i19" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i19.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
embedding
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/structurehomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homomorphismbetweenpartialalgebras">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/injectivefunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i20" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i20.p1">
<p class="ltx_p">
empty
</p>
</div>
</li>
<li class="ltx_item" id="I3.i21" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i21.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/envelope">
envelope
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i22" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i22.p1">
<p class="ltx_p">
equal
</p>
</div>
</li>
<li class="ltx_item" id="I3.i23" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i23.p1">
<p class="ltx_p">
evaluate, evaluation
</p>
</div>
</li>
<li class="ltx_item" id="I3.i24" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i24.p1">
<p class="ltx_p">
even
</p>
</div>
</li>
<li class="ltx_item" id="I3.i25" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i25.p1">
<p class="ltx_p">
evident
</p>
</div>
</li>
<li class="ltx_item" id="I3.i26" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i26.p1">
<p class="ltx_p">
evolute
</p>
</div>
</li>
<li class="ltx_item" id="I3.i27" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i27.p1">
<p class="ltx_p">
exact (as in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
exact sequence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ExactSequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/exactsequencetheoreminc3category">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/exactsequence1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/regularcategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
)
</p>
</div>
</li>
<li class="ltx_item" id="I3.i28" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i28.p1">
<p class="ltx_p">
exhaust, exhaustion
</p>
</div>
</li>
<li class="ltx_item" id="I3.i29" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i29.p1">
<p class="ltx_p">
existential
</p>
</div>
</li>
<li class="ltx_item" id="I3.i30" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i30.p1">
<p class="ltx_p">
expected
</p>
</div>
</li>
<li class="ltx_item" id="I3.i31" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i31.p1">
<p class="ltx_p">
explicit
</p>
</div>
</li>
<li class="ltx_item" id="I3.i32" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i32.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/expression">
expression
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i33" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i33.p1">
<p class="ltx_p">
extend,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
extension
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/substructure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extension">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/extensionofafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i34" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i34.p1">
<p class="ltx_p">
false
</p>
</div>
</li>
<li class="ltx_item" id="I3.i35" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i35.p1">
<p class="ltx_p">
field
</p>
</div>
</li>
<li class="ltx_item" id="I3.i36" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i36.p1">
<p class="ltx_p">
fix, fixed
</p>
</div>
</li>
<li class="ltx_item" id="I3.i37" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i37.p1">
<p class="ltx_p">
focus
</p>
</div>
</li>
<li class="ltx_item" id="I3.i38" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i38.p1">
<p class="ltx_p">
form
</p>
</div>
</li>
<li class="ltx_item" id="I3.i39" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i39.p1">
<p class="ltx_p">
formal
</p>
</div>
</li>
<li class="ltx_item" id="I3.i40" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i40.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
formula
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Formula.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logicallanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/firstorderlanguage">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, formulae
</p>
</div>
</li>
<li class="ltx_item" id="I3.i41" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i41.p1">
<p class="ltx_p">
fractal
</p>
</div>
</li>
<li class="ltx_item" id="I3.i42" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i42.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i43" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I3.i43.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functional
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/interpretationofintuitionisticlogicbymeansoffunctionals">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/intuitionisticlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functional">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
</ul>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx4">
<h2 class="ltx_title ltx_title_subsubsection">
G-L
</h2>
<div class="ltx_para" id="S0.SS0.SSSx4.p1">
<ul class="ltx_itemize" id="I4">
<li class="ltx_item" id="I4.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i1.p1">
<p class="ltx_p">
general,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
generalization
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/hilbertsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiomsystemforfirstorderlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i2.p1">
<p class="ltx_p">
generate, generating,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
generator
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/submodule">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/grothendieckcategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generatorofacategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/presentationofagroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/generator">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i3.p1">
<p class="ltx_p">
genus
</p>
</div>
</li>
<li class="ltx_item" id="I4.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i4.p1">
<p class="ltx_p">
graph
</p>
</div>
</li>
<li class="ltx_item" id="I4.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i5.p1">
<p class="ltx_p">
group
</p>
</div>
</li>
<li class="ltx_item" id="I4.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i6.p1">
<p class="ltx_p">
handwaving
</p>
</div>
</li>
<li class="ltx_item" id="I4.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i7.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/laplaceequation">
harmonic
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i8.p1">
<p class="ltx_p">
height
</p>
</div>
</li>
<li class="ltx_item" id="I4.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i9.p1">
<p class="ltx_p">
heuristics
</p>
</div>
</li>
<li class="ltx_item" id="I4.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i10.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
homogeneous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/arrowsrelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homogeneous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homogeneouslinearproblem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i11.p1">
<p class="ltx_p">
hyper-
</p>
</div>
</li>
<li class="ltx_item" id="I4.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i12.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
hypothesis
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Hypothesis.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/hypothesis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i13.p1">
<p class="ltx_p">
ideal
</p>
</div>
</li>
<li class="ltx_item" id="I4.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i14.p1">
<p class="ltx_p">
if and only if, iff
</p>
</div>
</li>
<li class="ltx_item" id="I4.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i15.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/imaginaries">
imaginary
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i16.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
implication
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Implication.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/implication">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, implies, imply
</p>
</div>
</li>
<li class="ltx_item" id="I4.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i17.p1">
<p class="ltx_p">
implicit
</p>
</div>
</li>
<li class="ltx_item" id="I4.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i18.p1">
<p class="ltx_p">
identification, identify
</p>
</div>
</li>
<li class="ltx_item" id="I4.i19" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i19.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
identity
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/identityinaclass">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/multivaluedfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/hypergroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/group">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i20" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i20.p1">
<p class="ltx_p">
independence,
<a class="nnexus_concept" href="http://planetmath.org/projectivebasis">
independent
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i21" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i21.p1">
<p class="ltx_p">
index
</p>
</div>
</li>
<li class="ltx_item" id="I4.i22" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i22.p1">
<p class="ltx_p">
induce
</p>
</div>
</li>
<li class="ltx_item" id="I4.i23" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i23.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
induction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Induction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/induction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i24" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i24.p1">
<p class="ltx_p">
inert, inertia
</p>
</div>
</li>
<li class="ltx_item" id="I4.i25" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i25.p1">
<p class="ltx_p">
inferior (as in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
limit inferior
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LimitInferior.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/limitinferior">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
)
</p>
</div>
</li>
<li class="ltx_item" id="I4.i26" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i26.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinite
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinite.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/infinite">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinity
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinity.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiomofinfinity">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i27" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i27.p1">
<p class="ltx_p">
inject, injection
</p>
</div>
</li>
<li class="ltx_item" id="I4.i28" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i28.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i29" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i29.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/lebesgueintegral">
integral
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i30" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i30.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/integralsign">
integrate
</a>
, integration
</p>
</div>
</li>
<li class="ltx_item" id="I4.i31" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i31.p1">
<p class="ltx_p">
interior
</p>
</div>
</li>
<li class="ltx_item" id="I4.i32" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i32.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/intersection">
intersect
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
intersection
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Intersection.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/partialorderingonsubobjectsofanobject">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i33" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i33.p1">
<p class="ltx_p">
invariance,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
invariant
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Invariant.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/invariant">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i34" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i34.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
inverse
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Inverse.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inversenumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/regularsemigroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inversestatement">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/inversefunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/matrixinverse">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i35" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i35.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/invertiblelineartransformation">
invertible
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i36" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i36.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/irrational">
irrational
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i37" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i37.p1">
<p class="ltx_p">
join
</p>
</div>
</li>
<li class="ltx_item" id="I4.i38" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i38.p1">
<p class="ltx_p">
kernel
</p>
</div>
</li>
<li class="ltx_item" id="I4.i39" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i39.p1">
<p class="ltx_p">
kurtosis
</p>
</div>
</li>
<li class="ltx_item" id="I4.i40" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i40.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
language
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/language">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/signature">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i41" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i41.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
lattice
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Lattice.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/latticeinmathbbrn">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lattice">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I4.i42" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i42.p1">
<p class="ltx_p">
leg
</p>
</div>
</li>
<li class="ltx_item" id="I4.i43" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i43.p1">
<p class="ltx_p">
lemma
</p>
</div>
</li>
<li class="ltx_item" id="I4.i44" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i44.p1">
<p class="ltx_p">
length
</p>
</div>
</li>
<li class="ltx_item" id="I4.i45" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i45.p1">
<p class="ltx_p">
lift, lifting
</p>
</div>
</li>
<li class="ltx_item" id="I4.i46" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i46.p1">
<p class="ltx_p">
limit
</p>
</div>
</li>
<li class="ltx_item" id="I4.i47" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i47.p1">
<p class="ltx_p">
line
</p>
</div>
</li>
<li class="ltx_item" id="I4.i48" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i48.p1">
<p class="ltx_p">
linear
</p>
</div>
</li>
<li class="ltx_item" id="I4.i49" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I4.i49.p1">
<p class="ltx_p">
logic
</p>
</div>
</li>
</ul>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx5">
<h2 class="ltx_title ltx_title_subsubsection">
M-Q
</h2>
<div class="ltx_para" id="S0.SS0.SSSx5.p1">
<ul class="ltx_itemize" id="I5">
<li class="ltx_item" id="I5.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i1.p1">
<p class="ltx_p">
manifold
</p>
</div>
</li>
<li class="ltx_item" id="I5.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i2.p1">
<p class="ltx_p">
map,
<a class="nnexus_concept" href="http://planetmath.org/mapping">
mapping
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i3.p1">
<p class="ltx_p">
matrix
</p>
</div>
</li>
<li class="ltx_item" id="I5.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i4.p1">
<p class="ltx_p">
meager
</p>
</div>
</li>
<li class="ltx_item" id="I5.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i5.p1">
<p class="ltx_p">
mean
</p>
</div>
</li>
<li class="ltx_item" id="I5.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i6.p1">
<p class="ltx_p">
measure,
<a class="nnexus_concept" href="http://planetmath.org/riemannmultipleintegral">
measurable
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i7.p1">
<p class="ltx_p">
median
</p>
</div>
</li>
<li class="ltx_item" id="I5.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i8.p1">
<p class="ltx_p">
model
</p>
</div>
</li>
<li class="ltx_item" id="I5.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i9.p1">
<p class="ltx_p">
module
</p>
</div>
</li>
<li class="ltx_item" id="I5.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i10.p1">
<p class="ltx_p">
mutatis mutandis
</p>
</div>
</li>
<li class="ltx_item" id="I5.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i11.p1">
<p class="ltx_p">
necessary
</p>
</div>
</li>
<li class="ltx_item" id="I5.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i12.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/positive">
negative
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i13.p1">
<p class="ltx_p">
nerve
</p>
</div>
</li>
<li class="ltx_item" id="I5.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i14.p1">
<p class="ltx_p">
nested
</p>
</div>
</li>
<li class="ltx_item" id="I5.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i15.p1">
<p class="ltx_p">
normal
</p>
</div>
</li>
<li class="ltx_item" id="I5.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i16.p1">
<p class="ltx_p">
notation
</p>
</div>
</li>
<li class="ltx_item" id="I5.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i17.p1">
<p class="ltx_p">
null
</p>
</div>
</li>
<li class="ltx_item" id="I5.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i18.p1">
<p class="ltx_p">
number
</p>
</div>
</li>
<li class="ltx_item" id="I5.i19" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i19.p1">
<p class="ltx_p">
obtuse
</p>
</div>
</li>
<li class="ltx_item" id="I5.i20" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i20.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/obvious">
obvious
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i21" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i21.p1">
<p class="ltx_p">
odd
</p>
</div>
</li>
<li class="ltx_item" id="I5.i22" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i22.p1">
<p class="ltx_p">
open
</p>
</div>
</li>
<li class="ltx_item" id="I5.i23" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i23.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i24" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i24.p1">
<p class="ltx_p">
order
</p>
</div>
</li>
<li class="ltx_item" id="I5.i25" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i25.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
orthogonal
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Orthogonal.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/orthomodularlattice">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/perpendicularityineuclideanplane">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/orthogonalmorphisms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/orthogonal">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i26" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i26.p1">
<p class="ltx_p">
oscillating
</p>
</div>
</li>
<li class="ltx_item" id="I5.i27" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i27.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Paradox.html">
Paradox
</a>
</span>
paradox
</p>
</div>
</li>
<li class="ltx_item" id="I5.i28" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i28.p1">
<p class="ltx_p">
partial
</p>
</div>
</li>
<li class="ltx_item" id="I5.i29" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i29.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
partition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/partition1">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integerpartition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/partition">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i30" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i30.p1">
<p class="ltx_p">
path
</p>
</div>
</li>
<li class="ltx_item" id="I5.i31" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i31.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
pathological
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Pathological.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/pathological">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i32" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i32.p1">
<p class="ltx_p">
PDE
</p>
</div>
</li>
<li class="ltx_item" id="I5.i33" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i33.p1">
<p class="ltx_p">
perimeter
</p>
</div>
</li>
<li class="ltx_item" id="I5.i34" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i34.p1">
<p class="ltx_p">
period,
<a class="nnexus_concept" href="http://planetmath.org/periodicgroup">
periodic
</a>
, periodicity
</p>
</div>
</li>
<li class="ltx_item" id="I5.i35" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i35.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/piecewise">
piecewise
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i36" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i36.p1">
<p class="ltx_p">
plane
</p>
</div>
</li>
<li class="ltx_item" id="I5.i37" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i37.p1">
<p class="ltx_p">
point
</p>
</div>
</li>
<li class="ltx_item" id="I5.i38" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i38.p1">
<p class="ltx_p">
polar
</p>
</div>
</li>
<li class="ltx_item" id="I5.i39" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i39.p1">
<p class="ltx_p">
positive
</p>
</div>
</li>
<li class="ltx_item" id="I5.i40" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i40.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
postulate
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Postulate.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiom">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i41" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i41.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/lamellarfield">
potential
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i42" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i42.p1">
<p class="ltx_p">
power
</p>
</div>
</li>
<li class="ltx_item" id="I5.i43" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i43.p1">
<p class="ltx_p">
prime
</p>
</div>
</li>
<li class="ltx_item" id="I5.i44" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i44.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
primitive
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/completelysimplesemigroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/primitive">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/antiderivative">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i45" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i45.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
product
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Product.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/product">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categoricaldirectproduct">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i46" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i46.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/sourcesandsinksofvectorfield">
productivity
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i47" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i47.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/projectionofpoint">
project
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
projection
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/directproductofalgebras">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/projection">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i48" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i48.p1">
<p class="ltx_p">
proper
</p>
</div>
</li>
<li class="ltx_item" id="I5.i49" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i49.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
proposition
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/booleanvaluedfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/propositionallogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i50" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i50.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
pyramid
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Pyramid.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/coneinmathbbr3">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i51" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i51.p1">
<p class="ltx_p">
QED, Q.E.D.
</p>
</div>
</li>
<li class="ltx_item" id="I5.i52" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i52.p1">
<p class="ltx_p">
QEF, Q.E.F.
</p>
</div>
</li>
<li class="ltx_item" id="I5.i53" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I5.i53.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/quotientgroup">
quotient
</a>
</p>
</div>
</li>
</ul>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx6">
<h2 class="ltx_title ltx_title_subsubsection">
R-S
</h2>
<div class="ltx_para" id="S0.SS0.SSSx6.p1">
<ul class="ltx_itemize" id="I6">
<li class="ltx_item" id="I6.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i1.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
radical
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/radical12">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/radicalofaninteger">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i2.p1">
<p class="ltx_p">
radius, radii
</p>
</div>
</li>
<li class="ltx_item" id="I6.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i3.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/ramificationindex">
ramification
</a>
, ramified, ramify
</p>
</div>
</li>
<li class="ltx_item" id="I6.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i4.p1">
<p class="ltx_p">
range
</p>
</div>
</li>
<li class="ltx_item" id="I6.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i5.p1">
<p class="ltx_p">
rank
</p>
</div>
</li>
<li class="ltx_item" id="I6.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i6.p1">
<p class="ltx_p">
ratio
</p>
</div>
</li>
<li class="ltx_item" id="I6.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i7.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/rationalnumber">
rational
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i8.p1">
<p class="ltx_p">
razor (as in Occam’s razor)
</p>
</div>
</li>
<li class="ltx_item" id="I6.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i9.p1">
<p class="ltx_p">
real
</p>
</div>
</li>
<li class="ltx_item" id="I6.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i10.p1">
<p class="ltx_p">
reduce,
<a class="nnexus_concept" href="http://planetmath.org/reducedautomaton">
reduced
</a>
,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
reduction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/diamondlemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reducedword">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i11.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
reflexive
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Reflexive.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reflexivenondegeneratesesquilinear">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/polyadicalgebrawithequality">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/dualspace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reflexiverelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i12.p1">
<p class="ltx_p">
region
</p>
</div>
</li>
<li class="ltx_item" id="I6.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i13.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
regular
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/regularpolygon">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/regularpolyhedron">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/cofinality">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i14.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
relation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relationonobjects">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i15.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/longdivision">
remainder
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i16.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/representablefunctor">
represent
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/grouprepresentation">
representation
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i17.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/resolventmatrix">
resolvent
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i18.p1">
<p class="ltx_p">
rest
</p>
</div>
</li>
<li class="ltx_item" id="I6.i19" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i19.p1">
<p class="ltx_p">
restrict,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
restriction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/subalgebraofanalgebraicsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/subfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/restrictionofafunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/constructingnearlinearspacesfromexistingones">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i20" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i20.p1">
<p class="ltx_p">
rig
</p>
</div>
</li>
<li class="ltx_item" id="I6.i21" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i21.p1">
<p class="ltx_p">
rigid
</p>
</div>
</li>
<li class="ltx_item" id="I6.i22" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i22.p1">
<p class="ltx_p">
ring
</p>
</div>
</li>
<li class="ltx_item" id="I6.i23" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i23.p1">
<p class="ltx_p">
root
</p>
</div>
</li>
<li class="ltx_item" id="I6.i24" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i24.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
rotation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Rotation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/euclideantransformation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i25" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i25.p1">
<p class="ltx_p">
rotor
</p>
</div>
</li>
<li class="ltx_item" id="I6.i26" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i26.p1">
<p class="ltx_p">
sector
</p>
</div>
</li>
<li class="ltx_item" id="I6.i27" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i27.p1">
<p class="ltx_p">
self-
</p>
</div>
</li>
<li class="ltx_item" id="I6.i28" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i28.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/interval">
segment
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i29" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i29.p1">
<p class="ltx_p">
semi-
</p>
</div>
</li>
<li class="ltx_item" id="I6.i30" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i30.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
sequence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Sequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categoricalsequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/sequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i31" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i31.p1">
<p class="ltx_p">
series
</p>
</div>
</li>
<li class="ltx_item" id="I6.i32" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i32.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
similar
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Similar.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/similarityingeometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalentmachines">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i33" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i33.p1">
<p class="ltx_p">
simple
</p>
</div>
</li>
<li class="ltx_item" id="I6.i34" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i34.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/singleton">
singleton
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i35" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i35.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/singular">
singular
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i36" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i36.p1">
<p class="ltx_p">
sink
</p>
</div>
</li>
<li class="ltx_item" id="I6.i37" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i37.p1">
<p class="ltx_p">
slope
</p>
</div>
</li>
<li class="ltx_item" id="I6.i38" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i38.p1">
<p class="ltx_p">
smash (as in smash product)
</p>
</div>
</li>
<li class="ltx_item" id="I6.i39" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i39.p1">
<p class="ltx_p">
smooth
</p>
</div>
</li>
<li class="ltx_item" id="I6.i40" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i40.p1">
<p class="ltx_p">
solid
</p>
</div>
</li>
<li class="ltx_item" id="I6.i41" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i41.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/equation">
solution
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i42" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i42.p1">
<p class="ltx_p">
source
</p>
</div>
</li>
<li class="ltx_item" id="I6.i43" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i43.p1">
<p class="ltx_p">
space
</p>
</div>
</li>
<li class="ltx_item" id="I6.i44" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i44.p1">
<p class="ltx_p">
spectral (as in
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SpectralRadius.html">
spectral radius
</a>
and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
spectral sequence
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SpectralSequence.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/spectralsequence">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
)
</p>
</div>
</li>
<li class="ltx_item" id="I6.i45" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i45.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/spectralvaluesclassification">
spectrum
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i46" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i46.p1">
<p class="ltx_p">
sphere, spherical
</p>
</div>
</li>
<li class="ltx_item" id="I6.i47" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i47.p1">
<p class="ltx_p">
split
</p>
</div>
</li>
<li class="ltx_item" id="I6.i48" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i48.p1">
<p class="ltx_p">
square
</p>
</div>
</li>
<li class="ltx_item" id="I6.i49" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i49.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/squareroot">
square root
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i50" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i50.p1">
<p class="ltx_p">
star, star-shaped
</p>
</div>
</li>
<li class="ltx_item" id="I6.i51" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i51.p1">
<p class="ltx_p">
strength
</p>
</div>
</li>
<li class="ltx_item" id="I6.i52" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i52.p1">
<p class="ltx_p">
strict
</p>
</div>
</li>
<li class="ltx_item" id="I6.i53" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i53.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structure
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i54" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i54.p1">
<p class="ltx_p">
subset
</p>
</div>
</li>
<li class="ltx_item" id="I6.i55" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i55.p1">
<p class="ltx_p">
subtract,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
subtraction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/subtraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/additionandsubtractionformulasforsineandcosine">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i56" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i56.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/necessaryandsufficient">
sufficient
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i57" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i57.p1">
<p class="ltx_p">
sum,
<a class="nnexus_concept" href="http://planetmath.org/summation">
summation
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I6.i58" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i58.p1">
<p class="ltx_p">
superior (as in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
limit superior
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LimitSuperior.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/limitsuperior">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
)
</p>
</div>
</li>
<li class="ltx_item" id="I6.i59" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i59.p1">
<p class="ltx_p">
suppose, supposition
</p>
</div>
</li>
<li class="ltx_item" id="I6.i60" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i60.p1">
<p class="ltx_p">
surface
</p>
</div>
</li>
<li class="ltx_item" id="I6.i61" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i61.p1">
<p class="ltx_p">
suspension
</p>
</div>
</li>
<li class="ltx_item" id="I6.i62" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i62.p1">
<p class="ltx_p">
symbol
</p>
</div>
</li>
<li class="ltx_item" id="I6.i63" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I6.i63.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
symmetry
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Symmetry.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/symmetry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/symmetricrelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantumsymmetry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
</ul>
</div>
</section>
<section class="ltx_subsubsection" id="S0.SS0.SSSx7">
<h2 class="ltx_title ltx_title_subsubsection">
T-Z
</h2>
<div class="ltx_para" id="S0.SS0.SSSx7.p1">
<ul class="ltx_itemize" id="I7">
<li class="ltx_item" id="I7.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i1.p1">
<p class="ltx_p">
tame
</p>
</div>
</li>
<li class="ltx_item" id="I7.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i2.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
tangent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Tangent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/trigonometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/tangentline">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/definitionsintrigonometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i3.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
tautology
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Tautology.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/tautology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i4.p1">
<p class="ltx_p">
tensor, tensorial
</p>
</div>
</li>
<li class="ltx_item" id="I7.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i5.p1">
<p class="ltx_p">
term
</p>
</div>
</li>
<li class="ltx_item" id="I7.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i6.p1">
<p class="ltx_p">
TFAE, the following are
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
equivalent
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Equivalent.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/filterbasis">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalenceofforcingnotions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equivalencerelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i7.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
theorem
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i8.p1">
<p class="ltx_p">
theoretical
</p>
</div>
</li>
<li class="ltx_item" id="I7.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i9.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
topology
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i10.p1">
<p class="ltx_p">
total
</p>
</div>
</li>
<li class="ltx_item" id="I7.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i11.p1">
<p class="ltx_p">
toy (as in
<a class="nnexus_concept" href="http://planetmath.org/toytheorem">
toy theorem
</a>
)
</p>
</div>
</li>
<li class="ltx_item" id="I7.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i12.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/curve">
trajectory
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i13.p1">
<p class="ltx_p">
transcend, transcendence,
<a class="nnexus_concept" href="http://planetmath.org/algebraicfunction">
transcendental
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i14.p1">
<p class="ltx_p">
transient
</p>
</div>
</li>
<li class="ltx_item" id="I7.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i15.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Translate.html">
translate
</a>
,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Translation.html">
translation
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i16.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
triangle
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Triangle.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/triangle">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, triangular
</p>
</div>
</li>
<li class="ltx_item" id="I7.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i17.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Trivial.html">
trivial
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i18.p1">
<p class="ltx_p">
true
</p>
</div>
</li>
<li class="ltx_item" id="I7.i19" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i19.p1">
<p class="ltx_p">
type
</p>
</div>
</li>
<li class="ltx_item" id="I7.i20" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i20.p1">
<p class="ltx_p">
uniform, uniformly
</p>
</div>
</li>
<li class="ltx_item" id="I7.i21" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i21.p1">
<p class="ltx_p">
unique, uniqueness
</p>
</div>
</li>
<li class="ltx_item" id="I7.i22" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i22.p1">
<p class="ltx_p">
unit, unity
</p>
</div>
</li>
<li class="ltx_item" id="I7.i23" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i23.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
universal
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/universalmappingproperty">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universalstructure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universalrelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i24" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i24.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
universe
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/universe">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universeofdiscourse">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i25" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i25.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
vacuous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/vacuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/quantifier">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i26" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i26.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
valuation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/valuation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/truthvaluesemanticsforclassicalpropositionallogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i27" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i27.p1">
<p class="ltx_p">
value
</p>
</div>
</li>
<li class="ltx_item" id="I7.i28" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i28.p1">
<p class="ltx_p">
vanish
</p>
</div>
</li>
<li class="ltx_item" id="I7.i29" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i29.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
variable
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Variable.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/variable">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i30" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i30.p1">
<p class="ltx_p">
variance
</p>
</div>
</li>
<li class="ltx_item" id="I7.i31" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i31.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/variation">
variation
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i32" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i32.p1">
<p class="ltx_p">
variational (as in variational calculus)
</p>
</div>
</li>
<li class="ltx_item" id="I7.i33" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i33.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
variety
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Variety.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/equationalclass">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/varietyofgroups">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i34" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i34.p1">
<p class="ltx_p">
vector, vectorial
</p>
</div>
</li>
<li class="ltx_item" id="I7.i35" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i35.p1">
<p class="ltx_p">
volume
</p>
</div>
</li>
<li class="ltx_item" id="I7.i36" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i36.p1">
<p class="ltx_p">
wedge (as in
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
wedge product
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/WedgeProduct.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/exterioralgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
)
</p>
</div>
</li>
<li class="ltx_item" id="I7.i37" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i37.p1">
<p class="ltx_p">
well defined,
<a class="nnexus_concept" href="http://planetmath.org/welldefined">
well-defined
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I7.i38" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i38.p1">
<p class="ltx_p">
wild
</p>
</div>
</li>
<li class="ltx_item" id="I7.i39" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i39.p1">
<p class="ltx_p">
WLOG, WOLOG, without loss of generality
</p>
</div>
</li>
<li class="ltx_item" id="I7.i40" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i40.p1">
<p class="ltx_p">
word
</p>
</div>
</li>
<li class="ltx_item" id="I7.i41" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I7.i41.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/Root
</span>
zero, zeroes, zeros
</p>
</div>
</li>
</ul>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/mathematicsvocabulary">
mathematics vocabulary
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
MathematicsVocabulary
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:39:43
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:39:43
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
94
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
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msc 00A20
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Classification
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msc 00A99
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Related topic
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TermsFromForeignLanguagesUsedInMathematics
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TermsFromForeignLanguagesUsedInMathematicsPageImagesVersion
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0 | Obvious | http://planetmath.org/Obvious | <!DOCTYPE html>
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obvious
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obvious
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Mathematicians use phrases like “it is obvious that”, “it is easy to see that”, “it is clear that”, and “is
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Trivial.html">
trivial
</a>
” to indicate that some steps have been omitted. The use of such
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
language
</a>
<sup style="display: none;">
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</sup>
may be classified under three headings — honest, dishonest, and pedagogical.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
The honest use of these phrases occurs when only a few steps have been omitted and these steps are simple enough that the average reader can easily fill in the gaps. Omitting such steps can be beneficial because it cuts down the length of an exposition and keeps the main ideas from getting lost amidst a morass of boring details and routine
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operations
</a>
<sup style="display: none;">
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</sup>
. By reminding the reader of small omissions in an unobtrusive fashion, such phrases help put the reader at ease — if they are left out, it is easy for the reader to be thrown off-course by a missing step or be left with an uneasy feeling that there might be a hole in a proof.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
The dishonest use of these phrases occurs when a somewhat lengthy calculation has been left out because the author was too lazy to write it down. By using these phrases, the author hopes to intimidate potential critics from pointing out that material is missing by insinuating that anyone who would point out that something is missing is too stupid to fill in a few obvious steps. Frequent dishonest use of these terms may be a symptom of mathematheosis.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
The pedagogical use of these phrases occurs when the author has deliberately left the filling-in of missing steps as an exercise to the reader.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
obvious
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Obvious
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:43:42
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:43:42
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
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<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
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</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
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</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
easy to see
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
clear
</td>
</tr>
</tbody>
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</div>
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0 | OnLineEncyclopediaOfIntegerSequences | http://planetmath.org/OnLineEncyclopediaOfIntegerSequences | <!DOCTYPE html>
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On-Line Encyclopedia of Integer Sequences
</title>
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<h1 class="ltx_title ltx_title_document">
On-Line Encyclopedia of Integer Sequences
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
An Internet
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Database.html">
database
</a>
of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Sequence.html">
sequences
</a>
of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
, together with formulas,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GeneratingFunction.html">
generating functions
</a>
,
<a class="nnexus_concept" href="http://planetmath.org/supercomputers">
computer
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Program.html">
programs
</a>
, keywords and cross-references to other sequences. Sometimes referred to by its acronym,
<span class="ltx_text ltx_font_italic">
OEIS
</span>
. It was started by Neil Sloane as an outgrowth of his printed catalogs of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/IntegerSequence.html">
integer sequences
</a>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Each sequence is assigned a unique identification number (the letter A followed by six digits in base 10) based on the date of its
<a class="nnexus_concept" href="http://planetmath.org/addition">
addition
</a>
to the database, though most entries contain information on their lexicographic
<a class="nnexus_concept" href="http://planetmath.org/conceptlattice">
context
</a>
. In 2010 the database had almost 200000 sequences.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
The most important sequences are tagged with the keyword “core.” Sequences of
<a class="nnexus_concept" href="http://planetmath.org/fraction">
fractions
</a>
are split into two sequences (numerators and denominators), while important mathematical constants like e are split into their base 10 digits. Zero is sometimes used as a placeholder value for non-existent terms.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
A few other important keywords include: “frac” for most fractions, “cofr” for
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
continued fractions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ContinuedFraction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuedfraction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, “eigen” for eigenvalues, “mult” for multiplicative sequences, “walk” for sequences pertaining to self-avoiding paths, etc.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
The sequences included in the database range in subject from Sloane’s love of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Combinatorics.html">
combinatorics
</a>
to
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
number theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NumberTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/numbertheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, geometry, calculus,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Cryptography.html">
cryptography
</a>
, quantum physics, etc., as well as many subjects that would seem to be far removed from mathematics, like archaeology and psychiatry.
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
In 2010 there was an effort to move the whole database to a Wiki platform, but this ultimately abandoned in favor of a custom-designed platform with a more rigorous editorial control. However, the OEIS does have a Wiki for subsidiary pages.
</p>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://oeis.org/
</span>
The
<a class="nnexus_concept" href="http://planetmath.org/onlineencyclopediaofintegersequences">
On-Line Encyclopedia of Integer Sequences
</a>
</p>
</div>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
A. Del Arte, “
<span class="ltx_text ltx_font_typewriter">
http://www.southend.wayne.edu/modules/news/article.php?storyid=553
</span>
Mathematician reaches 100k milestone of online integer archive”
<span class="ltx_text ltx_font_italic">
The South End
</span>
(Detroit), 11 Nov., Wayne State University (2004)
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
V. Origlio, “L’Enciclopedia delle sequenze intere”
<span class="ltx_text ltx_font_italic">
Biblioteche Oggi
</span>
, Jan.-Feb. (2006): 41 - 45
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p9">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
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Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
On-Line Encyclopedia of Integer Sequences
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
OnLineEncyclopediaOfIntegerSequences
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:47:36
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 15:47:36
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompositeFan (12809)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompositeFan (12809)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
11
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CompositeFan (12809)
</td>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
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</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
OEIS
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
Online Encyclopedia of Integer Sequences
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | OpenQuestion | http://planetmath.org/OpenQuestion | <!DOCTYPE html>
<html>
<head>
<title>
open question
</title>
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<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
open question
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The adjective
<em class="ltx_emph ltx_font_italic">
open
</em>
is used by mathematicians to a statement which has neither been proven to be true or to be false. (In the light of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
incompleteness
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Incompleteness.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/beyondformalismgodelsincompleteness">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
thorems, we should perhaps also add “not been proven to be
<a class="nnexus_concept" href="http://planetmath.org/projectivebasis">
independent
</a>
of the axioms”)
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Examples:
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
“It is an
<a class="nnexus_concept" href="http://planetmath.org/openquestion">
open question
</a>
whether there are an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinite
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinite.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/infinite">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
number of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
prime numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimeNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/prime">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
all of whose digits are 1.” means that it neither has been proven that there exist an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinity
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinity.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiomofinfinity">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of such primes nor been proven that there are only a finite number of such primes.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
“It is an
<a class="nnexus_concept" href="http://planetmath.org/openproblems">
open problem
</a>
to determine the smallest number of lines which contain all points of the set
<math alttext="S" class="ltx_Math" display="inline" id="p5.m1">
<mi>
S
</mi>
</math>
.” means that the number in question has not yet been determined.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
The term
<em class="ltx_emph ltx_font_italic">
conjecture
</em>
refers to a statement which the speaker has reason to believe the statement is correct even though the speaker cannot prove the statement. (Also, one should be warned about a common abuse of this term. Even after a statement has been proven, sometimes it is still referred to as a conjecture by force of habit.)
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
The
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
difference
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/difference">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/setdifference">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
between the terms
<em class="ltx_emph ltx_font_italic">
conjecture
</em>
and
<em class="ltx_emph ltx_font_italic">
open question
</em>
is that the term
<em class="ltx_emph ltx_font_italic">
open
</em>
is neutral — saying that a statement is open does not connote that the speaker is voicing an opinion regarding the truth or falsity of the statement.
</p>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
The attachment below gives some examples of famous open problems in mathematics.
<span class="ltx_text ltx_font_typewriter">
http://garden.irmacs.sfu.ca/
</span>
Open Problem Garden is an external that contains more examples of open problems in mathematics.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
open question
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
OpenQuestion
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:42:41
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:42:41
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
15
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
rspuzio (6075)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
Classification
</a>
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
open problem
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
open
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
conjecture
</td>
</tr>
</tbody>
</table>
</div>
</article>
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0 | OverviewOfTheContentOfPlanetMath | http://planetmath.org/OverviewOfTheContentOfPlanetMath | <!DOCTYPE html>
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overview of the content of PlanetMath
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<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
overview of the content of PlanetMath
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/TopicEntryOnFoundationsOfMathematics
</span>
<span class="ltx_text" style="font-size:144%;">
<a class="nnexus_concept" href="http://planetmath.org/foundationsofmathematicsoverview">
Foundations of Mathematics
</a>
</span>
</p>
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
Index for the Foundations of Mathematics
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/BibliographyForAxiomaticsAndMathematicsFoundationsInCategories
</span>
Bibliography for The Foundations of Mathematics
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/TopicEntryOnAlgebra
</span>
<span class="ltx_text" style="font-size:144%;">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Algebra
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/algebraiccategoriesandclassesofalgebras">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/Geometry
</span>
<span class="ltx_text" style="font-size:144%;">
Geometry
</span>
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/TopicEntryOnTopology
</span>
<span class="ltx_text" style="font-size:144%;">
<a class="nnexus_concept" href="http://planetmath.org/topologicalspace">
Topology
</a>
</span>
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/AlgebraicTopology
</span>
<span class="ltx_text" style="font-size:144%;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicTopology.html">
Algebraic Topology
</a>
</span>
</p>
<ol class="ltx_enumerate" id="I2">
<li class="ltx_item" id="I2.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I2.i1.p1">
<p class="ltx_p">
Index for algebraic topology
</p>
</div>
</li>
<li class="ltx_item" id="I2.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I2.i2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CategoricalOntologyABibliographyOfCategoryTheory
</span>
Bibliography for algebraic topology
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CategoryTheory
</span>
<span class="ltx_text" style="font-size:144%;">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Category Theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CategoryTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categorytheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bibliographyinalgebraictopologycategoriesandqat">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/graphtheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicsinalgebraictopology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</p>
<ol class="ltx_enumerate" id="I3">
<li class="ltx_item" id="I3.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I3.i1.p1">
<p class="ltx_p">
Index for category theory
</p>
</div>
</li>
<li class="ltx_item" id="I3.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I3.i2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/IndexOfCategories
</span>
<a class="nnexus_concept" href="http://planetmath.org/indexofcategories">
Index of categories
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I3.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I3.i3.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/BibliographyForMathematicalPhysicsFoundationsAxiomaticsAndCategories
</span>
Bibliography for axiomatics and category theory
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/HigherDimensionalAlgebraHDA
</span>
<span class="ltx_text" style="font-size:144%;">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Higher Dimensional Algebra
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/ncategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/2groupoid">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</p>
<ol class="ltx_enumerate" id="I4">
<li class="ltx_item" id="I4.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I4.i1.p1">
<p class="ltx_p">
Index for higher dimensional algebra
</p>
</div>
</li>
<li class="ltx_item" id="I4.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I4.i2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/HigherDimensionalAlgebraHDA
</span>
Bibliography for higher dimensional algebra
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="p9">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/AlgebraicGeometry
</span>
<span class="ltx_text" style="font-size:144%;">
Algebraic Geometry
</span>
</p>
</div>
<div class="ltx_para" id="p10">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/NumberTheory
</span>
<span class="ltx_text" style="font-size:144%;">
Number Theory
</span>
</p>
</div>
<div class="ltx_para" id="p11">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/TopicEntryOnDiscreteMathematics
</span>
<span class="ltx_text" style="font-size:144%;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DiscreteMathematics.html">
Discrete Mathematics
</a>
</span>
</p>
</div>
<div class="ltx_para" id="p12">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/TopicEntryOnAnalysis
</span>
<span class="ltx_text" style="font-size:144%;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Analysis.html">
Analysis
</a>
</span>
</p>
</div>
<div class="ltx_para" id="p13">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/DifferentialEquation
</span>
<span class="ltx_text" style="font-size:144%;">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Differential Equations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/DifferentialEquation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/differentialequation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</p>
<ol class="ltx_enumerate" id="I5">
<li class="ltx_item" id="I5.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I5.i1.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/indexofdifferentialequations">
Index of differential equations
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I5.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I5.i2.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/bibliographyforordinarydifferentialequations">
Bibliography for ordinary differential equations
</a>
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="p14">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/TopicEntryOnAppliedMathematics
</span>
<span class="ltx_text" style="font-size:144%;">
Applied Mathematics
</span>
</p>
<ol class="ltx_enumerate" id="I6">
<li class="ltx_item" id="I6.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I6.i1.p1">
<p class="ltx_p">
Index for Applied Mathematics
</p>
</div>
</li>
<li class="ltx_item" id="I6.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I6.i2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/FoundationsOfQuantumFieldTheories
</span>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Mathematical Foundations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/bibliographyforaxiomaticsandmathematicsfoundationsincategories">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/analyticsandformallogicsinmetamathematics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of Quantum Physics
</p>
</div>
</li>
<li class="ltx_item" id="I6.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I6.i3.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/AxiomaticAndCategoricalFoundationsOfMathematicsII
</span>
Axiomatic and
<a class="nnexus_concept" href="http://planetmath.org/categoricalquantumlmlogicalgebras">
Categorical
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiomoffoundation">
Foundations
</a>
of Mathematical Physics
</p>
</div>
</li>
<li class="ltx_item" id="I6.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I6.i4.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/BibliographyForMathematicalPhysicsFoundationsAxiomaticsAndCategories
</span>
Bibliography for Mathematical Physics and Physical Mathematics
</p>
</div>
</li>
</ol>
</div>
<div class="ltx_para" id="p15">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/TopicEntryOnOrderTheory
</span>
<span class="ltx_text" style="font-size:144%;">
Order Theory and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Logic Algebra
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/topicentryonthealgebraicfoundationsofmathematics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logicism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</p>
</div>
<div class="ltx_para" id="p16">
<p class="ltx_p">
<span class="ltx_text" style="font-size:144%;">
Mathematics Education and Educators
</span>
</p>
</div>
<div class="ltx_para" id="p17">
<ol class="ltx_enumerate" id="I7">
<li class="ltx_item" id="I7.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I7.i1.p1">
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<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/elementaryrecursivefunction">
Elementary
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2.
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</span>
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<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/highschoolmathematics">
High school mathematics
</a>
</span>
</p>
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</li>
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http://planetmath.org/TopicEntryOnMiscellaneousMathematics
</span>
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Miscellaneous
</span>
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</td>
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<td class="ltx_td ltx_align_left ltx_border_l">
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</a>
name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
OverviewOfTheContentOfPlanetMath
</td>
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matte (1858)
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msc 00A20
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0 | Pathological | http://planetmath.org/Pathological | <!DOCTYPE html>
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pathological
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
pathological
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
In mathematics, a
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Pathological.html">
pathological
</a>
object
</em>
is mathematical
object that has a highly unexpected .
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Pathological objects are typically percieved to, in some sense, be
badly behaving. On the other hand, they are perfectly properly
defined mathematical objects. Therefore this “bad behaviour” can
simply be seen as a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
contradiction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Contradiction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradiction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradictorystatement">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
with our intuitive
picture of how a certain object should behave.
</p>
</div>
<section class="ltx_subsubsection" id="S0.SS0.SSSx1">
<h2 class="ltx_title ltx_title_subsubsection">
Examples
</h2>
<div class="ltx_para" id="S0.SS0.SSSx1.p1">
<ul class="ltx_itemize" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
A very famous pathological
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
is the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Weierstrass function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/WeierstrassFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/nowheredifferentiable">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, which is a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
continuous function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ContinuousFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/classesofordinalsandenumeratingfunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
that is nowhere
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
differentiable
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Differentiable.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/differentiablefunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/totaldifferential">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
The
<a class="nnexus_concept" href="http://planetmath.org/peanospace">
Peano space
</a>
filling curve. This pathological curve
maps the unit interval
<math alttext="[0,1]" class="ltx_Math" display="inline" id="I1.i2.p1.m1">
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
0
</mn>
<mo>
,
</mo>
<mn>
1
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
</math>
continuously onto
<math alttext="[0,1]\times[0,1]" class="ltx_Math" display="inline" id="I1.i2.p1.m2">
<mrow>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
0
</mn>
<mo>
,
</mo>
<mn>
1
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
<mo>
×
</mo>
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
0
</mn>
<mo>
,
</mo>
<mn>
1
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
</math>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
The Cantor set. This is subset of the
<a class="nnexus_concept" href="http://planetmath.org/interval">
interval
</a>
<math alttext="[0,1]" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mrow>
<mo stretchy="false">
[
</mo>
<mn>
0
</mn>
<mo>
,
</mo>
<mn>
1
</mn>
<mo stretchy="false">
]
</mo>
</mrow>
</math>
has the pathological property that it is
<a class="nnexus_concept" href="http://planetmath.org/uncountable">
uncountable
</a>
yet its measure is zero.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
The Dirichlet’s function from
<math alttext="\mathbbmss{R}" class="ltx_Math" display="inline" id="I1.i4.p1.m1">
<mi>
ℝ
</mi>
</math>
to
<math alttext="\mathbbmss{R}" class="ltx_Math" display="inline" id="I1.i4.p1.m2">
<mi>
ℝ
</mi>
</math>
is continuous at every
irrational point and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
discontinuous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Discontinuous.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/discontinuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
at every rational point.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">
•
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Ackermann Function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AckermannFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/ackermannfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
</li>
</ul>
</div>
<div class="ltx_para" id="S0.SS0.SSSx1.p2">
<p class="ltx_p">
See also
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
]
</cite>
.
</p>
</div>
</section>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
<a class="nnexus_concept" href="http://planetmath.org/wikipedia">
Wikipedia
</a>
<span class="ltx_text ltx_font_typewriter">
http://en.wikipedia.org/wiki/Pathological (mathematics)
</span>
entry on pathological, mathematics.
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p3">
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<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
pathological
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Pathological
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:41:56
</td>
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Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:41:56
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
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Definition
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Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A20
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0 | QuotationsAboutMathematics | http://planetmath.org/QuotationsAboutMathematics | <!DOCTYPE html>
<html>
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<title>
quotations about mathematics
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
quotations about mathematics
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
There are many memorable quotations pertaining to mathematics, from mathematicians, scientists, artists, etc. The following small sampling is organized by what kind of person said the given quote, with first priority given to mathematicians.
</p>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
Mathematicians
</h2>
<div class="ltx_para" id="S1.p1">
<p class="ltx_p">
“God made the integers, and all the rest is the work of man.”
<br class="ltx_break"/>
</p>
<p class="ltx_p ltx_align_right">
—
<a class="nnexus_concept" href="http://planetmath.org/leopoldkronecker">
Leopold Kronecker
</a>
</p>
</div>
<div class="ltx_para" id="S1.p2">
<p class="ltx_p">
“A mathematician is a device for turning coffee into
<a class="nnexus_concept" href="http://planetmath.org/lemma">
theorems
</a>
.”
<br class="ltx_break"/>
</p>
<p class="ltx_p ltx_align_right">
— Pal Erdős
</p>
</div>
</section>
<section class="ltx_section" id="S2">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
2
</span>
Philosophers
</h2>
<div class="ltx_para" id="S2.p1">
<p class="ltx_p">
“Mathematics possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture.”
<br class="ltx_break"/>
</p>
<p class="ltx_p ltx_align_right">
— Bertrand Russell
</p>
</div>
<div class="ltx_para" id="S2.p2">
<p class="ltx_p">
“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”
<br class="ltx_break"/>
</p>
<p class="ltx_p ltx_align_right">
— Bertrand Russell
</p>
</div>
</section>
<section class="ltx_section" id="S3">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
3
</span>
Physicists
</h2>
<div class="ltx_para" id="S3.p1">
<p class="ltx_p">
“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”
<br class="ltx_break"/>
</p>
<p class="ltx_p ltx_align_right">
—
<a class="nnexus_concept" href="http://planetmath.org/alberteinstein">
Albert Einstein
</a>
</p>
</div>
</section>
<section class="ltx_section" id="S4">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
4
</span>
Politicians
</h2>
<div class="ltx_para" id="S4.p1">
<p class="ltx_p">
“I had a feeling once about Mathematics, that I saw it all — Depth beyond depth was revealed to me — the Byss and Abyss. I saw, as one might see the transit of Venus or even the Lord Mayor’s Show, a quantity
<a class="nnexus_concept" href="http://planetmath.org/incidencegeometry">
passing through
</a>
infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable: and how the one step involved all the others. It was like politics. But it was after dinner and I let it go!”
<br class="ltx_break"/>
</p>
<p class="ltx_p ltx_align_right">
— Winston Churchill
</p>
</div>
</section>
<section class="ltx_section" id="S5">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
5
</span>
Fictional characters
</h2>
<div class="ltx_para" id="S5.p1">
<p class="ltx_p">
“Math, my dear boy, is nothing more than the lesbian sister of biology.”
<br class="ltx_break"/>
</p>
<p class="ltx_p ltx_align_right">
— Peter Griffin,
<span class="ltx_text ltx_font_italic">
Family Guy
</span>
, “When You Wish Upon A Weinstein”
</p>
</div>
<div class="ltx_para" id="S5.p2">
<p class="ltx_p">
“How about we fire up the old Segway and find a nice quiet field to do
<a class="nnexus_concept" href="http://planetmath.org/longdivision">
long division
</a>
in? I mean, a nice quiet field
<span class="ltx_text ltx_font_italic">
in which to
</span>
do long division. Sorry, sorry, everybody.”
<br class="ltx_break"/>
</p>
<p class="ltx_p ltx_align_right">
— Neil Goldman,
<span class="ltx_text ltx_font_italic">
Family Guy
</span>
, “8 Simple Rules for Buying My Teenage Daughter”
</p>
</div>
</section>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
J. M. Cohen & J. C. Cohen,
<span class="ltx_text ltx_font_italic">
Dictionary of Twentieth-Century Quotations
</span>
. London: Penguin Books (1995): 115, 328
</span>
</li>
<li class="ltx_bibitem" id="bib.bib2">
<span class="ltx_bibtag ltx_role_refnum">
2
</span>
<span class="ltx_bibblock">
Paul J. Nahin,
<span class="ltx_text ltx_font_italic">
Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills
</span>
. Princeton: Princeton University Press (2006): xiv
</span>
</li>
<li class="ltx_bibitem" id="bib.bib3">
<span class="ltx_bibtag ltx_role_refnum">
3
</span>
<span class="ltx_bibblock">
Tom Stafford & Matt Webb,
<span class="ltx_text ltx_font_italic">
Mind Hacks: Tips and Tools for Using Your Brain
</span>
. New York: O’Reilly (2004): 312
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p2">
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<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/quotationsaboutmathematics">
quotations about mathematics
</a>
</td>
</tr>
<tr class="ltx_tr">
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<a class="nnexus_concept" href="http://planetmath.org/canonical">
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QuotationsAboutMathematics
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2013-03-22 18:11:25
</td>
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</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
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PrimeFan (13766)
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Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
7
</td>
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<tr class="ltx_tr">
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Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PrimeFan (13766)
</td>
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Entry type
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Feature
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Classification
</a>
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<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A20
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terms from foreign languages used in mathematics (page images version)
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<h1 class="ltx_title ltx_title_document">
terms from foreign languages used in mathematics (page images version)
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
This entry is best viewed in page . For the html version,
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/TermsFromForeignLanguagesUsedInMathematics
</span>
click here.
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<div class="ltx_para" id="p3">
<p class="ltx_p">
Following are from foreign that appear in mathematical literature. Each (TeX tabular) contains from the foreign indicated. The foreign are ordered according to how many appear in its corresponding . In each , the are listed in alphabetical .
</p>
</div>
<section class="ltx_section" id="S1">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
1
</span>
Latin
</h2>
<div class="ltx_para ltx_centering" id="S1.p1">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
abbr.
</span>
</td>
<td class="ltx_td ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
literal
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Literal.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/atomicformula">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
mathematical usage
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
a fortiori
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
with stronger reason
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
used in logic to denote an
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
to the effect that
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
because one ascertained fact exists;
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
therefore another which is
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
included in it or analogous to is
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
and is less improbable, unusual,
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
or surprising must also exist
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
a priori
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
from the former
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
already known/assumed
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
ad absurdum
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
to absurdity
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
assumption
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/deduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/derivationsinnaturaldeduction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is made in hopes of
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
obtaining a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
contradiction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Contradiction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradiction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/contradictorystatement">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
[
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
reductio ad absurdum
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/ReductioadAbsurdum.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/reductioadabsurdum">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is also used]
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
ad infinitum
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
to
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinity
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinity.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/axiomofinfinity">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
endlessly, infinitely
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CasusIrreducibilis.html">
casus irreducibilis
</a>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
not-reducible case
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
roots real but not
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
expressible via real
</span>
<span class="ltx_text ltx_font_typewriter" style="font-size:90%;">
http://planetmath.org/Radical5
</span>
<span class="ltx_text" style="font-size:90%;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Radical.html">
radicals
</a>
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
cf.
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
confer
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
compare
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
used to suggest that another
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
work might also be consulted
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
in to that
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
et al.
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
et alii
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
and others
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
used in multi-author references
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
but it is customary to include
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
all the authors in the first
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
citation and/or in the
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
bibliography
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
e.g.
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
exempli gratia
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
for example’s sake
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
for example
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
ibid.
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
ibidem
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
in the same
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
relates to the
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
immediately prior
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
i.e.
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
id est
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
that is
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
that is
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
inf
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
inferior
</span>
<span class="ltx_text" style="font-size:90%;">
,
</span>
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infimum
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infimum.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/infimum">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
lowest
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
limit inferior
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LimitInferior.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/limitinferior">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
;
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GreatestLowerBound.html">
greatest lower bound
</a>
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
inter alia
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
among other things
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
among other things
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
loc. cit.
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
loco citato
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
in the already mentioned
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
relates to before the
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
immediately prior citation
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
[probably less frequent
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
than op. cit.]
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
lb
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
logarithmus binaris
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/BinaryLogarithm.html">
binary logarithm
</a>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
log. in base 2
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
lg
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
logarithmus generalis
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
general
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
logarithm
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Logarithm.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/logarithm">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
log. in base 10
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
ln
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
logarithmus naturalis
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
natural logarithm
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NaturalLogarithm.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/naturallogarithm">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
log. in base
</span>
<math alttext="e" class="ltx_Math" display="inline" id="S1.p1.m1">
<mi mathsize="90%">
e
</mi>
</math>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
mutatis mutandis
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
once changing thing to be changed
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
repeat the
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
for the related case
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
N.B.
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
nota bene
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
note well
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
the following is important
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
op. cit.
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
opere citato
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
in the work already mentioned
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
relates to before the
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
immediately prior citation
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
[probably more frequent
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
than loc. cit.]
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
QED
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
quod erat demonstrandum
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
which was to be demonstrated
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
end of proof
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
QEF
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
quod erat faciendum
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
which was to be done
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
end of construction
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
regula falsi
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
rule of false position
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
Newton’s method
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
sine qua non
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
without which it could not be
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
an essential condition
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_border_r">
</td>
<td class="ltx_td ltx_align_center ltx_border_r">
<span class="ltx_text" style="font-size:90%;">
or
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Element.html">
element
</a>
; an indispensable thing
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
sup
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
superior
</span>
<span class="ltx_text" style="font-size:90%;">
,
</span>
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Supremum.html">
supremum
</a>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
uppermost
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
limit superior
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LimitSuperior.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/limitsuperior">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
,
</span>
<span class="ltx_text ltx_font_typewriter" style="font-size:90%;">
http://planetmath.org/LowestUpperBound
</span>
<span class="ltx_text" style="font-size:90%;">
least
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
upper bound
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/UpperBound.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/upperbound">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
viz
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic" style="font-size:90%;">
videlicet
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
that is to say, namely
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<span class="ltx_text" style="font-size:90%;">
a keynote abbreviation
</span>
</td>
</tr>
</tbody>
</table>
</div>
</section>
<section class="ltx_section" id="S2">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
2
</span>
German
</h2>
<div class="ltx_para ltx_centering" id="S2.p1">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
abbr.
</td>
<td class="ltx_td ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
literal
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
mathematical usage
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
Ansatz
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
approach, attempt
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
assumed form for an expression
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
eigen
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
, typical
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
eigenvalue; eigenvector
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
Grösse, Größe
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
size, magnitude
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
Grössencharacter
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
im kleinen
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
in the small
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
connected im kleinen
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
Nullstellensatz
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
zero point
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
zero point
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
Stufe
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
stair,
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
stufe of a field
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Urelement.html">
Urelement
</a>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
primeval element
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
set element which is not a set
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<math alttext="V" class="ltx_Math" display="inline" id="S2.p1.m1">
<mi>
V
</mi>
</math>
,
<math alttext="K_{4}" class="ltx_Math" display="inline" id="S2.p1.m2">
<msub>
<mi>
K
</mi>
<mn>
4
</mn>
</msub>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Vierergruppe.html">
Vierergruppe
</a>
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
four-group
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/klein4group">
Klein 4-group
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t">
<math alttext="\mathbb{Z}" class="ltx_Math" display="inline" id="S2.p1.m3">
<mi>
ℤ
</mi>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
Zahlen
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
numbers
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integers
</a>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r ltx_border_t">
<math alttext="Z" class="ltx_Math" display="inline" id="S2.p1.m4">
<mi>
Z
</mi>
</math>
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
Zentrum
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/GroupCentre
</span>
center
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/GroupCentre
</span>
center (of a group)
</td>
</tr>
</tbody>
</table>
</div>
</section>
<section class="ltx_section" id="S3">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
3
</span>
French
</h2>
<div class="ltx_para ltx_centering" id="S3.p1">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_l ltx_border_r ltx_border_t">
abbr.
</th>
<th class="ltx_td ltx_th ltx_th_column ltx_border_r ltx_border_t">
</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t">
literal
</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t">
mathematical usage
</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
espace
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
space
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
(topological) space [see Espace Étalé]
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
étale
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
slack
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
étale
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
fundamental group
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FundamentalGroup.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/fundamentalgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
;
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/Etale
</span>
étale
<a class="nnexus_concept" href="http://planetmath.org/structurehomomorphism">
morphism
</a>
; étale site
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_border_l ltx_border_r ltx_border_t">
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
étalé
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
spread out, displayed
</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/EtaleSpace3
</span>
Étalé space
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r ltx_border_t">
p.p.
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_italic">
presque partout
</span>
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
almost everywhere
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/AlmostSurely
</span>
almost everywhere
</td>
</tr>
</tbody>
</table>
</div>
</section>
<section class="ltx_section" id="S4">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">
4
</span>
Russian
</h2>
<div class="ltx_para ltx_centering" id="S4.p1">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t">
abbr.
</th>
<th class="ltx_td ltx_th ltx_th_column ltx_border_r ltx_border_t">
</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t">
literal
</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t">
mathematical usage
</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t">
<math alttext="\partial" class="ltx_Math" display="inline" id="S4.p1.m1">
<mo>
∂
</mo>
</math>
</th>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
italic ‘‘д’’ [may be pronounced ‘‘doh’’]
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
letter ‘‘d’’
</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t">
e.g. in
<math alttext="\displaystyle{\frac{\partial f}{\partial x}}" class="ltx_Math" display="inline" id="S4.p1.m2">
<mstyle displaystyle="true">
<mfrac>
<mrow>
<mo>
∂
</mo>
<mo>
</mo>
<mi>
f
</mi>
</mrow>
<mrow>
<mo>
∂
</mo>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mfrac>
</mstyle>
</math>
[see
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
partial derivative
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PartialDerivative.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/partialderivative">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
]
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para ltx_align_right" id="S4.p2">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
terms from foreign
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
languages
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/language">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/signature">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
used in mathematics (page images version)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
TermsFromForeignLanguagesUsedInMathematicspageImagesVersion
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:06:45
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:06:45
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Wkbj79 (1863)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Wkbj79 (1863)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
37
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Wkbj79 (1863)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Topic
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A99
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
TermsFromForeignLanguagesUsedInMathematics
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
MathematicsVocabulary
</td>
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topic entry on algebra
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
topic entry on algebra
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
The subject of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebra
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Algebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraiccategoriesandclassesofalgebras">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
may be defined as the study of
<a class="nnexus_concept" href="http://planetmath.org/algebraicsystem">
algebraic
systems
</a>
, where an algebraic system consists of a set together
with a certain number of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
operations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/operation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, which are
<a class="nnexus_concept" href="http://planetmath.org/function">
functions
</a>
(or
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
partial functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PartialFunction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/partialfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) on this set. A prototypical example of an
algebraic system is the
<a class="nnexus_concept" href="http://planetmath.org/integralclosure">
ring of integers
</a>
, which consists of the
set of integers,
<math alttext="\{\ldots,-2,-1,0,1,2,\ldots\}" class="ltx_Math" display="inline" id="p1.m1">
<mrow>
<mo stretchy="false">
{
</mo>
<mi mathvariant="normal">
…
</mi>
<mo>
,
</mo>
<mrow>
<mo>
-
</mo>
<mn>
2
</mn>
</mrow>
<mo>
,
</mo>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<mo>
,
</mo>
<mn>
0
</mn>
<mo>
,
</mo>
<mn>
1
</mn>
<mo>
,
</mo>
<mn>
2
</mn>
<mo>
,
</mo>
<mi mathvariant="normal">
…
</mi>
<mo stretchy="false">
}
</mo>
</mrow>
</math>
together
with the operations
<math alttext="+" class="ltx_Math" display="inline" id="p1.m2">
<mo>
+
</mo>
</math>
and
<math alttext="\times" class="ltx_Math" display="inline" id="p1.m3">
<mo>
×
</mo>
</math>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
In
<a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic">
addition
</a>
to studying
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Individual.html">
individual
</a>
systems, algebraists consider
classes of systems defined by common properties. For
<a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">
instance
</a>
,
the example cited above is an example of a ring, which is an algebraic
system with two operations which
<a class="nnexus_concept" href="http://planetmath.org/satisfactionrelation">
satisfy
</a>
certain axioms, such as
<a class="nnexus_concept" href="http://planetmath.org/distributivity">
distributivity
</a>
of one operation over the other.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
The reason for considering classes of systems
is in order to save work by stating and proving
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
theorems
</a>
at the
appropriate level of generality. For instance, while the statement
that every integer equals the sum of four squares is specific to the
ring of integers (there are many rings in which this is not the case)
and its proof makes use of specific facts about integers, the proof
of the fact that the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
product
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Product.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/product">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categoricaldirectproduct">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of two sums of integers equals the sum
of all products of numbers appearing in the first sum by numbers
appearing in the second sum only involves the distributive law, so
an analogous theorem will hold for any ring. Clearly, it is
wasteful to restate the same theorem and its proof for every ring
so we state and prove it once as a theorem about rings, then apply
it to specific instances of rings.
</p>
</div>
<div class="ltx_para" id="p4">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/node/ConceptsInAbstractAlgebra
</span>
<a class="nnexus_concept" href="http://planetmath.org/conceptsinabstractalgebra">
Concepts in abstract algebra
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
topics on
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GroupTheory.html">
group theory
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
topics on ring theory
</p>
</div>
</li>
<li class="ltx_item" id="I1.i4" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
4.
</span>
<div class="ltx_para" id="I1.i4.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/topicsonideals">
topics on ideal
</a>
theory
</p>
</div>
</li>
<li class="ltx_item" id="I1.i5" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
5.
</span>
<div class="ltx_para" id="I1.i5.p1">
<p class="ltx_p">
topics on field theory
</p>
</div>
</li>
<li class="ltx_item" id="I1.i6" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
6.
</span>
<div class="ltx_para" id="I1.i6.p1">
<p class="ltx_p">
topics on
<a class="nnexus_concept" href="http://mathworld.wolfram.com/HomologicalAlgebra.html">
homological algebra
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i7" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
7.
</span>
<div class="ltx_para" id="I1.i7.p1">
<p class="ltx_p">
topics on
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
category theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CategoryTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categorytheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bibliographyinalgebraictopologycategoriesandqat">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/graphtheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicsinalgebraictopology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i8" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
8.
</span>
<div class="ltx_para" id="I1.i8.p1">
<p class="ltx_p">
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
algebraic k-theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AlgebraicK-Theory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/algebraicktheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i9" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
9.
</span>
<div class="ltx_para" id="I1.i9.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/specialnotationsinalgebra">
Special notations in algebra
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i10" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
10.
</span>
<div class="ltx_para" id="I1.i10.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/topicsonpolynomials">
Topics on polynomials
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i11" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
11.
</span>
<div class="ltx_para" id="I1.i11.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/topicsonfieldextensionsandgaloistheory">
Topics on field extensions and Galois theory
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i12" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
12.
</span>
<div class="ltx_para" id="I1.i12.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/entriesonfinitelygeneratedideals">
Entries on finitely generated ideals
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i13" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
13.
</span>
<div class="ltx_para" id="I1.i13.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/node/2530
</span>
Topic entry on
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
linear algebra
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/LinearAlgebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/linearalgebra">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i14" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
14.
</span>
<div class="ltx_para" id="I1.i14.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/node/5663
</span>
<a class="nnexus_concept" href="http://planetmath.org/conceptsinlinearalgebra">
Concepts in linear algebra
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i15" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
15.
</span>
<div class="ltx_para" id="I1.i15.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/matricesofspecialform">
Matrices of special form
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i16" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
16.
</span>
<div class="ltx_para" id="I1.i16.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/MatrixFactorization
</span>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Matrix decompositions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/MatrixDecomposition.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/matrixfactorization">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i17" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
17.
</span>
<div class="ltx_para" id="I1.i17.p1">
<p class="ltx_p">
<a class="nnexus_concept" href="http://planetmath.org/bibliographyforgrouptheory">
Bibliography for group theory
</a>
</p>
</div>
</li>
<li class="ltx_item" id="I1.i18" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
18.
</span>
<div class="ltx_para" id="I1.i18.p1">
<p class="ltx_p">
topics on
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
universal algebra
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/UniversalAlgebra.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicentryonthealgebraicfoundationsofmathematics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
</p>
</div>
</li>
</ol>
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topic entry on algebra
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rspuzio (6075)
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rspuzio (6075)
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11
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rspuzio (6075)
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<title>
vacuous
</title>
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<h1 class="ltx_title ltx_title_document">
vacuous
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
Suppose
<math alttext="X" class="ltx_Math" display="inline" id="p1.m1">
<mi>
X
</mi>
</math>
is a set and
<math alttext="P" class="ltx_Math" display="inline" id="p1.m2">
<mi>
P
</mi>
</math>
is a
<a class="nnexus_concept" href="http://planetmath.org/property">
property
</a>
defined as follows:
</p>
<table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A0.EGx1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex1">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<span class="ltx_text ltx_markedasmath">
<math alttext="X" class="ltx_Math" display="inline" id="S0.Ex1.m3.m1">
<mi>
X
</mi>
</math>
has property
<math alttext="P" class="ltx_Math" display="inline" id="S0.Ex1.m3.m2">
<mi>
P
</mi>
</math>
if and only if
</span>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S0.Ex2">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<span class="ltx_text ltx_markedasmath">
<math alttext="\forall Y[" class="ltx_Math" display="inline" id="S0.Ex2.m3.m1">
<mrow>
<mo>
∀
</mo>
<mi>
Y
</mi>
<mo stretchy="false">
[
</mo>
</mrow>
</math>
<math alttext="Y" class="ltx_Math" display="inline" id="S0.Ex2.m3.m2">
<mi>
Y
</mi>
</math>
<a class="nnexus_concept" href="http://planetmath.org/satisfactionrelation">
satisfies
</a>
condition
<math alttext="1]\Rightarrow" class="ltx_Math" display="inline" id="S0.Ex2.m3.m3">
<mrow>
<mn>
1
</mn>
<mo stretchy="false">
]
</mo>
<mo>
⇒
</mo>
</mrow>
</math>
<math alttext="Y" class="ltx_Math" display="inline" id="S0.Ex2.m3.m4">
<mi>
Y
</mi>
</math>
satisfies condition
<math alttext="2" class="ltx_Math" display="inline" id="S0.Ex2.m3.m5">
<mn>
2
</mn>
</math>
</span>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
where condition
<math alttext="1" class="ltx_Math" display="inline" id="p1.m3">
<mn>
1
</mn>
</math>
and condition
<math alttext="2" class="ltx_Math" display="inline" id="p1.m4">
<mn>
2
</mn>
</math>
define the property.
If condition
<math alttext="1" class="ltx_Math" display="inline" id="p1.m5">
<mn>
1
</mn>
</math>
is never satisfied then
<math alttext="X" class="ltx_Math" display="inline" id="p1.m6">
<mi>
X
</mi>
</math>
satisfies property
<math alttext="P" class="ltx_Math" display="inline" id="p1.m7">
<mi>
P
</mi>
</math>
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/vacuous">
vacuously
</a>
</em>
.
</p>
</div>
<section class="ltx_subsubsection" id="S0.SS0.SSSx1">
<h2 class="ltx_title ltx_title_subsubsection">
Examples
</h2>
<div class="ltx_para" id="S0.SS0.SSSx1.p1">
<ol class="ltx_enumerate" id="I1">
<li class="ltx_item" id="I1.i1" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
1.
</span>
<div class="ltx_para" id="I1.i1.p1">
<p class="ltx_p">
If
<math alttext="X" class="ltx_Math" display="inline" id="I1.i1.p1.m1">
<mi>
X
</mi>
</math>
is the set
<math alttext="\{1,2,3\}" class="ltx_Math" display="inline" id="I1.i1.p1.m2">
<mrow>
<mo stretchy="false">
{
</mo>
<mn>
1
</mn>
<mo>
,
</mo>
<mn>
2
</mn>
<mo>
,
</mo>
<mn>
3
</mn>
<mo stretchy="false">
}
</mo>
</mrow>
</math>
and
<math alttext="P" class="ltx_Math" display="inline" id="I1.i1.p1.m3">
<mi>
P
</mi>
</math>
is the property defined as above with condition
<math alttext="1=" class="ltx_Math" display="inline" id="I1.i1.p1.m4">
<mrow>
<mn>
1
</mn>
<mo>
=
</mo>
<mi>
</mi>
</mrow>
</math>
<math alttext="Y" class="ltx_Math" display="inline" id="I1.i1.p1.m5">
<mi>
Y
</mi>
</math>
is a
<a class="nnexus_concept" href="http://planetmath.org/infinite">
infinite subset
</a>
of
<math alttext="X" class="ltx_Math" display="inline" id="I1.i1.p1.m6">
<mi>
X
</mi>
</math>
, and condition
<math alttext="2=" class="ltx_Math" display="inline" id="I1.i1.p1.m7">
<mrow>
<mn>
2
</mn>
<mo>
=
</mo>
<mi>
</mi>
</mrow>
</math>
<math alttext="Y" class="ltx_Math" display="inline" id="I1.i1.p1.m8">
<mi>
Y
</mi>
</math>
contains
<math alttext="7" class="ltx_Math" display="inline" id="I1.i1.p1.m9">
<mn>
7
</mn>
</math>
. Then
<math alttext="X" class="ltx_Math" display="inline" id="I1.i1.p1.m10">
<mi>
X
</mi>
</math>
has property
<math alttext="P" class="ltx_Math" display="inline" id="I1.i1.p1.m11">
<mi>
P
</mi>
</math>
vacously; every infinite subset of
<math alttext="\{1,2,3\}" class="ltx_Math" display="inline" id="I1.i1.p1.m12">
<mrow>
<mo stretchy="false">
{
</mo>
<mn>
1
</mn>
<mo>
,
</mo>
<mn>
2
</mn>
<mo>
,
</mo>
<mn>
3
</mn>
<mo stretchy="false">
}
</mo>
</mrow>
</math>
contains the number
<math alttext="7" class="ltx_Math" display="inline" id="I1.i1.p1.m13">
<mn>
7
</mn>
</math>
<cite class="ltx_cite ltx_citemacro_cite">
[
<a class="ltx_ref" href="#bib.bib1" title="">
1
</a>
]
</cite>
.
</p>
</div>
</li>
<li class="ltx_item" id="I1.i2" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
2.
</span>
<div class="ltx_para" id="I1.i2.p1">
<p class="ltx_p">
The
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
empty set
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/EmptySet.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/emptyset">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is a
<a class="nnexus_concept" href="http://planetmath.org/hausdorffspace">
Hausdorff space
</a>
(vacuously).
</p>
</div>
</li>
<li class="ltx_item" id="I1.i3" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">
3.
</span>
<div class="ltx_para" id="I1.i3.p1">
<p class="ltx_p">
Suppose property
<math alttext="P" class="ltx_Math" display="inline" id="I1.i3.p1.m1">
<mi>
P
</mi>
</math>
is defined by the statement :
<br class="ltx_break"/>
<em class="ltx_emph ltx_font_italic">
The present King of France does not exist.
</em>
<br class="ltx_break"/>
Then either of the following
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
propositions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/booleanvaluedfunction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/propositionallogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is satisfied vacuously.
<br class="ltx_break"/>
<em class="ltx_emph ltx_font_italic">
The present king of France is bald.
</em>
<br class="ltx_break"/>
<em class="ltx_emph ltx_font_italic">
The present King of France is not bald.
</em>
</p>
</div>
</li>
</ol>
</div>
</section>
<section class="ltx_bibliography" id="bib">
<h2 class="ltx_title ltx_title_bibliography">
References
</h2>
<ul class="ltx_biblist">
<li class="ltx_bibitem" id="bib.bib1">
<span class="ltx_bibtag ltx_role_refnum">
1
</span>
<span class="ltx_bibblock">
Wikipedia
<span class="ltx_text ltx_font_typewriter">
http://en.wikipedia.org/wiki/Vacuous_truth
</span>
entry on Vacuous truth.
</span>
</li>
</ul>
</section>
<div class="ltx_para ltx_align_right" id="p2">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/quantifier">
vacuous
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Vacuous
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:42:27
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 14:42:27
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
9
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
matte (1858)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/definition">
Definition
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A20
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
vacuously
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
<a class="nnexus_concept" href="http://planetmath.org/implication">
vacuously true
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Synonym
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
vacuous truth
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | ComplexSystemsBiologyCSB | http://planetmath.org/ComplexSystemsBiologyCSB | <!DOCTYPE html>
<html>
<head>
<title>
complex systems biology (CSB)
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
complex systems biology (CSB)
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
</div>
<section class="ltx_subsection" id="S0.SS1">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
0.1
</span>
Introduction
</h2>
<div class="ltx_para" id="S0.SS1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
Complex systems biology (
<math alttext="CSB" class="ltx_Math" display="inline" id="S0.SS1.p1.m1">
<mrow>
<mi>
C
</mi>
<mo mathvariant="italic">
</mo>
<mi>
S
</mi>
<mo mathvariant="italic">
</mo>
<mi>
B
</mi>
</mrow>
</math>
)
</span>
is generally described as a non-reductionist, mathematical theory
of emergent living organisms or biosystems in terms of a
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Network.html">
network
</a>
, graph or
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
category
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Category.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/category">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of integrated interactions between their structural and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functional
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/interpretationofintuitionisticlogicbymeansoffunctionals">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/intuitionisticlogic">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functional">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<a class="nnexus_concept" href="http://planetmath.org/connectedgraph">
components
</a>
or subsystems. This is often abbreviated to
<span class="ltx_text ltx_font_typewriter">
http://planetphysics.org/?op=getobj&from=books&id=248
</span>
<a class="nnexus_concept" href="http://planetmath.org/complexsystemsbiologycsb">
systems biology
</a>
in entries that should be described in fact as
<span class="ltx_text ltx_font_italic">
complex systems biology
</span>
. Notably, several mathematical physicists or mathematicians,
such as von Neumann believed that all
<a class="nnexus_concept" href="http://planetmath.org/noncommutativedynamicmodelingdiagrams">
complex systems
</a>
can be ultimately ‘decomposed’ or disassembled into their simpler, physical components, whereas others, such as Elsasser argued that the heterogeneous logical class of biosystems makes them
<a class="nnexus_concept" href="http://planetmath.org/irreducible1">
irreducible
</a>
to their physical components of logically
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
homogeneous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/arrowsrelation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homogeneous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homogeneousideal">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
classes; the latter view was
also shared by
<a class="nnexus_concept" href="http://planetmath.org/robertrosen">
Robert Rosen
</a>
who also produced encoding and dynamic reasons for which reductionism would not work
for biosystems. Thus, it would seem that there is a fundamental, physical and mathematical controvercy regarding the
essential nature of Life.
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Resolution.html">
Resolution
</a>
of this fundamental controvercy in terms of mathematics is denied by many biologists who argue that the dynamics and physiology of biosystems is not formalizable in either mathematical terms or physical theory. The key question:
<span class="ltx_text ltx_font_italic">
“What is Life?”
</span>
is also the title of the widely-read book published in 1945 by the famous quantum theoretician Erwin
<math alttext='Schr\"{o}dinger' class="ltx_Math" display="inline" id="S0.SS1.p1.m2">
<mrow>
<mi>
S
</mi>
<mo>
</mo>
<mi>
c
</mi>
<mo>
</mo>
<mi>
h
</mi>
<mo>
</mo>
<mi>
r
</mi>
<mo>
</mo>
<mtext>
ö
</mtext>
<mo>
</mo>
<mi>
d
</mi>
<mo>
</mo>
<mi>
i
</mi>
<mo>
</mo>
<mi>
n
</mi>
<mo>
</mo>
<mi>
g
</mi>
<mo>
</mo>
<mi>
e
</mi>
<mo>
</mo>
<mi>
r
</mi>
</mrow>
</math>
, Nobel laureate and inventor of the equation that carries his name.
</p>
</div>
<div class="ltx_para" id="S0.SS1.p2">
<p class="ltx_p">
To address this fundamental question of life, applied mathematicians, as well as mathematical and theoretical biologists have been developing for over a century precise mathematical models of biosystems or organisms of increasing sophistication and generality. One can select the birth of cybernetics, biocybernetics, and the application of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
category theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CategoryTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categorytheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bibliographyinalgebraictopologycategoriesandqat">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/graphtheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicsinalgebraictopology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, as well as
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
set theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/SetTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/settheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, to biosystems as the starting point of
<math alttext="CSB" class="ltx_Math" display="inline" id="S0.SS1.p2.m1">
<mrow>
<mi>
C
</mi>
<mo>
</mo>
<mi>
S
</mi>
<mo>
</mo>
<mi>
B
</mi>
</mrow>
</math>
which is currently the branch of inter-disciplinary science, between mathematics and biology, as well as sociology, that aims to define in precise, mathematical terms the nature of dynamic and organizational complexity both in living organisms and in societies.
The famous topologist and Fields medalist René Thom was one of the more recent contributors to this field from the
viewpoint of
<a class="nnexus_concept" href="http://planetmath.org/topologicalspace">
topology
</a>
and Poincaré’ s
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Qualitative Dynamics
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/categoriesandsupercategoriesinrelationalbiology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/organismicsupercategoriesandsupercomplexsystemsbiodynamics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. Grothendieck is also said to have a keen interest in such complexity problems related to living organisms. Defining the main problems and approaches in
<math alttext="CSB" class="ltx_Math" display="inline" id="S0.SS1.p2.m2">
<mrow>
<mi>
C
</mi>
<mo>
</mo>
<mi>
S
</mi>
<mo>
</mo>
<mi>
B
</mi>
</mrow>
</math>
remains
a monumental task for multi-disciplinary teams of applied mathematicians, biologists, biochemists, physicists, biophysicists, sociologists, computer scientists, and so on. The underlying logical problems are also formidable
as most complexity problems do not have easy, or simple, Boolean logic solutions,
and are not amenable to linear engineering
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Analysis.html">
analysis
</a>
, either direct or reverse.
</p>
</div>
</section>
<section class="ltx_subsection" id="S0.SS2">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
0.2
</span>
Complex systems biology
</h2>
<div class="ltx_theorem ltx_theorem_definition" id="S0.Thmdefinition1">
<h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem">
Definition 0.1
</span>
.
</h6>
<div class="ltx_para" id="S0.Thmdefinition1.p1">
<p class="ltx_p">
The
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/categoricalquantumlmlogicalgebras">
categorical ontology
</a>
theory of levels
</em>
is often defined as the classification theory in
<a class="nnexus_concept" href="http://planetmath.org/logicism">
ontology
</a>
, or the theory of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Existence.html">
existence
</a>
of items (or objects–defined in the mathematical or logical sense) by means of the mathematical theory of categories into three levels of dynamic systems pertaining to: the physical/chemical level, the biological level, and the psychological level (or human mind).
<a class="nnexus_concept" href="http://planetmath.org/doublegroupoidwithconnection">
Connections
</a>
between the three levels of reality and their
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
transformations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/functorialmorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/transformation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
are represented, respectively, by
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
morphisms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Morphism.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structurehomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
/
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functors
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Functor.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functor">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and
<a class="nnexus_concept" href="http://planetmath.org/naturaltransformation">
natural transformations
</a>
defined for
<a class="nnexus_concept" href="http://planetmath.org/categoryofmolecularsets">
categories of molecular sets
</a>
,
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/CategoryOfMRSystems3
</span>
categories of
<math alttext="(M,R)" class="ltx_Math" display="inline" id="S0.Thmdefinition1.p1.m1">
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
M
</mi>
<mo>
,
</mo>
<mi>
R
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
-systems and
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
organismic supercategories
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/categoriesandsupercategoriesinrelationalbiology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/supercategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/nsupercategorytheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categoricaldynamics">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
</div>
<div class="ltx_para" id="S0.SS2.p1">
<p class="ltx_p">
From a categorical ontology theory of levels viewpoint, however, the term complex may appear to be misplaced because
<span class="ltx_text ltx_font_italic">
systems with chaos
</span>
, or chaotic dynamics, are currently defined by physicists as
<span class="ltx_text ltx_font_italic">
‘complex systems’
</span>
, which may have placed a role in the emergence of living systems that are, in fact,
<span class="ltx_text ltx_font_italic">
super-complex
</span>
. Therefore, the more appropriate
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">
classification
</a>
of this relatively new area in mathematical or
<a class="nnexus_concept" href="http://planetmath.org/mathematicalbiologyandtheoreticalbiophysicsofdna">
theoretical biology
</a>
and Biophysics is
<a class="nnexus_concept" href="http://planetmath.org/artificialintelligence">
super-complex systems
</a>
biology,
<math alttext="s" class="ltx_Math" display="inline" id="S0.SS2.p1.m1">
<mi>
s
</mi>
</math>
-complex systems biology, or simply “systems biology”–as a more general approach where the focus may be not on the super-complexity aspects of living systems but on computer modeling of physiological, or functional genomics, integration of physiological flows, signaling pathways or interactomics. However, unlike the case of purely functional
<math alttext="(M,R)" class="ltx_Math" display="inline" id="S0.SS2.p1.m2">
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
M
</mi>
<mo>
,
</mo>
<mi>
R
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
-systems theory in
<a class="nnexus_concept" href="http://planetmath.org/arbsymmetryandgroupoidrepresentations">
abstract relational biology
</a>
(ARB), complex systems biology (or systems biology) proponents are primarily concerned with the integration of data from a multitude of bioinformatics and genomic/proteomic/post-genomic (primarily structural) data;
<math alttext="CSB" class="ltx_Math" display="inline" id="S0.SS2.p1.m3">
<mrow>
<mi>
C
</mi>
<mo>
</mo>
<mi>
S
</mi>
<mo>
</mo>
<mi>
B
</mi>
</mrow>
</math>
scientists also aim to study
<span class="ltx_text ltx_font_italic">
interactomics
</span>
or
<span class="ltx_text ltx_font_italic">
metabolomics
</span>
primarily through computer-based data analysis, and often Bayesian-based attempts at integration. branches of mathematics that find applications in
<math alttext="CSB" class="ltx_Math" display="inline" id="S0.SS2.p1.m4">
<mrow>
<mi>
C
</mi>
<mo>
</mo>
<mi>
S
</mi>
<mo>
</mo>
<mi>
B
</mi>
</mrow>
</math>
are, for example: computer modeling,
<a class="nnexus_concept" href="http://planetmath.org/hypergraph">
colored graphs
</a>
, graph-theoretical based approaches, biotopology, genetic, metabolic and signaling network theories,
<a class="nnexus_concept" href="http://planetmath.org/geneticnets">
Bayesian models
</a>
, biostatistics, correlation techniques, and less frequently: abstract algebra, group theory,
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
groupoid
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/quantumoperatoralgebrasinquantumfieldtheories">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/groupoidandgrouprepresentationsrelatedtoquantumsymmetries">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/groupoids">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
and category theory modeling of cell-cell interactions and biodynamics.
</p>
</div>
<div class="ltx_para ltx_align_right" id="S0.SS2.p2">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
complex systems biology (CSB)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Canonical name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
ComplexSystemsBiologyCSB
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:11:43
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified on
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:11:43
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Owner
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Last modified by
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Numerical id
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
44
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Author
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
bci1 (20947)
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Entry type
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Topic
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A30
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 18A40
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 37F99
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 11Y16
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 18A05
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Classification
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 03D15
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
systems biology
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
CSB
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Synonym
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
abstract relational biology
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Category
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
SystemDefinitions
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
CategoryOfMRSystems3
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
CategoricalOntology
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
OrganismicSets2
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
FunctionalBiology
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
RosettaGroupoids
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
NaturalTransformationsOfOrganismicStructures
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
MolecularSetVariable
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
CategoryOfMolecularSets
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
SupercategoryOfVariableMolecularSets
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Related topic
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
Mat
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
categorical ontology of levels
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
complex system biology modeling and ontology
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l">
Defines
</td>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
CSB
</td>
</tr>
</tbody>
</table>
</div>
</section>
</article>
</div>
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0 | MatheRealism | http://planetmath.org/MatheRealism | <!DOCTYPE html>
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MatheRealism
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<h1 class="ltx_title ltx_title_document">
MatheRealism
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
<em class="ltx_emph ltx_font_italic">
MatheRealism
</em>
is the position that the amount of
<a class="nnexus_concept" href="http://planetmath.org/fisherinformationmatrix">
information
</a>
in
the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
universe
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/universe">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/universeofdiscourse">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
places a limit on the possible contents of mathematics.
Its supporters claim it as a philosophical
<a class="nnexus_concept" href="http://planetmath.org/foundationsofmathematicsoverview">
foundation of mathematics
</a>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
argument
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Argument.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/argument">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
proceeds as follows. The number of atoms is about
10^80 and will remain so forever, limited by the horizon of
<a class="nnexus_concept" href="http://planetmath.org/datatypesinstatistics">
observation
</a>
and notwithstanding the expansion of the universe; the
remaining part of the universe is causally disconnected from us. The
number of elementary particles is less than 10^100. Although every
atom has infinitely many eigenstates, only a finite number of them can
be distinguished (due to the uncertainty
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
relation
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/presentationofagroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/relation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of quantum
physics). Further the eigenstates, with exception of the ground state,
have a limited life time. All this leads to the result that only a finite number of bits
can be stored in the universe. A
<a class="nnexus_concept" href="http://planetmath.org/lamellarfield">
conservative
</a>
<a class="nnexus_concept" href="http://planetmath.org/estimator">
estimate
</a>
is 10^100
bits, but in any case there is an
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
upper limit
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/UpperLimit.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/definiteintegral">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
of information
<em class="ltx_emph ltx_font_italic">
L
</em>
.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
According to MatheRealism only such numbers exist which are
<a class="nnexus_concept" href="http://planetmath.org/computablenumber">
computable
</a>
or can be identified uniquely and addressed individually by any other means. In particular, the supporters of
this view claim that any irrational number which cannot be
represented by less information than is
<a class="nnexus_concept" href="http://planetmath.org/superset">
contained
</a>
by its
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
infinite
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinite.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/infinite">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
string of bits, cannot get addressed at all and that even most
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
natural
numbers
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/NaturalNumber.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/naturalnumber">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
cannot get addressed because their most economical
<a class="nnexus_concept" href="http://planetmath.org/grouprepresentation">
representation
</a>
requires more information than
<em class="ltx_emph ltx_font_italic">
L
</em>
. According to
MatheRealism, a number which cannot be represented, addressed, or used
otherwise does not exist. This implies that
<a class="nnexus_concept" href="http://mathworld.wolfram.com/InfiniteSet.html">
infinite sets
</a>
do not exist.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Note: Although all available numbers have a finite contents of information, there is not a
<a class="nnexus_concept" href="http://planetmath.org/minimalandmaximalnumber">
greatest number
</a>
, because, by useful abbreviations, numbers as large as desired can be represented by means of little information.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
The expression MatheRealism is touching on materialism. It may not be mixed up with the notion ”realism” of current philosophy of mathematics which in fact is an idealism.
</p>
</div>
<div class="ltx_para" id="p6">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Literature
</span>
W. Mückenheim: Die Mathematik des Unendlichen, Shaker, Aachen 2006.
</p>
</div>
<div class="ltx_para ltx_align_right" id="p7">
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<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
MatheRealism
</td>
</tr>
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<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
MatheRealism
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
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16
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msc 00A30
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Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
DecimalExpansion
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0 | UltracomplexSystems | http://planetmath.org/UltracomplexSystems | <!DOCTYPE html>
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<title>
ultra-complex systems
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
ultra-complex systems
</h1>
<span class="ltx_ERROR undefined">
\xyoption
</span>
<div class="ltx_para" id="p1">
<p class="ltx_p">
curve
</p>
</div>
<section class="ltx_subsection" id="S0.SS1">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">
0.1
</span>
Ultra-complex systems
</h2>
<div class="ltx_para" id="S0.SS1.p1">
<p class="ltx_p">
An
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/ultracomplexsystems">
ultra-complex system
</a>
</em>
<a class="nnexus_concept" href="http://planetmath.org/representablefunctor">
represents
</a>
the human mind from the standpoint
of a generalized
<a class="nnexus_concept" href="http://planetmath.org/categoricalquantumlmlogicalgebras">
Categorical Ontology
</a>
Theory of Levels as the highest level of complexity that
emerged through biological and social coevolution over the last
<math alttext="2.2" class="ltx_Math" display="inline" id="S0.SS1.p1.m1">
<mn>
2.2
</mn>
</math>
million years on Earth.
</p>
</div>
<section class="ltx_subsubsection" id="S0.SS1.SSS1">
<h3 class="ltx_title ltx_title_subsubsection">
<span class="ltx_tag ltx_tag_subsubsection">
0.1.1
</span>
Preliminary Data:
</h3>
<div class="ltx_para" id="S0.SS1.SSS1.p1">
<p class="ltx_p">
One can represent in square categorical
<a class="nnexus_concept" href="http://planetmath.org/commutativediagram">
diagrams
</a>
the emergence of ultra-complex
dynamics from the super-complex dynamics of human organisms coupled
<span class="ltx_text ltx_font_italic">
via
</span>
social interactions
in
<a class="nnexus_concept" href="http://planetmath.org/characteristicsubgroup">
characteristic
</a>
patterns represented by
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/RosettaGroupoids
</span>
Rosetta biogroupoids, together with the complex–albeit inanimate–systems with ‘chaos’. With the emergence of the ultra-complex system of the human mind– based on the super-complex human organism– there is always an associated
progression towards
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
higher dimensional algebras
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/ncategory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/2groupoid">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
from the lower
dimensions of human neural network dynamics and the simple algebra
of physical dynamics, as shown in the following, essentially
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/commutative">
non-commutative
</a>
</em>
categorical diagram.
</p>
</div>
<div class="ltx_theorem ltx_theorem_definition" id="S0.Thmdefinition1">
<h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem">
Definition 0.1
</span>
.
</h6>
<div class="ltx_para" id="S0.Thmdefinition1.p1">
<p class="ltx_p">
An
<em class="ltx_emph ltx_font_italic">
ultra-complex system,
<math alttext="U_{CS}" class="ltx_Math" display="inline" id="S0.Thmdefinition1.p1.m1">
<msub>
<mi>
U
</mi>
<mrow>
<mi>
C
</mi>
<mo mathvariant="italic">
</mo>
<mi>
S
</mi>
</mrow>
</msub>
</math>
</em>
is defined as an object
<a class="nnexus_concept" href="http://planetmath.org/grouprepresentation">
representation
</a>
in the following non-commutative
diagram of systems and dynamic system
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
morphisms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Morphism.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/category">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
or ‘dynamic
<a class="nnexus_concept" href="http://planetmath.org/functorialmorphism">
transformations
</a>
’:
</p>
</div>
<div class="ltx_para" id="S0.Thmdefinition1.p2">
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\xymatrix@C=5pc{[SUPER-COMPLEX]\ar[r]^{(\textbf{Higher
Dim})}\ar[d]_{\Lambda}&~{}~{}~{}(U_{CS}=ULTRA-COMPLEX)\ar[d]^{onto}\\
COMPLEX&\ar[l]^{(\textbf{Generic Map})}[SIMPLE]}" class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\xymatrix
</mtext>
</merror>
<mi mathvariant="normal">
@
</mi>
<mi>
C
</mi>
<mo>
=
</mo>
<mn>
5
</mn>
<mi>
p
</mi>
<mi>
c
</mi>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
S
</mi>
<mi>
U
</mi>
<mi>
P
</mi>
<mi>
E
</mi>
<mi>
R
</mi>
<mo>
-
</mo>
<mi>
C
</mi>
<mi>
O
</mi>
<mi>
M
</mi>
<mi>
P
</mi>
<mi>
L
</mi>
<mi>
E
</mi>
<mi>
X
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\ar
</mtext>
</merror>
<msup>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
r
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mtext mathvariant="bold">
Higher
Dim
</mtext>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\ar
</mtext>
</merror>
<msub>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
d
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
<mi mathvariant="normal">
Λ
</mi>
</msub>
<mpadded width="+9.9pt">
<mi mathvariant="normal">
&
</mi>
</mpadded>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
U
</mi>
<mrow>
<mi>
C
</mi>
<mo>
</mo>
<mi>
S
</mi>
</mrow>
</msub>
<mo>
=
</mo>
<mi>
U
</mi>
<mi>
L
</mi>
<mi>
T
</mi>
<mi>
R
</mi>
<mi>
A
</mi>
<mo>
-
</mo>
<mi>
C
</mi>
<mi>
O
</mi>
<mi>
M
</mi>
<mi>
P
</mi>
<mi>
L
</mi>
<mi>
E
</mi>
<mi>
X
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\ar
</mtext>
</merror>
<msup>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
d
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mi>
o
</mi>
<mo>
</mo>
<mi>
n
</mi>
<mo>
</mo>
<mi>
t
</mi>
<mo>
</mo>
<mi>
o
</mi>
</mrow>
</msup>
<mi>
C
</mi>
<mi>
O
</mi>
<mi>
M
</mi>
<mi>
P
</mi>
<mi>
L
</mi>
<mi>
E
</mi>
<mi>
X
</mi>
<mi mathvariant="normal">
&
</mi>
<merror class="ltx_ERROR undefined undefined">
<mtext>
\ar
</mtext>
</merror>
<msup>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
l
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mtext mathvariant="bold">
Generic Map
</mtext>
<mo stretchy="false">
)
</mo>
</mrow>
</msup>
<mrow>
<mo stretchy="false">
[
</mo>
<mi>
S
</mi>
<mi>
I
</mi>
<mi>
M
</mi>
<mi>
P
</mi>
<mi>
L
</mi>
<mi>
E
</mi>
<mo stretchy="false">
]
</mo>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
</div>
<div class="ltx_para" id="S0.SS1.SSS1.p2">
<p class="ltx_p">
Note that the above diagram is indeed not ‘natural’ (i.e. it is not
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
commutative
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/abeliangroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/commutativesemigroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
) for reasons
related to the emergence of the higher dimensions of the super–complex
(biological/organismic) and/or ultra–complex (psychological/neural network dynamic) levels in comparison with
the low dimensions of either simple (physical/classical) or complex (chaotic) dynamic systems.
</p>
</div>
<div class="ltx_para ltx_align_right" id="S0.SS1.SSS1.p3">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_t">
Title
</td>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
ultra-complex systems
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Canonical name
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
UltracomplexSystems
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
Date of creation
</td>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:11:40
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l">
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0 | ErrorsCanCancelEachOtherOut | http://planetmath.org/ErrorsCanCancelEachOtherOut | <!DOCTYPE html>
<html>
<head>
<title>
errors can cancel each other out
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
errors can cancel each other out
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
If one uses the
<span class="ltx_text ltx_font_typewriter">
http://planetmath.org/ChangeOfVariableInDefiniteIntegral
</span>
change of variable
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx1">
<tbody id="S0.E1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle\tan{x}\;:=\;t,\quad dx\;=\;\frac{dt}{1\!+\!t^{2}},\quad\cos^{2}x%
\;=\;\frac{1}{1\!+\!t^{2}}" class="ltx_Math" display="inline" id="S0.E1.m1">
<mrow>
<mrow>
<mrow>
<mi>
tan
</mi>
<mo>
</mo>
<mpadded width="+2.8pt">
<mi>
x
</mi>
</mpadded>
</mrow>
<mo rspace="5.3pt">
:=
</mo>
<mi>
t
</mi>
</mrow>
<mo rspace="12.5pt">
,
</mo>
<mrow>
<mrow>
<mrow>
<mi>
d
</mi>
<mo>
</mo>
<mpadded width="+2.8pt">
<mi>
x
</mi>
</mpadded>
</mrow>
<mo rspace="5.3pt">
=
</mo>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<mi>
d
</mi>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
<mrow>
<mpadded width="-1.7pt">
<mn>
1
</mn>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mfrac>
</mstyle>
</mrow>
<mo rspace="12.5pt">
,
</mo>
<mrow>
<mrow>
<msup>
<mi>
cos
</mi>
<mn>
2
</mn>
</msup>
<mo>
</mo>
<mpadded width="+2.8pt">
<mi>
x
</mi>
</mpadded>
</mrow>
<mo rspace="5.3pt">
=
</mo>
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<mrow>
<mpadded width="-1.7pt">
<mn>
1
</mn>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mrow>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(1)
</span>
</td>
</tr>
</tbody>
</table>
<p class="ltx_p">
for finding the value of the
<a class="nnexus_concept" href="http://planetmath.org/definiteintegral">
definite integral
</a>
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="I\;:=\;\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac{dx}{2\cos^{2}x+1}," class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mpadded width="+2.8pt">
<mi>
I
</mi>
</mpadded>
<mo rspace="5.3pt">
:=
</mo>
<mrow>
<msubsup>
<mo largeop="true" symmetric="true">
∫
</mo>
<mfrac>
<mi>
π
</mi>
<mn>
4
</mn>
</mfrac>
<mfrac>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
<mn>
4
</mn>
</mfrac>
</msubsup>
<mfrac>
<mrow>
<mi>
d
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mrow>
<msup>
<mi>
cos
</mi>
<mn>
2
</mn>
</msup>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
the following calculation looks appropriate and faultless:
</p>
<table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S0.EGx2">
<tbody id="S0.E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_td ltx_align_right ltx_eqn_cell">
<math alttext="\displaystyle I\;=\;\int_{1}^{-1}\!\!\frac{dt}{3\!+\!t^{2}}\;=\;\frac{1}{\sqrt%
{3}}\!\!\operatornamewithlimits{\Big{/}}_{\!\!\!1}^{\,\;\quad-1}\!\arctan\frac%
{t}{\sqrt{3}}\;=\;\frac{1}{\sqrt{3}}\!\left(\frac{5\pi}{6}-\frac{\pi}{6}\right%
)\;=\;\frac{2\pi}{3\sqrt{3}}" class="ltx_Math" display="inline" id="S0.E2.m1">
<mrow>
<mpadded width="+2.8pt">
<mi>
I
</mi>
</mpadded>
<mo rspace="5.3pt">
=
</mo>
<mrow>
<mpadded width="-3.3pt">
<mstyle displaystyle="true">
<msubsup>
<mo largeop="true" symmetric="true">
∫
</mo>
<mn>
1
</mn>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</msubsup>
</mstyle>
</mpadded>
<mpadded width="+2.8pt">
<mstyle displaystyle="true">
<mfrac>
<mrow>
<mi>
d
</mi>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
<mrow>
<mpadded width="-1.7pt">
<mn>
3
</mn>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mfrac>
</mstyle>
</mpadded>
</mrow>
<mo rspace="5.3pt">
=
</mo>
<mrow>
<mpadded width="-3.3pt">
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<msqrt>
<mn>
3
</mn>
</msqrt>
</mfrac>
</mstyle>
</mpadded>
<mo>
</mo>
<mrow>
<mrow>
<mpadded width="-1.7pt">
<munderover>
<mo mathsize="160%" movablelimits="false" stretchy="false">
/
</mo>
<mpadded lspace="-5pt" width="-5pt">
<mn>
1
</mn>
</mpadded>
<mrow>
<mo lspace="16.9pt">
-
</mo>
<mn>
1
</mn>
</mrow>
</munderover>
</mpadded>
<mo>
</mo>
<mi>
arctan
</mi>
</mrow>
<mo>
</mo>
<mpadded width="+2.8pt">
<mstyle displaystyle="true">
<mfrac>
<mi>
t
</mi>
<msqrt>
<mn>
3
</mn>
</msqrt>
</mfrac>
</mstyle>
</mpadded>
</mrow>
</mrow>
<mo rspace="5.3pt">
=
</mo>
<mrow>
<mpadded width="-1.7pt">
<mstyle displaystyle="true">
<mfrac>
<mn>
1
</mn>
<msqrt>
<mn>
3
</mn>
</msqrt>
</mfrac>
</mstyle>
</mpadded>
<mo>
</mo>
<mrow>
<mo>
(
</mo>
<mrow>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<mn>
5
</mn>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
<mn>
6
</mn>
</mfrac>
</mstyle>
<mo>
-
</mo>
<mstyle displaystyle="true">
<mfrac>
<mi>
π
</mi>
<mn>
6
</mn>
</mfrac>
</mstyle>
</mrow>
<mo rspace="5.3pt">
)
</mo>
</mrow>
</mrow>
<mo rspace="5.3pt">
=
</mo>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<msqrt>
<mn>
3
</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">
<span class="ltx_tag ltx_tag_equation ltx_align_right">
(2)
</span>
</td>
</tr>
</tbody>
</table>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
The result is quite . Unfortunately, the calculation two errors, the effects of which cancel each other out.
<br class="ltx_break"/>
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
The crucial error in (2) is using the substitution (1) when
<math alttext="\tan{x}" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mi>
tan
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</math>
is
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
discontinuous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Discontinuous.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/discontinuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in the point
<math alttext="x=\frac{\pi}{2}" class="ltx_Math" display="inline" id="p3.m2">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mfrac>
<mi>
π
</mi>
<mn>
2
</mn>
</mfrac>
</mrow>
</math>
on the
<a class="nnexus_concept" href="http://planetmath.org/interval">
interval
</a>
<math alttext="[\frac{\pi}{4},\,\frac{3\pi}{4}]" class="ltx_Math" display="inline" id="p3.m3">
<mrow>
<mo stretchy="false">
[
</mo>
<mfrac>
<mi>
π
</mi>
<mn>
4
</mn>
</mfrac>
<mo rspace="4.2pt">
,
</mo>
<mfrac>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
<mn>
4
</mn>
</mfrac>
<mo stretchy="false">
]
</mo>
</mrow>
</math>
of integration. The error is however canceled out by the second error using the value
<math alttext="\frac{5\pi}{6}" class="ltx_Math" display="inline" id="p3.m4">
<mfrac>
<mrow>
<mn>
5
</mn>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
<mn>
6
</mn>
</mfrac>
</math>
for
<math alttext="\arctan\frac{-1}{\sqrt{3}}" class="ltx_Math" display="inline" id="p3.m5">
<mrow>
<mi>
arctan
</mi>
<mo>
</mo>
<mfrac>
<mrow>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
<msqrt>
<mn>
3
</mn>
</msqrt>
</mfrac>
</mrow>
</math>
, when the right value were
<math alttext="-\frac{\pi}{6}" class="ltx_Math" display="inline" id="p3.m6">
<mrow>
<mo>
-
</mo>
<mfrac>
<mi>
π
</mi>
<mn>
6
</mn>
</mfrac>
</mrow>
</math>
(the values of arctan lie only between
<math alttext="-\frac{\pi}{2}" class="ltx_Math" display="inline" id="p3.m7">
<mrow>
<mo>
-
</mo>
<mfrac>
<mi>
π
</mi>
<mn>
2
</mn>
</mfrac>
</mrow>
</math>
and
<math alttext="\frac{\pi}{2}" class="ltx_Math" display="inline" id="p3.m8">
<mfrac>
<mi>
π
</mi>
<mn>
2
</mn>
</mfrac>
</math>
; see
<a class="nnexus_concept" href="http://planetmath.org/cyclometricfunctions">
cyclometric functions
</a>
). The value
<math alttext="\frac{5\pi}{6}" class="ltx_Math" display="inline" id="p3.m9">
<mfrac>
<mrow>
<mn>
5
</mn>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
<mn>
6
</mn>
</mfrac>
</math>
belongs to a different branch of the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/InverseTangent.html">
inverse tangent
</a>
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/function">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
than
<math alttext="\frac{\pi}{6}" class="ltx_Math" display="inline" id="p3.m10">
<mfrac>
<mi>
π
</mi>
<mn>
6
</mn>
</mfrac>
</math>
; parts of two distinct branches cannot together form the
<a class="nnexus_concept" href="http://planetmath.org/antiderivative">
antiderivative
</a>
which must be
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
continuous
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Continuous.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
<br class="ltx_break"/>
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
What were a right way to calculate
<math alttext="I" class="ltx_Math" display="inline" id="p4.m1">
<mi>
I
</mi>
</math>
? The
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
universal trigonometric substitution
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/weierstrasssubstitutionformulas">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integrationofrationalfunctionofsineandcosine">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
produces an awkward
<a class="nnexus_concept" href="http://planetmath.org/integralsign">
integrand
</a>
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\frac{2\!+\!2t^{2}}{3\!-\!2t^{2}\!+\!3t^{4}}" class="ltx_Math" display="block" id="S0.Ex2.m1">
<mfrac>
<mrow>
<mpadded width="-1.7pt">
<mn>
2
</mn>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<mpadded width="-1.7pt">
<mn>
3
</mn>
</mpadded>
<mo rspace="0.8pt">
-
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mpadded width="-1.7pt">
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mpadded>
</mrow>
</mrow>
<mo rspace="0.8pt">
+
</mo>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<msup>
<mi>
t
</mi>
<mn>
4
</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
and
<math alttext="\sqrt{2}-1" class="ltx_Math" display="inline" id="p4.m2">
<mrow>
<msqrt>
<mn>
2
</mn>
</msqrt>
<mo>
-
</mo>
<mn>
1
</mn>
</mrow>
</math>
and
<math alttext="1-\sqrt{2}" class="ltx_Math" display="inline" id="p4.m3">
<mrow>
<mn>
1
</mn>
<mo>
-
</mo>
<msqrt>
<mn>
2
</mn>
</msqrt>
</mrow>
</math>
, therefore it is unusable. It is now better to change the interval of integration, using the
<a class="nnexus_concept" href="http://planetmath.org/property">
properties
</a>
of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
trigonometric functions
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/4#PT3">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TrigonometricFunctions.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/definitionsintrigonometry">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
Since the (graph of) cosine
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Squared.html">
squared
</a>
is
<a class="nnexus_concept" href="http://planetmath.org/symmetry">
symmetric about
</a>
the line
<math alttext="x=\frac{\pi}{2}" class="ltx_Math" display="inline" id="p5.m1">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mfrac>
<mi>
π
</mi>
<mn>
2
</mn>
</mfrac>
</mrow>
</math>
, we could integrate only over
<math alttext="[\frac{\pi}{4},\,\frac{\pi}{2}]" class="ltx_Math" display="inline" id="p5.m2">
<mrow>
<mo stretchy="false">
[
</mo>
<mfrac>
<mi>
π
</mi>
<mn>
4
</mn>
</mfrac>
<mo rspace="4.2pt">
,
</mo>
<mfrac>
<mi>
π
</mi>
<mn>
2
</mn>
</mfrac>
<mo stretchy="false">
]
</mo>
</mrow>
</math>
and multiply the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
integral
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://dlmf.nist.gov/1.4#iv">
<img alt="Dlmf" src="http://dlmf.nist.gov/style/DLMF-16.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lebesgueintegral">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
by 2 (cf.
<a class="nnexus_concept" href="http://planetmath.org/integralsofevenandoddfunctions">
integral of even and odd functions
</a>
):
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="I\;=\;2\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{dx}{2\cos^{2}x+1}." class="ltx_Math" display="block" id="S0.Ex3.m1">
<mrow>
<mrow>
<mpadded width="+2.8pt">
<mi>
I
</mi>
</mpadded>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mrow>
<msubsup>
<mo largeop="true" symmetric="true">
∫
</mo>
<mfrac>
<mi>
π
</mi>
<mn>
4
</mn>
</mfrac>
<mfrac>
<mi>
π
</mi>
<mn>
2
</mn>
</mfrac>
</msubsup>
<mfrac>
<mrow>
<mi>
d
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mrow>
<msup>
<mi>
cos
</mi>
<mn>
2
</mn>
</msup>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</mfrac>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
We can also get rid of the inconvenient
<a class="nnexus_concept" href="http://mathworld.wolfram.com/UpperLimit.html">
upper limit
</a>
<math alttext="\frac{\pi}{2}" class="ltx_Math" display="inline" id="p5.m3">
<mfrac>
<mi>
π
</mi>
<mn>
2
</mn>
</mfrac>
</math>
by changing over to the sine in virtue of the
<a class="nnexus_concept" href="http://planetmath.org/goniometricformulas">
complement formula
</a>
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="\cos(\frac{\pi}{2}\!-\!x)\;=\;\sin{x}," class="ltx_Math" display="block" id="S0.Ex4.m1">
<mrow>
<mrow>
<mrow>
<mi>
cos
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mpadded width="-1.7pt">
<mfrac>
<mi>
π
</mi>
<mn>
2
</mn>
</mfrac>
</mpadded>
<mo rspace="0.8pt">
-
</mo>
<mi>
x
</mi>
</mrow>
<mo rspace="5.3pt" stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo rspace="5.3pt">
=
</mo>
<mrow>
<mi>
sin
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo>
,
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
getting
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="I\;=\;2\int_{0}^{\frac{\pi}{4}}\frac{dx}{2\sin^{2}x+1}." class="ltx_Math" display="block" id="S0.Ex5.m1">
<mrow>
<mrow>
<mpadded width="+2.8pt">
<mi>
I
</mi>
</mpadded>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mrow>
<msubsup>
<mo largeop="true" symmetric="true">
∫
</mo>
<mn>
0
</mn>
<mfrac>
<mi>
π
</mi>
<mn>
4
</mn>
</mfrac>
</msubsup>
<mfrac>
<mrow>
<mi>
d
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mrow>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mrow>
<msup>
<mi>
sin
</mi>
<mn>
2
</mn>
</msup>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
</mfrac>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
Then (1) is usable, and because
<math alttext="\sin^{2}x=\frac{t^{2}}{1\!+\!t^{2}}" class="ltx_Math" display="inline" id="p5.m4">
<mrow>
<mrow>
<msup>
<mi>
sin
</mi>
<mn>
2
</mn>
</msup>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
=
</mo>
<mfrac>
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
<mrow>
<mpadded width="-1.7pt">
<mn>
1
</mn>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mfrac>
</mrow>
</math>
, we obtain
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="I\;=\;2\int_{0}^{1}\frac{dt}{\left(\frac{2t^{2}}{1\!+\!t^{2}}+1\right)(1\!+\!t%
^{2})}\;=\;2\int_{0}^{1}\frac{dt}{3t^{2}\!+\!1}\;=\;\frac{2}{\sqrt{3}}\!\!%
\operatornamewithlimits{\Big{/}}_{\!\!\!0}^{\,\;\quad 1}\!\arctan{t\sqrt{3}}\;%
=\;\frac{2\pi}{3\sqrt{3}}." class="ltx_Math" display="block" id="S0.Ex6.m1">
<mrow>
<mrow>
<mpadded width="+2.8pt">
<mi>
I
</mi>
</mpadded>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mrow>
<msubsup>
<mo largeop="true" symmetric="true">
∫
</mo>
<mn>
0
</mn>
<mn>
1
</mn>
</msubsup>
<mpadded width="+2.8pt">
<mfrac>
<mrow>
<mi>
d
</mi>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
<mrow>
<mrow>
<mo>
(
</mo>
<mrow>
<mfrac>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mrow>
<mpadded width="-1.7pt">
<mn>
1
</mn>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mrow>
</mfrac>
<mo>
+
</mo>
<mn>
1
</mn>
</mrow>
<mo>
)
</mo>
</mrow>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<mpadded width="-1.7pt">
<mn>
1
</mn>
</mpadded>
<mo rspace="0.8pt">
+
</mo>
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</mfrac>
</mpadded>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mrow>
<msubsup>
<mo largeop="true" symmetric="true">
∫
</mo>
<mn>
0
</mn>
<mn>
1
</mn>
</msubsup>
<mpadded width="+2.8pt">
<mfrac>
<mrow>
<mi>
d
</mi>
<mo>
</mo>
<mi>
t
</mi>
</mrow>
<mrow>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mpadded width="-1.7pt">
<msup>
<mi>
t
</mi>
<mn>
2
</mn>
</msup>
</mpadded>
</mrow>
<mo rspace="0.8pt">
+
</mo>
<mn>
1
</mn>
</mrow>
</mfrac>
</mpadded>
</mrow>
</mrow>
<mo rspace="5.3pt">
=
</mo>
<mrow>
<mpadded width="-3.3pt">
<mfrac>
<mn>
2
</mn>
<msqrt>
<mn>
3
</mn>
</msqrt>
</mfrac>
</mpadded>
<mo>
</mo>
<mrow>
<mrow>
<mpadded width="-1.7pt">
<munderover>
<mo mathsize="160%" movablelimits="false" stretchy="false">
/
</mo>
<mpadded lspace="-5pt" width="-5pt">
<mn>
0
</mn>
</mpadded>
<mpadded lspace="14.4pt" width="+14.4pt">
<mn>
1
</mn>
</mpadded>
</munderover>
</mpadded>
<mo>
</mo>
<mi>
arctan
</mi>
</mrow>
<mo>
</mo>
<mrow>
<mi>
t
</mi>
<mo>
</mo>
<mpadded width="+2.8pt">
<msqrt>
<mn>
3
</mn>
</msqrt>
</mpadded>
</mrow>
</mrow>
</mrow>
<mo rspace="5.3pt">
=
</mo>
<mfrac>
<mrow>
<mn>
2
</mn>
<mo>
</mo>
<mi>
π
</mi>
</mrow>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<msqrt>
<mn>
3
</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
</div>
<div class="ltx_para ltx_align_right" id="p6">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
<a class="nnexus_concept" href="http://planetmath.org/errorscancanceleachotherout">
errors can cancel each other out
</a>
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
ErrorsCanCancelEachOtherOut
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:59:39
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 18:59:39
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
13
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
pahio (2872)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Example
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A35
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 26A06
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 97D70
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
UniversalTrigonometricSubstitution
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
SubstitutionNotation
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Related topic
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
IntegrationOfRationalFunctionOfSineAndCosine
</td>
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0 | Needleinthehaystack | http://planetmath.org/Needleinthehaystack | <!DOCTYPE html>
<html>
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<title>
needle-in-the-haystack
</title>
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<div class="ltx_page_main">
<div class="ltx_page_content">
<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
needle-in-the-haystack
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
A common problem in mathematics is to prove the
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Existence.html">
existence
</a>
and uniqueness of a solution,
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Element.html">
element
</a>
, object in a category, etc. The typical approach for such a proof is to establish the existence of the solution, then determine the uniqueness. In
<a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">
instances
</a>
like this, uniqueness is often proved by assuming there are two solutions and concluding they are indeed equal.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
Needle-in-a-haystack Heuristic
</span>
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
An alternative and often more constructive approach is to prove uniqueness first, then prove existence – we call this the
<em class="ltx_emph ltx_font_italic">
<a class="nnexus_concept" href="http://planetmath.org/needleinthehaystack">
needle-in-the-haystack
</a>
</em>
heuristic because it is much easier to find a needle in a haystack when you first prove you know where it will be if it is there at all.
<span class="ltx_note ltx_role_footnote">
<sup class="ltx_note_mark">
1
</sup>
<span class="ltx_note_outer">
<span class="ltx_note_content">
<sup class="ltx_note_mark">
1
</sup>
This terminology is due to F. R. Beyl
</span>
</span>
</span>
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
We give a basic example from elementary mathematics.
</p>
</div>
<div class="ltx_theorem ltx_theorem_prop" id="Thmthm1">
<h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem">
Proposition 1
</span>
.
</h6>
<div class="ltx_para" id="Thmthm1.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
Two distinct non-parallel lines in a the
<math alttext="xy" class="ltx_Math" display="inline" id="Thmthm1.p1.m1">
<mrow>
<mi>
x
</mi>
<mo mathvariant="italic">
</mo>
<mi>
y
</mi>
</mrow>
</math>
-plane
<a class="nnexus_concept" href="http://planetmath.org/intersection">
intersect
</a>
at exactly one point.
</span>
</p>
</div>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
We pause to observe this is an existence and uniqueness question: a point of
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Intersection.html">
intersection
</a>
must exist, and it must be unique. Because we care only about
illustrating the heuristic we assume our meaning of lines excludes vertical
lines, so each line is given by a
<a class="nnexus_concept" href="http://planetmath.org/function">
function
</a>
of the form
<math alttext="y=mx+b" class="ltx_Math" display="inline" id="p5.m1">
<mrow>
<mi>
y
</mi>
<mo>
=
</mo>
<mrow>
<mrow>
<mi>
m
</mi>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<mi>
b
</mi>
</mrow>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_proof">
<h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">
Proof.
</h6>
<div class="ltx_para" id="p6">
<p class="ltx_p">
Let
<math alttext="y_{1}=m_{1}x+b_{1}" class="ltx_Math" display="inline" id="p6.m1">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
</mrow>
</math>
and
<math alttext="y_{2}=m_{2}x+b_{2}" class="ltx_Math" display="inline" id="p6.m2">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
2
</mn>
</msub>
<mo>
=
</mo>
<mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mrow>
</math>
be distinct lines which are non-parallel; hence,
<math alttext="m_{1}\neq m_{2}" class="ltx_Math" display="inline" id="p6.m3">
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
≠
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</math>
. If
<math alttext="y_{1}=y_{2}" class="ltx_Math" display="inline" id="p6.m4">
<mrow>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo>
=
</mo>
<msub>
<mi>
y
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</math>
for some
<math alttext="x" class="ltx_Math" display="inline" id="p6.m5">
<mi>
x
</mi>
</math>
then
<math alttext="m_{1}x+b_{1}=m_{2}x+b_{2}" class="ltx_Math" display="inline" id="p6.m6">
<mrow>
<mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
+
</mo>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mrow>
</math>
so
<math alttext="(m_{1}-m_{2})x=b_{2}-b_{1}" class="ltx_Math" display="inline" id="p6.m7">
<mrow>
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
<mo>
=
</mo>
<mrow>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
</mrow>
</math>
.
This leads to
<math alttext="\displaystyle x=\frac{b_{2}-b_{1}}{m_{1}-m_{2}}" class="ltx_Math" display="inline" id="p6.m8">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mfrac>
</mstyle>
</mrow>
</math>
.
Thus we have uniquely characterized any intersection point as the point
<math alttext="\displaystyle(x,y_{1}(x))" class="ltx_Math" display="inline" id="p6.m9">
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo>
,
</mo>
<mrow>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
, with
<math alttext="\displaystyle x=\frac{b_{2}-b_{1}}{m_{1}-m_{2}}" class="ltx_Math" display="inline" id="p6.m10">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mfrac>
</mstyle>
</mrow>
</math>
,
if an intersection exists.
</p>
</div>
<div class="ltx_para" id="p7">
<p class="ltx_p">
To see that an intersection exists we appeal to the unique
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
characterization
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Characterization.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/characterisation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
<table class="ltx_equation ltx_eqn_table" id="S0.Ex1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft">
</td>
<td class="ltx_eqn_cell ltx_align_center">
<math alttext="y_{1}\left(\frac{b_{2}-b_{1}}{m_{1}-m_{2}}\right)=m_{1}\frac{b_{2}-b_{1}}{m_{1%
}-m_{2}}+b_{1}=\frac{m_{1}b_{2}-m_{2}b_{1}}{m_{1}-m_{2}}=m_{2}\frac{b_{2}-b_{1%
}}{m_{1}-m_{2}}+b_{2}=y_{2}\left(\frac{b_{2}-b_{1}}{m_{1}-m_{2}}\right)." class="ltx_Math" display="block" id="S0.Ex1.m1">
<mrow>
<mrow>
<mrow>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<mrow>
<mo>
(
</mo>
<mfrac>
<mrow>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mfrac>
<mo>
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<mfrac>
<mrow>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>
+
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mo>
=
</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
</mrow>
<mo>
-
</mo>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
<mo>
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
</mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mfrac>
<mo>
=
</mo>
<mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
<mo>
</mo>
<mfrac>
<mrow>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>
+
</mo>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
</mrow>
<mo>
=
</mo>
<mrow>
<msub>
<mi>
y
</mi>
<mn>
2
</mn>
</msub>
<mo>
</mo>
<mrow>
<mo>
(
</mo>
<mfrac>
<mrow>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mfrac>
<mo>
)
</mo>
</mrow>
</mrow>
</mrow>
<mo>
.
</mo>
</mrow>
</math>
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright">
</td>
</tr>
</table>
<p class="ltx_p">
∎
</p>
</div>
</div>
<div class="ltx_theorem ltx_theorem_remark" id="Thmthm2">
<h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem">
Remark 2
</span>
.
</h6>
<div class="ltx_para" id="Thmthm2.p1">
<p class="ltx_p">
<span class="ltx_text ltx_font_italic">
After establishing a unique characterization of a solution it is often so convincing as to leave no need for the verification of the solution, so the existence stage of the proof may be omitted. [Indeed, a standard pre-calculus course would not hesitate to stop the proof at end of the uniqueness stage.]
</span>
</p>
</div>
</div>
<div class="ltx_para" id="p8">
<p class="ltx_p">
If we chose to write the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
theorem
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/lemma">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
in the traditional manner and introduce the existence first, then we would have begun with a rather unpredictable beginning:
</p>
<blockquote class="ltx_quote">
<p class="ltx_p">
Let
<math alttext="\displaystyle x=\frac{b_{2}-b_{1}}{m_{1}-m_{2}}" class="ltx_Math" display="inline" id="p8.m1">
<mrow>
<mi>
x
</mi>
<mo>
=
</mo>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<msub>
<mi>
b
</mi>
<mn>
2
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
b
</mi>
<mn>
1
</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>
m
</mi>
<mn>
1
</mn>
</msub>
<mo>
-
</mo>
<msub>
<mi>
m
</mi>
<mn>
2
</mn>
</msub>
</mrow>
</mfrac>
</mstyle>
</mrow>
</math>
.
</p>
</blockquote>
<p class="ltx_p">
Such an approach is no less logical but can leave a reader confused as to the reason these choices are being made.
</p>
</div>
<div class="ltx_para" id="p9">
<p class="ltx_p">
Furthermore, to prove the uniqueness of a solution by means of assuming there are two solutions can be a misleading approach for some problems. For example, the intersection of
<math alttext="f(x)=x^{2}-8x" class="ltx_Math" display="inline" id="p9.m1">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<msup>
<mi>
x
</mi>
<mn>
2
</mn>
</msup>
<mo>
-
</mo>
<mrow>
<mn>
8
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
</mrow>
</math>
with
<math alttext="g(x)=-3x-4" class="ltx_Math" display="inline" id="p9.m2">
<mrow>
<mrow>
<mi>
g
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
x
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
=
</mo>
<mrow>
<mrow>
<mo>
-
</mo>
<mrow>
<mn>
3
</mn>
<mo>
</mo>
<mi>
x
</mi>
</mrow>
</mrow>
<mo>
-
</mo>
<mn>
4
</mn>
</mrow>
</mrow>
</math>
is unique, but given two solutions
<math alttext="(x_{1},y_{1})" class="ltx_Math" display="inline" id="p9.m3">
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
1
</mn>
</msub>
<mo>
,
</mo>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
,
<math alttext="(x_{2},y_{2})" class="ltx_Math" display="inline" id="p9.m4">
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
2
</mn>
</msub>
<mo>
,
</mo>
<msub>
<mi>
y
</mi>
<mn>
2
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
, it becomes rather difficult to conclude
<math alttext="(x_{1},y_{1})=(x_{2},y_{2})" class="ltx_Math" display="inline" id="p9.m5">
<mrow>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
1
</mn>
</msub>
<mo>
,
</mo>
<msub>
<mi>
y
</mi>
<mn>
1
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
<mo>
=
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<msub>
<mi>
x
</mi>
<mn>
2
</mn>
</msub>
<mo>
,
</mo>
<msub>
<mi>
y
</mi>
<mn>
2
</mn>
</msub>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
directly. With the needle-in-the-haystack heuristic one begins the proof by showing the only solution is
<math alttext="(4,-16)" class="ltx_Math" display="inline" id="p9.m6">
<mrow>
<mo stretchy="false">
(
</mo>
<mn>
4
</mn>
<mo>
,
</mo>
<mrow>
<mo>
-
</mo>
<mn>
16
</mn>
</mrow>
<mo stretchy="false">
)
</mo>
</mrow>
</math>
, if there is a solution. Thus we never compare two systems of equations in several
<a class="nnexus_concept" href="http://planetmath.org/variable">
variables
</a>
.
</p>
</div>
<div class="ltx_para" id="p10">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
When to use the heuristic
</span>
</p>
</div>
<div class="ltx_para" id="p11">
<p class="ltx_p">
When an existence and uniqueness is claimed where there are accompanying equations or
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
structures
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Structure.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structure">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
that must be satisfied by the solution, it is often possible to apply the heuristic. Often such problems can be worded as two sets intersecting at a single point. If the possible point of intersection is characterized before we know the intersection is non-empty, then we have an immediate candidate with which we may prove the intersection is non-empty.
</p>
</div>
<div class="ltx_para" id="p12">
<p class="ltx_p">
<span class="ltx_text ltx_font_bold">
When not to use the heuristic
</span>
</p>
</div>
<div class="ltx_para" id="p13">
<p class="ltx_p">
There are, however, many examples where the traditional approach of establishing existence first is preferable. For instance, in a proof that every
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">
integer
</a>
has a
<a class="nnexus_concept" href="http://planetmath.org/ufd">
unique factorization
</a>
, establishing some factorization exists is nearly
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Trivial.html">
trivial
</a>
: choose a prime
<math alttext="p" class="ltx_Math" display="inline" id="p13.m1">
<mi>
p
</mi>
</math>
which divides
<math alttext="n" class="ltx_Math" display="inline" id="p13.m2">
<mi>
n
</mi>
</math>
, then by
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
induction
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Induction.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/induction">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
choose a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
prime factorization
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimeFactorization.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/integerfactorization">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
<math alttext="q_{1}\cdots q_{s}" class="ltx_Math" display="inline" id="p13.m3">
<mrow>
<msub>
<mi>
q
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
</mo>
<msub>
<mi>
q
</mi>
<mi>
s
</mi>
</msub>
</mrow>
</math>
of
<math alttext="n/p" class="ltx_Math" display="inline" id="p13.m4">
<mrow>
<mi>
n
</mi>
<mo>
/
</mo>
<mi>
p
</mi>
</mrow>
</math>
, and then
<math alttext="n=pq_{1}\cdots q_{s}" class="ltx_Math" display="inline" id="p13.m5">
<mrow>
<mi>
n
</mi>
<mo>
=
</mo>
<mrow>
<mi>
p
</mi>
<mo>
</mo>
<msub>
<mi>
q
</mi>
<mn>
1
</mn>
</msub>
<mo>
</mo>
<mi mathvariant="normal">
⋯
</mi>
<mo>
</mo>
<msub>
<mi>
q
</mi>
<mi>
s
</mi>
</msub>
</mrow>
</mrow>
</math>
is a prime factorization of
<math alttext="n" class="ltx_Math" display="inline" id="p13.m6">
<mi>
n
</mi>
</math>
. However, establishing the uniqueness of a prime factorization is a far more delicate matter.
</p>
</div>
<div class="ltx_para" id="p14">
<p class="ltx_p">
The heuristic applies best when the
<a class="nnexus_concept" href="http://planetmath.org/definition">
definition
</a>
of the solution, element, or object in question is not immediately presentable from the definition. When a definition provides a blue-print for establishing existence then the traditional method may be preferable.
</p>
</div>
<div class="ltx_para ltx_align_right" id="p15">
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
needle-in-the-haystack
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
<a class="nnexus_concept" href="http://planetmath.org/canonical">
Canonical
</a>
name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Needleinthehaystack
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:07:34
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 16:07:34
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Algeboy (12884)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Algeboy (12884)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
10
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Algeboy (12884)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
msc 00A35
</td>
</tr>
</tbody>
</table>
</div>
</article>
</div>
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0 | PreservationAndReflection | http://planetmath.org/PreservationAndReflection | <!DOCTYPE html>
<html>
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<title>
preservation and reflection
</title>
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<article class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">
preservation and reflection
</h1>
<div class="ltx_para" id="p1">
<br class="ltx_break"/>
<p class="ltx_p">
In mathematics, the word “preserve” usually means the “preservation of properties”. Loosely speaking, whenever a mathematical
<a class="nnexus_concept" href="http://planetmath.org/concretecategory">
construct
</a>
<math alttext="A" class="ltx_Math" display="inline" id="p1.m1">
<mi>
A
</mi>
</math>
has some property
<math alttext="P" class="ltx_Math" display="inline" id="p1.m2">
<mi>
P
</mi>
</math>
, after
<math alttext="A" class="ltx_Math" display="inline" id="p1.m3">
<mi>
A
</mi>
</math>
is somehow “transformed” into
<math alttext="A^{\prime}" class="ltx_Math" display="inline" id="p1.m4">
<msup>
<mi>
A
</mi>
<mo>
′
</mo>
</msup>
</math>
, the transformed object
<math alttext="A^{\prime}" class="ltx_Math" display="inline" id="p1.m5">
<msup>
<mi>
A
</mi>
<mo>
′
</mo>
</msup>
</math>
also has property
<math alttext="P" class="ltx_Math" display="inline" id="p1.m6">
<mi>
P
</mi>
</math>
. The constructs usually refer to sets and the
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
transformations
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/functorialmorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/transformation">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
typically are
<a class="nnexus_concept" href="http://planetmath.org/function">
functions
</a>
or something
<a class="nnexus_concept" href="http://planetmath.org/equivalentmachines">
similar
</a>
.
</p>
</div>
<div class="ltx_para" id="p2">
<p class="ltx_p">
Here is a simple example, let
<math alttext="f" class="ltx_Math" display="inline" id="p2.m1">
<mi>
f
</mi>
</math>
be a function from a set
<math alttext="A" class="ltx_Math" display="inline" id="p2.m2">
<mi>
A
</mi>
</math>
to
<math alttext="B" class="ltx_Math" display="inline" id="p2.m3">
<mi>
B
</mi>
</math>
. Let
<math alttext="A" class="ltx_Math" display="inline" id="p2.m4">
<mi>
A
</mi>
</math>
be a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
finite set
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FiniteSet.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/finite">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. Let
<math alttext="P" class="ltx_Math" display="inline" id="p2.m5">
<mi>
P
</mi>
</math>
be the property of a set being finite. Then
<math alttext="f" class="ltx_Math" display="inline" id="p2.m6">
<mi>
f
</mi>
</math>
preserves
<math alttext="P" class="ltx_Math" display="inline" id="p2.m7">
<mi>
P
</mi>
</math>
, since
<math alttext="f(A)" class="ltx_Math" display="inline" id="p2.m8">
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
A
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is finite. Note that we are not saying that
<math alttext="B" class="ltx_Math" display="inline" id="p2.m9">
<mi>
B
</mi>
</math>
is finite. We are merely saying that the portion of
<math alttext="B" class="ltx_Math" display="inline" id="p2.m10">
<mi>
B
</mi>
</math>
that is the
<em class="ltx_emph ltx_font_italic">
image
</em>
of
<math alttext="A" class="ltx_Math" display="inline" id="p2.m11">
<mi>
A
</mi>
</math>
(the transformed portion) is finite.
</p>
</div>
<div class="ltx_para" id="p3">
<p class="ltx_p">
Here is another example. The property of being
<a class="nnexus_concept" href="http://planetmath.org/connectedspace">
connected
</a>
in a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
topological space
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/TopologicalSpace.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topologicalspace">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
is preserved under a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
continuous function
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://planetmath.org/classesofordinalsandenumeratingfunctions">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/continuous">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. Here, the constructs are topological spaces, and the transformation is a continuous function. In other words, if
<math alttext="f:X\to Y" class="ltx_Math" display="inline" id="p3.m1">
<mrow>
<mi>
f
</mi>
<mo>
:
</mo>
<mrow>
<mi>
X
</mi>
<mo>
→
</mo>
<mi>
Y
</mi>
</mrow>
</mrow>
</math>
is a continuous function from
<math alttext="X" class="ltx_Math" display="inline" id="p3.m2">
<mi>
X
</mi>
</math>
to
<math alttext="Y" class="ltx_Math" display="inline" id="p3.m3">
<mi>
Y
</mi>
</math>
. If
<math alttext="X" class="ltx_Math" display="inline" id="p3.m4">
<mi>
X
</mi>
</math>
is connected, so is
<math alttext="f(X)\subseteq Y" class="ltx_Math" display="inline" id="p3.m5">
<mrow>
<mrow>
<mi>
f
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
X
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
<mo>
⊆
</mo>
<mi>
Y
</mi>
</mrow>
</math>
.
</p>
</div>
<div class="ltx_para" id="p4">
<p class="ltx_p">
Many more examples can be found in
<a class="nnexus_concept" href="http://mathworld.wolfram.com/AbstractAlgebra.html">
abstract algebra
</a>
.
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
Group homomorphisms
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/GroupHomomorphism.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/grouphomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, for example, preserve
<a class="nnexus_concept" href="http://planetmath.org/commutative">
commutativity
</a>
, as well as the property of being
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
finitely generated
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FinitelyGenerated.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/subalgebraofanalgebraicsystem">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/finitelygeneratedgroup">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
.
</p>
</div>
<div class="ltx_para" id="p5">
<p class="ltx_p">
The word “reflect” is the dual notion of “preserve”. It means that if the transformed object has property
<math alttext="P" class="ltx_Math" display="inline" id="p5.m1">
<mi>
P
</mi>
</math>
, then the original object also has property
<math alttext="P" class="ltx_Math" display="inline" id="p5.m2">
<mi>
P
</mi>
</math>
. This usage is rarely found outside of
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
category theory
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/CategoryTheory.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/categorytheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/bibliographyinalgebraictopologycategoriesandqat">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/graphtheory">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/topicsinalgebraictopology">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
, and is almost exclusively reserved for
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
functors
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Functor.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/functor">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
. For example, a
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
faithful functor
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/FaithfulFunctor.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/faithfulfunctor">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
reflects
<a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">
isomorphism
</a>
<sup style="display: none;">
<a class="nnexus_concept" href="http://mathworld.wolfram.com/Isomorphism.html">
<img alt="Mathworld" src="http://mathworld.wolfram.com/favicon_mathworld.png"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homomorphismbetweenalgebraicsystems">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/structurehomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/homomorphismbetweenpartialalgebras">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/isomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/typesofmorphisms">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/semiautomatonhomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
<a class="nnexus_concept" href="http://planetmath.org/ringhomomorphism">
<img alt="Planetmath" src="http://planetmath.org/sites/default/files/fab-favicon.ico"/>
</a>
</sup>
: if
<math alttext="F" class="ltx_Math" display="inline" id="p5.m3">
<mi>
F
</mi>
</math>
is a faithful functor from
<math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="p5.m4">
<mi class="ltx_font_mathcaligraphic">
𝒞
</mi>
</math>
to
<math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="p5.m5">
<mi class="ltx_font_mathcaligraphic">
𝒟
</mi>
</math>
, and the object
<math alttext="F(A)" class="ltx_Math" display="inline" id="p5.m6">
<mrow>
<mi>
F
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
A
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
is isomorphic to the object
<math alttext="F(B)" class="ltx_Math" display="inline" id="p5.m7">
<mrow>
<mi>
F
</mi>
<mo>
</mo>
<mrow>
<mo stretchy="false">
(
</mo>
<mi>
B
</mi>
<mo stretchy="false">
)
</mo>
</mrow>
</mrow>
</math>
in
<math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="p5.m8">
<mi class="ltx_font_mathcaligraphic">
𝒟
</mi>
</math>
, then
<math alttext="A" class="ltx_Math" display="inline" id="p5.m9">
<mi>
A
</mi>
</math>
is isomorphic to
<math alttext="B" class="ltx_Math" display="inline" id="p5.m10">
<mi>
B
</mi>
</math>
in
<math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="p5.m11">
<mi class="ltx_font_mathcaligraphic">
𝒞
</mi>
</math>
.
</p>
<table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">
Title
</th>
<td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">
preservation and reflection
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Canonical name
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
PreservationAndReflection
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Date of creation
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:12:18
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified on
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
2013-03-22 17:12:18
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Owner
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Last modified by
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Numerical id
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
5
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Author
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
CWoo (3771)
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Entry type
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
Definition
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Classification
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
msc 00A35
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_r">
preserve
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">
Defines
</th>
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">
reflect
</td>
</tr>
</tbody>
</table>
<p class="ltx_p ltx_align_right">
</p>
</div>
</article>
</div>
<footer class="ltx_page_footer">
<div class="ltx_page_logo">
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