aarjaneiro/planetmath huggingface dataset:
integral sign

integral sign

The

integral sign

∫

is a stylised version of the

long s

letter.

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.

Gottfried Wilhelm Leibniz introduced the integral sign as the first letter s of the Latin word

summa

(‘sum’).  The long shape of

∫

may be thought to symbolically depict the fact that

http://planetmath.org/DefiniteIntegral

integral is a limiting case of sum.

A variant

∮

of the integral sign is used in integrals taken along a

closed curve

in

ℝ

2

or about a closed surface in

ℝ

3

; see e.g.

Cauchy integral theorem

,

derivation of heat equation

.

The

function

given after the integral sign, i.e. the function to be

integrated

, is the

integrand

.

Title

integral sign

Canonical name

IntegralSign

Date of creation

2013-03-22 18:04:00

Last modified on

2013-03-22 18:04:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 00A05

Classification

msc 00A06

Related topic

RiemannIntegral

Related topic

RiemannStieltjesIntegral

Related topic

Integral2

Defines

integrand

Defines

integrate

Generated on Tue Feb 6 22:18:14 2018 by

LaTeXML
introducing 0th power

introducing 0th power

Let

a

be a number not equal to zero. Then for all

n

∈

ℕ

, we have that

a

n

is the product of

a

with itself

n

. Using the fact that the

integer

1 is a

multiplicative identity

, (

a

⋅

1

=

a

for any

a

), we can write

a

n

⋅

1

=

a

n

=

a

n

+

0

=

a

n

⋅

a

0

,

where we have used the

properties

of

exponents

under

multiplication

. Now, after canceling a factor of

a

n

from both sides of the above

equation

, we derive that

a

0

=

1

for any non-zero number.

Title

introducing 0th power

Canonical

name

Introducing0thPower

Date of creation

2013-03-22 13:24:20

Last modified on

2013-03-22 13:24:20

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Topic

Classification

msc 00A05

Related topic

EmptyProduct

Generated on Tue Feb 6 22:18:15 2018 by

LaTeXML
Knuth’s up arrow notation

Knuth’s up arrow notation

Knuth’s up arrow noation

is a way of writing numbers which would be unwieldy in standard decimal notation. It expands on the

exponential

notation

m

↑

n

=

m

n

. Define

m

↑

↑

0

=

1

and

m

↑

↑

n

=

m

↑

(

m

↑

↑

[

n

-

1

]

)

.

Obviously

m

↑

↑

1

=

m

1

=

m

, so

3

↑

↑

2

=

3

3

⁣

↑

⁣

↑

1

=

3

3

=

27

, but

2

↑

↑

3

=

2

2

⁣

↑

⁣

↑

2

=

2

2

2

⁣

↑

⁣

↑

1

=

2

(

2

2

)

=

16

.

In general,

m

↑

↑

n

=

m

m

⋯

m

, a tower of height

n

.

Clearly, this process can be extended:

m

↑

↑

↑

0

=

1

and

m

↑

↑

↑

n

=

m

↑

↑

(

m

↑

↑

↑

[

n

-

1

]

)

.

An alternate notation is to write

m

(

i

)

⁢

n

for

m

⁢

↑

⋯

↑

⏟

i

-

2

⁢

times

⁢

n

. (

i

-

2

times because then

m

(

2

)

⁢

n

=

m

⋅

n

and

m

(

1

)

⁢

n

=

m

+

n

.) Then in general we can define

m

(

i

)

⁢

n

=

m

(

i

-

1

)

⁢

(

m

(

i

)

⁢

(

n

-

1

)

)

.

To get a sense of how quickly these numbers grow,

3

↑

↑

↑

2

=

3

↑

↑

3

is more than seven and a half trillion, and the numbers continue to grow much more than exponentially.

Title

Knuth’s up arrow notation

Canonical

name

KnuthsUpArrowNotation

Date of creation

2013-03-22 12:58:43

Last modified on

2013-03-22 12:58:43

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 00A05

Synonym

up-arrow

Synonym

up arrow

Synonym

up-arrow notation

Synonym

up arrow notation

Synonym

Knuth notation

Related topic

ConwaysChainedArrowNotation

Generated on Tue Feb 6 22:18:17 2018 by

LaTeXML
lemma

lemma

There is no technical distinction a lemma, a

proposition

, and a

theorem

. A

lemma

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.

Even less

well-defined

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

proposition

to denote a significant result that is still shy of deserving a proper name. In contrast, a

theorem

under this format would a major result, and would often be named in to mathematicians who worked on or solved the problem in question.

The Greek word “lemma” itself means “anything which is received, such as a gift, profit, or a bribe.” According to

[

1

]

, 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

relation

is that a theorem is a mathematical fact which you see to be true (and can now show others!).

A somewhat more distinct

concept

(though still subject to author discretion) is that of a

corollary

, which is a result that can be considered an immediate

consequence

of a previous theorem (typically, the preceding theorem in the text).

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

http://planetmath.org/ConverseTheorem

attached entry for more on this last example.

References

1

N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.
(pp. 16)

Title

lemma

Canonical

name

Lemma

Date of creation

2013-03-22 13:46:56

Last modified on

2013-03-22 13:46:56

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

19

Author

mathcam (2727)

Entry type

Definition

Classification

msc 00A05

Defines

proposition

Defines

theorem

Defines

corollary

Generated on Tue Feb 6 22:18:19 2018 by

LaTeXML
Mac OS Calculator

Mac OS Calculator

The

Mac OS Calculator

is a software calculator that comes bundled with the Apple Mac OS operating system. As late as Mac OS 9, the

Calculator

was a very basic calculator with only the

arithmetic operations

. In Mac OS X, the Calculator program was upgraded, with not just the

addition

of Scientific mode (a

scientific calculator

) but also Programmer mode with

functions

useful for

computer

programmers, such as Unicode character lookup by numerical value. The currently displayed value is not lost at a change of mode.

Like the

Windows Calculator

, for the Mac OS Calculator

0

0

=

1

.

References

1

Maria Langer,

Mac OS 8: Visual Quickstart Guide

Berkeley: Peachpit Press (1997): 108

Title

Mac OS Calculator

Canonical

name

MacOSCalculator

Date of creation

2013-03-22 16:39:26

Last modified on

2013-03-22 16:39:26

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Classification

msc 01A07

Defines

Mac OS 9 Calculator

Defines

Mac OS X Calculator

Generated on Tue Feb 6 22:18:21 2018 by

LaTeXML
metric system

metric system

The

metric system

is a system of weights and measures first proposed in France and gradually coming closer to worldwide acceptance in which for each given

dimension

, 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

decimal point

to the right (and adding significant zeroes as necessary), and the

difference

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

constant

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

consistent

prefixes to add to the basic units to make them larger or smaller.

The basic units are:

http://

planetmath

.org/BasicLength

Length

Meter

Mass

Gram

Time

Second

Temperature

Kelvin

The standard prefixes are:

Kilo

1000.0000

Hecto

100.0000

Deca

10.0000

(none)

1.0000

Deci

0.1000

Centi

0.0100

Milli

0.0010

Micro

0.0001

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

United States

remains an important group of hold-outs.

Title

metric system

Canonical

name

MetricSystem

Date of creation

2013-03-22 18:29:37

Last modified on

2013-03-22 18:29:37

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Related topic

DecimalPlace

Generated on Tue Feb 6 22:18:22 2018 by

LaTeXML
mock-up of a basic calculator

mock-up of a basic calculator

This mock-up of a basic

calculator

is realistic in that it has almost every

function

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.

MC

OFF

C

ON

MR

÷

M+

7

8

9

×

STO

4

5

6

-

x

1

2

3

+

%

0

.

±

=

A basic calculator can be counted on to have buttons for the basic

arithmetic operations

(

addition

,

subtraction

,

multiplication

and

division

). A button for

square root

is sometimes provided, its

operation

usually being “postfix”. The

percentage

key, when provided, does not always behave in a standard way. For example, on some calculators, a 15% tip on a meal costing

$

⁡

42.54

might be computed as

[

4

]

[

2

]

[

.

]

[

5

]

[

4

]

[

+

]

[

1

]

[

5

]

[

%

]

[

=

]

.

Title

mock-up of a basic calculator

Canonical

name

MockupOfABasicCalculator

Date of creation

2013-03-22 16:53:48

Last modified on

2013-03-22 16:53:48

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 00A05

Classification

msc 01A07

Generated on Tue Feb 6 22:18:24 2018 by

LaTeXML
mock-up of scientific calculator

mock-up of scientific calculator

This mock-up of a

scientific calculator

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

random number

generator on the key for

π

or the

decimal point

).

BIN

EE

OFF

ON

OCT

d

c

RAND

C

DEC

a

⁢

b

c

Σ

+

Σ

-

y

⁢

σ

⁢

n

-

1

HEX

nCr

x

^

y

^

y

⁢

σ

⁢

n

x!

nPr

x

¯

y

¯

x

⁢

σ

⁢

n

-

1

e

x

π

P

→

R

R

→

P

x

⁢

σ

⁢

n

ln

DEG

GRAD

RAD

MR

10

x

sin

-

1

cos

-

1

tan

-

1

M+

log

sin

cos

tan

STO

x

y

D

E

F

%

x

y

A

B

C

÷

x

3

7

8

9

×

x

3

4

5

6

-

x

1

2

3

+

x

2

0

.

±

=

1

x

Not all scientific calculators support binary, octal and

hexadecimal

display.

Fractions

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

percentage

key is more of a rarity on scientific calculators, something reflected by the

Windows Calculator

, which has percentage in Standard mode but not Scientific (puzzlingly, this is also true of the

square root

key).

Square root and cubic root are usually “postfix” operations, e.g., meaning that to compute

2209

one would enter

[

2

]

⁢

[

2

]

⁢

[

0

]

⁢

[

9

]

⁢

[

x

]

. 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.

Title

mock-up of scientific calculator

Canonical

name

MockupOfScientificCalculator

Date of creation

2013-03-22 16:53:50

Last modified on

2013-03-22 16:53:50

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 00A05

Classification

msc 01A65

Generated on Tue Feb 6 22:18:26 2018 by

LaTeXML
parametre

parametre

Parametre

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 “

variable

constant”.  But in giving a curve or a surface in

parametric form

, the parametres work as proper variables which determine the values of the

coordinates

of the points; then we can describe the parametres as “auxiliary variables”.

The parametric

{

x

=

a

⁢

cos

⁡

t

y

=

a

⁢

sin

⁡

t

of the origin-centered circle of parametres:

a

(the radius) is a variable constant which is held constant all the time when one considers one circle;

t

is an auxiliary variable which has to get all real values (e.g. from the

interval

[

0

,

2

⁢

π

]

)  for obtaining all points of the perimetre.

In the

analytic geometry

, one speaks of the

parametre of

parabola

(a.k.a.

latus rectum

):  it means the chord of the parabola which is

perpendicular

to the axis and goes through the focus; it is the quantity

2

⁢

p

in the standard equation

x

2

=

2

⁢

p

⁢

y

of the parabola (

p

is the

distance

of the focus and the

directrix

).

Title

parametre

Canonical

name

Parametre

Date of creation

2013-03-22 17:06:59

Last modified on

2013-03-22 17:06:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Definition

Classification

msc 00A05

Synonym

parameter

Related topic

Indeterminate

Related topic

DerivativeForParametricForm

Related topic

Curve

Related topic

PerimeterOfAstroid

Related topic

CissoidOfDiocles

Related topic

Variable

Related topic

SurfaceNormal

Defines

auxiliary variable

Defines

parametric form

Defines

parametric presentation

Defines

parameter of parabola

Generated on Tue Feb 6 22:18:30 2018 by

LaTeXML
percentage

percentage

A

percentage

is a ratio expressed in terms of a unit being 100. A percentage is usually denoted by the symbol “%.” For example,

20

%

of

$

⁡

700.47

is

$

⁡

175.12

(using

fixed point arithmetic

to two

decimal places

for display).

On most

calculators

, one sure way to

calculate

a percentage is by entering a

decimal point

before the desired percentage and multiplying that by the amount one wishes to calculate the percentage of. Some calculators have a percentage key.

When tipping at most restaurants in the

United States

, 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.

Note that the percentage symbol % (Shift-5 in most American keyboard layouts) is

overloaded

in TeX as a comment start indicator and in

Mathematica

as a shortcut for referring to the previous output.

Title

percentage

Canonical

name

Percentage

Date of creation

2013-03-22 16:37:56

Last modified on

2013-03-22 16:37:56

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 00A05

Generated on Tue Feb 6 22:18:31 2018 by

LaTeXML
proof

proof

A

proof

is an

argument

designed to show that a statement (usually a

theorem

) is true.

The mathematical

concept

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

sufficient

. Thus, a valid mathematical proof assures there are no

counterexamples

to the proven statement.

There are several kinds of proofs, one commonly used one being

proof by contradiction

. A proof by contradiction starts by assuming that the opposite of the theorem is true, and then proceeds to work out the

consequences

of that

assumption

until encountering a

contradiction

, thus proving the theorem.

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

π

or

5

) also tend to prove the irrationality of the number by at first assuming that the number is in fact

rational

and that there are two integers which form a ratio for the given number.

Another kind of proof is the proof by induction, which starts by showing the statement is true for a small case (such as

n

=

1

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

n

it is also true for

n

+

1

). 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.

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

compass

and straightedge an

isosceles triangle

with an angle that is half of any of the two other angles, a

constructive proof

would give the instructions on how to draw such a

triangle

.

References

1

Paul J. Nahin,

Dr. Euler’s Fabulous

Formula

: Cures Many Mathematical Ills

. Princeton: Princeton University Press (2006): 8

2

Thomas A. Whitelaw,

Introduction

to Abstract Algebra

. New York: CRC Press (1995): 11

Title

proof

Canonical

name

Proof

Date of creation

2013-03-22 18:10:22

Last modified on

2013-03-22 18:10:22

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Generated on Tue Feb 6 22:18:33 2018 by

LaTeXML
property

property

Let

X

be a set. A

property

p

of

X

is a

function

p

:

X

→

{

𝑡𝑟𝑢𝑒

,

𝑓𝑎𝑙𝑠𝑒

}

.

An

element

x

∈

X

is said to

have

or

does not have the property

p

depending on whether

p

⁢

(

x

)

=

𝑡𝑟𝑢𝑒

or

p

⁢

(

x

)

=

𝑓𝑎𝑙𝑠𝑒

. Any property gives rise in a natural way to the set

X

⁢

(

p

)

:=

{

x

∈

X

|

x

⁢

has property

⁢

p

}

and the corresponding

http://planetmath.org/node/CharacteristicFunction

characteristic function

1

X

⁢

(

p

)

. The identification of

p

with

X

⁢

(

p

)

⊆

X

enables us to think of a property of

X

as a 1-ary, or a

unary relation

on

X

. Therefore, one may treat all these notions equivalently.

Usually, a property

p

of

X

can be identified with a so-called

propositional function

, or a

predicate

φ

⁢

(

v

)

, where

v

is a variable or a tuple of variables whose values range over

X

. The values of a propositional function is a

proposition

, which can be interpreted as being either “true” or “false”, so that

X

⁢

(

p

)

=

{

x

∣

φ

⁢

(

x

)

⁢

is

⁢

𝑡𝑟𝑢𝑒

}

.

Below are a few examples:

•

Let

X

=

ℤ

. Let

φ

⁢

(

v

)

be the propositional function “

v

is divisible by

3

”. If

p

is the property identified with

φ

⁢

(

v

)

, then

X

⁢

(

p

)

=

3

⁢

ℤ

.

•

Again, let

X

=

ℤ

. Let

φ

⁢

(

v

1

,

v

2

)

:=

“

v

1

is divisible by

v

2

” and

p

the corresponding property. Then

X

⁢

(

p

)

=

{

(

m

,

n

)

∣

m

=

n

⁢

p

⁢

, for some

⁢

p

∈

ℤ

}

,

which is a subset of

X

×

X

. So

p

is a property of

X

×

X

.

•

The reflexive property of a

binary relation

on

X

can be identified with the propositional function

φ

⁢

(

V

)

:=

`

⁢

`

⁢

∀

a

∈

X

⁢

,

⁢

(

a

,

a

)

∈

V

”, and therefore

X

⁢

(

p

)

=

{

R

⊆

X

×

X

∣

φ

⁢

(

R

)

⁢

is

⁢

𝑡𝑟𝑢𝑒

}

,

which is a subset of

2

X

×

X

. Thus,

p

is a property of

2

X

×

X

.

•

In point set

topology

, we often encounter the

finite intersection property

on a family of subsets of a given set

X

. Let

φ

(

𝒱

)

:=

`

`

∀

n

∈

ℕ

,

∀

E

1

∈

𝒱

,

…

,

∀

E

n

∈

𝒱

,

∃

x

∈

X

(

x

∈

E

1

∩

⋯

∩

E

n

)

”

and

p

the corresponding property, then

X

⁢

(

p

)

=

{

ℱ

⊆

2

X

∣

φ

⁢

(

ℱ

)

⁢

is

⁢

𝑡𝑟𝑢𝑒

}

,

which is a subset of

2

2

X

. Thus

p

is a property of

2

2

X

.

Title

property

Canonical name

Property

Date of creation

2013-03-22 14:01:29

Last modified on

2013-03-22 14:01:29

Owner

drini (3)

Last modified by

drini (3)

Numerical id

15

Author

drini (3)

Entry type

Definition

Classification

msc 00A05

Synonym

attribute

Synonym

propositional function

Related topic

Subset

Related topic

CharacteristicFunction

Related topic

Relation

Related topic

ClosureOfARelationWithRespectToAProperty

Defines

unary relation

Defines

predicate

Generated on Tue Feb 6 22:18:35 2018 by

LaTeXML