aarjaneiro/planetmath huggingface dataset:
Trice sequence

Trice sequence

The

Trice sequence

consists of the nine numbers 21, 36, 55, 60, 67, 68, 92, 93, 125 in

ascending order

. It was published in Clifford Pickover’s

Mazes of the Mind

in 1992 and added to the

On-Line Encyclopedia of Integer Sequences

early on in its history (see A001491), but for years people couldn’t figure out what the sequence was because most of them were searching for a mathematical explanation. The first three numbers are both triangular and hexagonal, but none of the following numbers are. According to Neil Sloane, “this was a mystery for many years; even Clifford Pickover could not recall the explanation .” It wasn’t until 1998 that Derek Holt identified the sequence and let Sloane know that it consists of the

opus numbers

of Beethoven’s nine symphonies.

Title

Trice sequence

Canonical

name

TriceSequence

Date of creation

2013-03-22 16:45:00

Last modified on

2013-03-22 16:45:00

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A99

Generated on Tue Feb 6 22:21:50 2018 by

LaTeXML
triskaidekaphobia

triskaidekaphobia

Triskaidekaphobia

is the irrational fear of the number 13. This fear is recognized as a mental disorder by the Diagnostic and Statistical Manual of Mental Disorders IV (DSM-IV). Few mathematicians suffer from triskaidekaphobia; generally it’s artists who are afflicted. For example, the composer Arnold Schoenberg and the pianist Glenn Gould. However,

Benjamin Franklin

was mildly afflicted: if there were 13 people at his dinner table, he would call for his secretary to join the party. He was apparently not bothered by the fact that the

United States

at the time consisted of 13 states and the flag had 13 stars and 13 stripes. The rationale for Franklin’s affliction seems to have been that there were 13 people at the Last Supper of Jesus Christ. Architects for the most part don’t have to worry about triskaidekaphobia when building new buildings, but when working on existing buildings they have to remember to subtract one if the building’s 13th floor was renumbered 14.

References

1

W. Clarke,

http://faculty.lasierra.edu/ wclarke/hist.pdf

Some Mathematical History: The History of Numbers

(2003): 7

Title

triskaidekaphobia

Canonical

name

Triskaidekaphobia

Date of creation

2013-03-22 17:23:48

Last modified on

2013-03-22 17:23:48

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A99

Generated on Tue Feb 6 22:21:51 2018 by

LaTeXML
WLOG

WLOG

“WLOG” (or “WOLOG”) is an acronym which stands for “without loss of generality.”

WLOG is invoked in situations where some property of a model or system is

invariant

under the particular choice of

instance

attributes

, but for the sake of demonstration, these attributes must be fixed.

For example, we might want to prove something about

open intervals

(

a

,

b

)

of the real number line. But the proof might become too tedious if

a

and

b

were arbitrary real numbers, so in the proof we simply assume that

a

=

0

and

b

=

1

, and

without loss of generality

, the same

arguments

apply to general

intervals

(

a

,

b

)

. Depending on the proof, the loss of generality might be accomplished by translating and

scaling

the interval to

(

0

,

1

)

before

carrying out the argument, and then translating and rescaling back to

(

a

,

b

)

afterwards.

WLOG can also be invoked to shorten proofs where there are a number of choices of

configuration

, but the proof is “the same” for each of them. We need only walk through the proof for one of these configurations, and “WLOG” serves as a note that we haven’t weakened the argument. For example, the proof of the

fundamental theorem of arithmetic

uses this notion, in essence settling on a “

canonical form

” for

prime factorizations

to simplify the argument.

For more examples, see

http://planetmath.org/?op=search&term=WLOG+without+loss+generality

approximate index of PM entries invoking WLOG.

Title

WLOG

Canonical name

WLOG

Date of creation

2013-03-22 12:40:17

Last modified on

2013-03-22 12:40:17

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

6

Author

akrowne (2)

Entry type

Definition

Classification

msc 00A99

Synonym

WOLOG

Synonym

without loss of generality

Generated on Tue Feb 6 22:21:53 2018 by

LaTeXML
topic on major resources for mathematics and mathematical physics

topic on major resources for mathematics and mathematical physics

1

Mathematics resources, international institutes and major facilities

This is a new contributed topic on major facilities and resources
for mathematics and mathematical/theoretical physics, including computational/supercomputing
and theoretical paper web-archiving:

•

Princeton Advanced Theoretical Research Institute

•

IHES, in Paris, France

•

http://www.phy.cam.ac.uk/

The Cavendish Laboratory,Department of Physics,

http://en.wikipedia.org/wiki/Cavendish_Laboratory

University of Cambridge, UK

•

Supercomputing National Centers, e.g.: NCSA at Urbana, IL

•

Beckmann Advanced Research Institute, Urbana-Champaign, Illinois

•

RAL=

http://www.scitech.ac.uk/About/Find/RAL/Introduction.aspx

Rutherford Appleton Laboratory
near Didcot, Oxfordshire

•

Max Planck Institute, Germany

•

Institute for Theoretical Physics Research, Germany

•

Trieste Advanced Theoretical Physics Research Institute

•

High-density Energy Research Institute, Kyoto, Japan

•

CERN, International preprint archives in Geneva

•

ANL web-archiving center of

mathematical physics

/theoretical papers

•

National Centers for Nanotechnology, Material Science Centers

•

IHEP in Russia

•

Institute of Mathematics of the Romanian Academy, in Bucharest, Romania

•

XYZ

•

XYZ

Title

topic on major resources for mathematics and mathematical physics

Canonical

name

TopicOnMajorResourcesForMathematicsAndMathematicalPhysics

Date of creation

2013-03-22 18:44:11

Last modified on

2013-03-22 18:44:11

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

21

Author

bci1 (20947)

Entry type

Topic

Classification

msc 00B60

Classification

msc 00B10

Classification

msc 00A99

Generated on Tue Feb 6 22:21:57 2018 by

LaTeXML
Greek alphabet

Greek alphabet

The modern Greek 24 letters. Ancient Greeks only used the capital letters; the lower case letters were not invented until the time of Alexander the Great. The Greek letters digamma and qoppa were incorporated into the Roman as F and Q, respectively, but are not part of the modern Greek .

Following is a of letters in the modern Greek in alphabetical :

lower case

capital

name

comments

α

A

alpha

β

B

beta

bita in modern Greek

γ

Γ

gamma

δ

Δ

delta

ϵ

or

ε

E

epsilon

ζ

Z

zeta

zita in modern Greek

η

H

eta

ita in modern Greek

θ

or

ϑ

Θ

theta

thita in modern Greek

ι

I

iota

κ

or

ϰ

K

kappa

λ

Λ

lambda

μ

M

mu

mi in modern Greek

ν

N

nu

ni in modern Greek

ξ

Ξ

xi

o

O

omicron

o-small in modern Greek

π

or

ϖ

Π

pi

ρ

or

ϱ

P

rho

σ

or

ς

Σ

sigma

ς

is used only at the end of Greek

τ

T

tau

υ

Υ

upsilon

ipsilon in modern Greek

ϕ

or

φ

Φ

phi

χ

X

chi

ψ

Ψ

psi

ω

Ω

omega

o-big in modern Greek

Title

Greek alphabet

Canonical name

GreekAlphabet

Date of creation

2013-03-22 15:40:25

Last modified on

2013-03-22 15:40:25

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

20

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 01-00

Classification

msc 01A20

Synonym

Greek letters

Generated on Thu Feb 8 19:06:17 2018 by

LaTeXML
table of common logarithms

table of common logarithms

This table is for

integers

in the range

-

1

<

n

<

100

. To find the

common logarithm

(that is, with base 10, the same base of our

numeral system

) for the desired

n

, look for the ten’s place digit (0 in the case of

-

1

<

n

<

10

) in the leftmost column and the one’s place digit in the topmost row. The

intersection

x

of that row and column thus gives

log

10

⁡

n

to four or five

decimal places

.

n

0

1

2

3

4

5

6

7

8

9

0

-

∞

0.00000

0.30103

0.477121

0.60206

0.69897

0.778151

0.845098

0.90309

0.954243

1

1.

1.04139

1.07918

1.11394

1.14613

1.17609

1.20412

1.23045

1.25527

1.27875

2

1.30103

1.32222

1.34242

1.36173

1.38021

1.39794

1.41497

1.43136

1.44716

1.4624

3

1.47712

1.49136

1.50515

1.51851

1.53148

1.54407

1.5563

1.5682

1.57978

1.59106

4

1.60206

1.61278

1.62325

1.63347

1.64345

1.65321

1.66276

1.6721

1.68124

1.6902

5

1.69897

1.70757

1.716

1.72428

1.73239

1.74036

1.74819

1.75587

1.76343

1.77085

6

1.77815

1.78533

1.79239

1.79934

1.80618

1.81291

1.81954

1.82607

1.83251

1.83885

7

1.8451

1.85126

1.85733

1.86332

1.86923

1.87506

1.88081

1.88649

1.89209

1.89763

8

1.90309

1.90849

1.91381

1.91908

1.92428

1.92942

1.9345

1.93952

1.94448

1.94939

9

1.95424

1.95904

1.96379

1.96848

1.97313

1.97772

1.98227

1.98677

1.99123

1.99564

Title

table of common logarithms

Canonical

name

TableOfCommonLogarithms

Date of creation

2013-03-22 18:08:50

Last modified on

2013-03-22 18:08:50

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 01-08

Classification

msc 65A05

Classification

msc 26-00

Classification

msc 26A09

Generated on Thu Feb 8 19:06:19 2018 by

LaTeXML
MathML

MathML

MathML

(or

Mathematical Markup Language

) is an application of XML customized for the transmission of documents pertaining to mathematics. The official Document Type

Definition

was created by the

World Wide Web

Consortium.

MathML is concerned as much with the meaning of mathematical formulas as it is with their

presentation

. This means, for example, that any instances of the

tacit multiplication operator

must be stated in the MathML source even though they are not meant to be shown on the display.

To give one example of the famous Euler identity

e

i

⁢

π

=

-

1

in MathML:

There are programs available that convert TeX to MathML.

Title

MathML

Canonical

name

MathML

Date of creation

2013-03-22 16:54:19

Last modified on

2013-03-22 16:54:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 01A07

Synonym

Mathematical Markup Language

Generated on Thu Feb 8 19:06:21 2018 by

LaTeXML
Ishango bone

Ishango bone

The

Ishango bone

is an ancient baboon bone with numerical markings on it which was unearthed in a bank of Edward Lake in the Congo by geologist Jean de Heinzelin de Braucourt. Carbon dating shows the bone is at least 20,000 years old. Now on exhibit at the museum of the Royal Belgian Institute of Natural Sciences, it is characterized as “the oldest mathematical artifact.”

The bone has three columns of numbers, the middle column reads: 3, 6, 4, 8, 10, 5, 5, 7 (this

sequence

is A100000 in Sloane’s OEIS). The other two columns read: 11, 13, 17, 19 (a

prime quadruplet

); and 11, 21, 19, 9. The columns add up to 60, 48 and 60.

Mathematicians and scientists have speculated on the meaning of the numbers on the bone. Alexander Marshack believes the bone might be a

lunar calendar

, while Claudia Zaslavsky speculates the author was a woman tracking her menstrual cycle. Number theorists note that 48 and 60 are both

multiples

of 12 and cite this as evidence of the early humans’ ability to multiply.

References

1

D. Huylebrouck, “L’Afrique, berceau des mathematiques”, in

Mathematiques exotiques Dossier No. 47, Pour La Science

(2005) 46 - 50

2

G. G. Joseph,

The Crest of the Peacock: Non-European Roots of Mathematics

London: Penguin Books (1992).

3

A. Marshack,

The Roots of Civilization

Mount Kisco (1991)

4

D. Olivastro, “The First Etches”

Ancient Puzzles

New York: Bantam Books (1993): 7 - 30

5

V. Pletser & D. Huylebrouck, “The Ishango artifact: the missing base 12 link”,

Proc. Katachi Univ. Symmetry Congress (KUS2), Paper C11

, (1999) 339 - 346.

6

Claudia Zaslavsky,

Africa Counts

New York: Lawrence Hill Books (1973)

Title

Ishango bone

Canonical name

IshangoBone

Date of creation

2013-03-22 17:20:26

Last modified on

2013-03-22 17:20:26

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 01A10

Generated on Thu Feb 8 19:06:25 2018 by

LaTeXML
Book of Numbers

Book of Numbers

The

Book of Numbers

is the fourth book of the

Holy Bible

. The original Hebrew title is “Ba-Midbar”, which means “In the Desert.” In the Greek

translation

of the Old Testament, the book was referred to as “Arithmoi”, giving rise to its name in most modern languages. Chapter 1 lists the results of the Census taken after the Jews escaped from Egypt, “every male from twenty years old and upward, all that were able to go forth to war,” categorized by tribes (but the Levites were excluded from the Census). In the King James Version, the numbers are given rounded off by

hundreds

.

Reuben

46500

Simeon

59300

Gad

5650

Judah

74600

Issachar

54400

Zebulun

57400

Ephraim

4500

Manasseh

32200

Benjamin

35400

Dan

62700

Asher

41500

Naphtali

53400

Levi

not counted

Unlike other uses of numbers in the

Bible

, the numbers in the Book of Numbers are understood to be actual enumerations and not chosen for their symbolic value because of their mathematical or numerological

properties

(as is the case with the Book of Revelations).

The title

Book of Numbers

was used by John Conway and Richard Guy for their book on

number theory

. Title aside, it contains no biblical references.

Title

Book of Numbers

Canonical

name

BookOfNumbers

Date of creation

2013-03-22 16:38:22

Last modified on

2013-03-22 16:38:22

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 01A20

Generated on Thu Feb 8 19:07:43 2018 by

LaTeXML
Hypatia

Hypatia

Hypatia

of Alexandria (350 - 415) Greek mathematician and educator.

The daughter of Theon of Alexandria, Hypatia was allowed to study subjects most contemporary women were not allowed to. Theon wrote commentaries on the works of Diophantus, Ptolemy, and Apollonius. Some historians believe some of these may have been written by Hypatia, but there has been no proof of this. Hypatia was murdered in an intrigue involving a skirmish between Jews and Christians. The one woman in Raphael’s painting

The School of Athens

is said to be Hypatia.

References

1

I. Mueller “Hypatia” in

Women of Mathematics: A Bibliographic Sourcebook

L. Grinstein, P. Cambpell, ed.s New York: Greenwood Press (1987): 74 - 79

Title

Hypatia

Canonical name

Hypatia

Date of creation

2013-03-22 17:00:47

Last modified on

2013-03-22 17:00:47

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

5

Author

Mravinci (12996)

Entry type

Biography

Classification

msc 01A20

Classification

msc 01A16

Synonym

Hypatia of Alexandria

Generated on Thu Feb 8 19:07:57 2018 by

LaTeXML
numerology

numerology

Modern

numerology

is the divination of the future by means of reducing the name and birthdate of the person for whom the chart is being cast for to single base 10 digits by means of

modular arithmetic

. The digits are assigned abstract human qualities beforehand, and once the digits are obtained, arithmetic (or any kind of mathematics) no longer plays any rôle in the process. Numerology is considered with as much disdain by mathematicians as astrology is by astronomers, if not more so.

The letters of the person’s name are assigned numerical values, (e.g., A = 1, B = 2, C = 3, etc.) and added up, then the

digital root

of the sum is obtained. (It doesn’t matter if J = 10 or J = 1, the overall result will be the same.) The same procedure applies for the birthdate (e.g., January = 1, February = 2, etc.) Any

computer algebra system

capable of string manipulation by ASCII or Unicode

character

codes is capable of performing calculations for numerology. For example,

Mathematica

provides the

FromCharacterCode

and

ToCharacterCode

functions. One only needs to remember to subtract 64 or 96 (depending on case), to use only one case and to not use spaces.

Ancient Greeks such as Pythagoras believed that numbers indeed possessed abstract human qualities based on their mathematical properties, or perhaps even occult significance relevant to mystery religions. There is no vestige of this in modern numerology, but mathematics carries vestiges of this in some of its terminology, if nothing else (for example, the

amicable numbers

).

The term “numerology” is sometimes used derogatorily to refer to what could more accurately be described as “number folklore.” Richard Guy’s

strong law of small numbers

“suggests that number folklore is likely to develop” as a result of people liking to find patterns from finite amounts of data. (Slone, 2007)

References

1

K. Lagerquist and L. Lenard,

The

Complete

Idiot’s Guide to Numerology

Indianapolis: Macmillan (1999)

2

M. Slone, personal communication (2007)

Title

numerology

Canonical name

Numerology

Date of creation

2013-03-22 16:29:06

Last modified on

2013-03-22 16:29:06

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

7

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 01A20

Classification

msc 01A65

Generated on Thu Feb 8 19:07:59 2018 by

LaTeXML
Roman numerals

Roman numerals

Roman numerals

are a method of writing numbers employed primarily by the ancient Romans. It place of digits, the Romans used letters to represent the numbers

central

to the system:

I

1

V

5

X

10

L

50

C

100

D

500

M

1000

Larger numbers can be made by writing a bar over the letter, which means one

thousand

times as much. For instance

V

¯

is

5000

.

Other numbers were written by putting letters together. For instance

I

⁢

I

means

2

. Larger letters go on the left, so

L

⁢

I

⁢

I

is

52

, but

I

⁢

I

⁢

L

is not a valid Roman numeral.

One additional rule allows a letter to the left of a larger letter to signify subtracting the smaller from the larger. For instance

I

⁢

V

is

4

. This can only be done once;

3

is written

I

⁢

I

⁢

I

, not

I

⁢

I

⁢

V

. Also, it is generally required that the smaller letter be the one immediately smaller than the larger, so

1999

is usually written

M

⁢

C

⁢

M

⁢

X

⁢

C

⁢

I

⁢

X

, not

M

⁢

I

⁢

M

.

It is worth noting that today it is usually considered incorrect to repeat a letter four times, so

I

⁢

V

is preferred to

I

⁢

I

⁢

I

⁢

I

. However many older monuments do not use the

subtraction

rule at all, so

44

was written

X

⁢

X

⁢

X

⁢

X

⁢

I

⁢

I

⁢

I

⁢

I

instead of the now preferable

X

⁢

L

⁢

I

⁢

V

.

Title

Roman numerals

Canonical

name

RomanNumerals

Date of creation

2013-03-22 12:57:31

Last modified on

2013-03-22 12:57:31

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 01A20

Generated on Thu Feb 8 19:08:03 2018 by

LaTeXML
Zeno’s paradox

Zeno’s paradox

Imagine the great Greek hero Achilles starting a race with a turtle. Achilles is a fast runner,

10

metres per second, while the turtle is slow and runs at one metre per second. Therefore Achilles agrees to give the turtle some advantage and the turtle starts 10 metres in front of Achilles. The ancient Greek philosopher Zeno found the following “paradox”.

If Achilles wants to get in front of the turtle he first has to run to where the turtle started. But in that time the turtle has bridged some

distance

, which Achilles now has to run in to take up. But in this time again the turtle has gone for some distance and Achilles is still in behind of the turtle. This process continues forever and apparently Achilles cannot pass the turtle.

To solve this paradox we have to take a look at the times needed to run these . It takes Achilles one second to get to where the turtle started. In this time the turtle runs one metre. It takes only the tenth of a second for Achilles to get there as well. The turtle now runs 10 centimetres, which Achilles passes in one

hundredth

of a second and so on. So Achilles reaches the turtle after

1

⁢

s

+

0.1

⁢

s

+

0.01

⁢

s

+

…

=

1.1111

⁢

…

⁢

s

.

The paradox can be solved, if we take into consideration the fact that an series (a sum of infinitely many numbers) may well converge.

To see that the paradox actually never arises, we consider a race, where the turtle gets an advantage of

d

and runs at a speed

v

. Achilles runs at a speed

x

⁢

v

with

x

>

1

. Then the time

t

needed for Achilles to reach the turtle is given as:

t

=

d

v

⁢

∑

j

=

1

∞

1

x

j

,

which converges if and only if

x

>

1

, so in any possible race Achilles can catch up with the turtle, as was expected.

Title

Zeno’s paradox

Canonical

name

ZenosParadox

Date of creation

2013-03-22 14:07:50

Last modified on

2013-03-22 14:07:50

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

10

Author

mathwizard (128)

Entry type

Topic

Classification

msc 01A20

Generated on Thu Feb 8 19:08:12 2018 by

LaTeXML
abacus

abacus

The

abacus

is a computational tool of ancient history. In its simplest form, it is a frame with a number of rods, and on each rod there are beads to represent numbers. Usually, the beads on the leftmost rod represent units, those on the second leftmost rod represent tens, then

hundreds

, etc. The Chinese added a separator near the top, leaving two beads in the “heaven” part of the device to represent five of a given place value, e.g., a “heaven” bead on the third rod from the left represents 500 while a bead on the “earth” side of the same rod stands for just 100. For calculations involving money, where the unit of currency is usually divided into a hundred cents, the two leftmost rods can be used to represent the cent values while the third rod for the left is then the unit rod.

References

1

J. Dilson,

The Abacus: A Pocket

Computer

. New York: St. Martin’s Press (1994)

Title

abacus

Canonical

name

Abacus

Date of creation

2013-03-22 17:06:25

Last modified on

2013-03-22 17:06:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 01A25

Classification

msc 00A07

Generated on Thu Feb 8 19:08:13 2018 by

LaTeXML
Pietro Cataldi

Pietro Cataldi

Pietro Antonio Cataldi

(1552 - 1626) Italian mathematician, astronomer, author and educator.

Still a teenager, Cataldi started his teaching career in 1567, and contrary to the custom of the day, he conducted class in the Bolognese dialect of Italian rather than in Latin. In

addition

to teaching, Cataldi also consulted with the military, his work on the calculation of roots, using

continued fractions

, was intended for practical artillery applications. After teaching in Florence and Perugia, Cataldi returned to Bologna in 1584, where he then taught astronomy in addition to mathematics, until his death. In 1588, Cataldi discovered two consecutive

Mersenne primes

, 131071 and 524287, and was for almost two centuries the discoverer of the largest known prime number.

In addition to his own books on

arithmetic

,

number theory

, and

geometry

, Cataldi also worked on an Italian edition of Euclid’s

Elements

.

Title

Pietro Cataldi

Canonical

name

PietroCataldi

Date of creation

2013-03-22 18:05:07

Last modified on

2013-03-22 18:05:07

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

4

Author

Mravinci (12996)

Entry type

Biography

Classification

msc 01A40

Classification

msc 01A45

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LaTeXML