the first great number theorist - the history of...

The First GreatNumber Theorist

Garner

AncientFascination withNumbersFriendly Numbers

Polygonal Numbers

Diophantus andHis Works

RhetoricalAlgebraMetrodorus

SyncopatedAlgebraDiophantus

Brahmagupta

Pacioli

Recorde

Bombelli

Viéte

Homework

The First Great Number TheoristThe History of Mathematics, Part 11

Chuck Garner, Ph.D.

Department of Mathematics

Rockdale Magnet School for Science and Technology

February 20, 2014

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OutlineAncient Fascination with Numbers

Friendly Numbers

Polygonal Numbers

Diophantus and His Works

Rhetorical AlgebraMetrodorus

Syncopated AlgebraDiophantus

Brahmagupta

Pacioli

Recorde

Bombelli

Viéte

Homework

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Polygonal Numbers

Brahmagupta

Pacioli

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Friendly Numbers

Polygonal Numbers

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Two Aspects of Numbers

1. Relationship among numbers — “Arithmetic”

(Today we call this number theory)

2. Computing with numbers — “Logistic”

(Today we call this arithmetic)

Greek tendency to think of aspects in pairs:

I Even/Odd

I Measure/Number

I Magnitude/Multitude

I Arithmetic/Logistic

I Commensurable/Incommensurable

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Amicable Numbers

Pythagoreans defined amicable numbers: Two numbers

are amicable if each is the sum of the proper divisors of

the other.

220 and 284 are amicable:

I 220 = 1 + 2 + 4 + 71 + 142

I 284 = 1+2+4+5+10+11+20+22+44+55+110

1636 Fermat found the next pair: 17296 and

1638 Descartes found 9363584 and 9437056

1747 Euler found 30 pairs

1866 Paganini found 1184 and 1210

2014 Millions known today

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Amicable Numbers

Pythagoreans defined amicable numbers: Two numbers

are amicable if each is the sum of the proper divisors of

the other.

220 and 284 are amicable:

I 220 = 1 + 2 + 4 + 71 + 142

I 284 = 1+2+4+5+10+11+20+22+44+55+110

1636 Fermat found the next pair: 17296 and

1638 Descartes found 9363584 and 9437056

1747 Euler found 30 pairs

1866 Paganini found 1184 and 1210

2014 Millions known today

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Other NumbersI Perfect

I Deficient

I Abundant

I Semiperfect

I Weird

I Practical

I �asiperfect

I Sociable Chain

I Crowd

I Evil

I Vampire

I Apocalyptic

I Happy

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Triangular Numbers

The nth triangular number is

n(n + 1)

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Square Numbers

The nth square number is n2

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Pentagonal Numbers

The nth pentagonal number is

n(3n− 1)

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Hexagonal Numbers

The nth hexagonal number is n(2n− 1)

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Polygonal Number Theorems

TheoremAny square number is the sum of two consecutivetriangular numbers.

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TheoremThe sum of any number of consecutive odd integers,starting with 1, is a square.

TheoremThe expression for each k-gonal number is given by

(k − 2)n2 + (4− k)n2

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TheoremThe sum of any number of consecutive odd integers,starting with 1, is a square.

TheoremThe expression for each k-gonal number is given by

(k − 2)n2 + (4− k)n2

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Friendly Numbers

Polygonal Numbers

Brahmagupta

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Diophantus

200 ad-284 ad

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Diophantus

I Not much is known about his life

I Lived, worked in Alexandria

I Gave “logistic” solutions in his work

I Was the first the solve problems in this manner

I Some consider him “father of algebra”

I Wrote the Arithmetica

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The Arithmetica

First Latin edition (1621)

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The Arithmetica

I Contains solutions to 130 problems in 13 books

I No general methods; methods are to be inferred

from specific examples

I Solves

I linear and quadratic equations

I a special cubic equation

I linear and quadratic indeterminate equations

I Only positive rational answers considered

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The Arithmetica

Problem I-1 Find numbers such that the sum is 100

and the di�erence is 40.

and the sum of the third and the fi�h is 30.

and the sum of the squares is 208.

Solution: Let numbers be x and y . Set x = 10 + z and

y = 10− z . Then (10 + z)2 + (10− z)2 = 208, or

200 + 2z2 = 208. Then z = 2 so z = 12 and y = 8.

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The Arithmetica

and the di�erence is 40.

and the sum of the third and the fi�h is 30.

and the sum of the squares is 208.

Solution: Let numbers be x and y . Set x = 10 + z and

y = 10− z . Then (10 + z)2 + (10− z)2 = 208, or

200 + 2z2 = 208. Then z = 2 so z = 12 and y = 8.

Garner

Polygonal Numbers

Brahmagupta

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Friendly Numbers

Polygonal Numbers

Brahmagupta

Pacioli

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Bombelli

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Three Stages of Algebra

1. Rhetorical Algebra

2. Syncopated Algebra

3. Symbolic Algebra

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Metrodorus’ Greek Anthology

I Nearly nothing is know of Metrodorus’ life

I He was a grammarian and mathematician

I Believed to have lived in 6th century ad

I Compiled Greek Anthology, c. 500 ad

I All problems are rhetorical

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Friendly Numbers

Polygonal Numbers

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Diophantus

Diophantus used the following abbreviations in the

Arithmetica

the unknown ς sigma variant

unknown squared ∆Υ dunamis (∆ΥNAMIΣ), powerunknown cubed KΥ kubos (KΥBOΣ), cube

fourth power ∆Υ∆ literally, square-squarefi�h power ∆KΥ square-cubesixth power KΥK cube-cube

minus Λ| leipis (ΛEIΨIΣ), lacking

the constant

◦M monades (MONA∆EΣ), units

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Diophantus

x3 + 13x2 + 8x would be found in Diophantus as

KΥα∆Υιγςη.

x3 − 8x2 + 2x − 3 would be found in Diophantus as

KΥαςβΛ|∆Υη◦

I First instance of syncopated algebra

I Took a while to catch on

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Diophantus

x3 + 13x2 + 8x would be found in Diophantus as

KΥα∆Υιγςη.

x3 − 8x2 + 2x − 3 would be found in Diophantus as

KΥαςβΛ|∆Υη◦

I First instance of syncopated algebra

I Took a while to catch on

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Brahmagupta

598-670

“As the sun eclipses the stars by its brilliancy, so the manof knowledge will eclipse the fame of others in assembliesof the people if he proposes algebraic problems, and still

more if he solves them.”

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Brahmagupta

I Addition indicated by juxtaposition

I Subtraction by a dot over the subtrahend

I Multiplication by “bha” wri�en a�er factors (from

bhavita, product)

I Division by writing divisor beneath dividend

I Square root by “ka” before number (from karana,

irrational)

I Unknown by “ya” (from yava�avat, so much as)

I Knowns prefixed by “ru” (from rupa, absolute

number)

I Additional unknowns by color; a second unknown

might be “ka” (from kalaka, black)

8xy +√

10− 7 may appear as ya ka 8 bha ka 10 ru 7.

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Brahmagupta

bhavita, product)

irrational)

number)

8xy +√

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Brahmagupta

bhavita, product)

irrational)

number)

8xy +√

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Brahmagupta

bhavita, product)

irrational)

number)

8xy +√

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Brahmagupta

bhavita, product)

irrational)

number)

8xy +√

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Brahmagupta

bhavita, product)

irrational)

number)

8xy +√

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Brahmagupta

bhavita, product)

irrational)

number)

8xy +√

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Brahmagupta

bhavita, product)

irrational)

number)

8xy +√

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Brahmagupta

bhavita, product)

irrational)

number)

8xy +√

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Brahmagupta

bhavita, product)

irrational)

number)

8xy +√

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Pacioli

Luca Pacioli

1445-1517

“The second thing necessary in business is to be a goodbookkeeper and ready mathematician.”

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Pacioli

I Revolutionized bookkeeping

I Wrote Summa de arithmetica in 1494

I Used abbreviations:

I p for plus (from piu, more)

I m for minus (from meno, less)

I co for the unknown (from cosa, thing)

I ce for square (from censo)

I cu for cube (from cuba)

I cece for fourth power

I ae for equals (from aequalis)

x3 − 8x2 + 2x − 3 = 9 would be cu m 8 ce p 2 co m 3 ae 9.

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Pacioli

I Revolutionized bookkeeping

I Wrote Summa de arithmetica in 1494

I Used abbreviations:

I p for plus (from piu, more)

I m for minus (from meno, less)

I co for the unknown (from cosa, thing)

I ce for square (from censo)

I cu for cube (from cuba)

I cece for fourth power

I ae for equals (from aequalis)

x3 − 8x2 + 2x − 3 = 9 would be cu m 8 ce p 2 co m 3 ae 9.

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Pacioli

I Used R for square root (from radix)

I RV as a grouping symbol (from radix universalis)

√7 +√

15 would be R V 7 p R 15.

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Pacioli

I Used R for square root (from radix)

I RV as a grouping symbol (from radix universalis)√7 +√

15 would be R V 7 p R 15.

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Recorde

Robert Recorde

1510-1558

“Besides the mathematical arts there is no infallibleknowledge, except that it be borrowed from them.”

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Recorde

I Wrote clear, readable math texts in English

I Worked as surveyor of the mines and the mint in

Ireland

I Wrote Whetstone of Wi�e in 1557

I Gave us our modern equals sign (=); took over a

century to be established

I Also used + and −

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Recorde’s The Whetstone of Wi�e

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Recorde’s The Whetstone of Wi�e

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Bombelli

Rafael Bombelli

1526-1572

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Bombelli

I Improved algebraic notation

I Gave rules for arithmetic with negatives

I Used grouping symbols

I Rq for square root and Rc for cube root

√7 +√

15 would be Rq b7 p Rq 15c.

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Bombelli

I Improved algebraic notation

I Gave rules for arithmetic with negatives

I Used grouping symbols

I Rq for square root and Rc for cube root√7 +√

15 would be Rq b7 p Rq 15c.

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Viéte

François Viéte

1540-1603

“There is no problem that cannot be solved.”

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Viéte

I Before Viéte, di�erent le�ers used for di�erent

powers

I Viéte used the same le�er properly qualified

I Our x is AI Our x2

is A quadratum, also A qI Our x3

is A cubum, also A c

I Viéte used vowels for unknowns and consonants

for knowns

x3− 8x2 + 2x− 3 = 9 would be A c− 8 A q + 2 A− 3 ae 9.

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Viéte

powers

for knowns

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Viéte

powers

for knowns

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Friendly Numbers

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HomeworkI Reminder: Weekender #4 due February 24

I Weekender #5 due February 28

I The same problem, 3000 years apart;

Reader, 5.C5

I A selection of 10 problems from the Arithmetica;

Reader, 5.D

I Old English mathematics from the Whetstone ofWi�e;

Reader, 9.A

I More algebraic symbolism from the Renaissance;

Math Through the Ages, Sketch 8

Next: Be�er Scratches, Fewer Grunts

the first great number theorist - the history of...

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