III

Alogos

Small states (republic or tyranny) freedom of thoughtseafaring people, connecting Asia and Europeno privileged class of priests fighting against new ideas

600 - 500 BCGreek enlightenment

New idea: The world can be understood!

parallelsChina: Confucius India: Buddha

Thales of Milet (624 - 545)

Measuring the hight of a pyramid

All matter is assumed to be livingKnew magnetite and elektron (amber)Predicted the solar eclipse of 28 May 585 BC

Pythagoras (570 - 500) = mouth of Apollo

Mother: Parthenis (= Virgin) impregnated by the God of the Sun Apollo and called, in his honour, Pythais

SamosEgyptCretePersiaBabylonSicily

Pythagoras (570 - 500)

Eudemos pupil of Aristotle wrote the first catalogue of mathematicians:Pythagoras was the first mathematician. Theorem and proof.

SamosEgyptCretePersiaBabylonSicily

Founder of a „school“ (secret society)existing until 370 BCidentification badge: regular pentagonbasic principle: all is numberPythagoras was not called by his name. “that man““he has said it himself“ ended every debatebound to secrecy towards every outsider

PythagorasMarble bust 4th century BC

Look !

c2 = 4 * ab/2 + (a - b)2 = a2 + b2

abc

a2 + b2 = c2

ma2 + mb2 = mc2

ma2 + mb2 = mc2

a2 + b2 = c2

ma2 + mb2 = mc2

a2 + b2 = c2

ma2 + mb2 = mc2

a2 + b2 = c2

Vegetarian (believed in transmigration of souls)

No written records

No gravestone

All is number

2, 3, 4,...

and proportion

Raphael: School of Pythagoras (detail)

ca

Hippasos of Metapont:First recognition of the irrational (ca. 450 BC)Hippasos published his knowledgeand was punished

Recognition of the irrational by means of the isosceles triangle 2a2 = c2 or the golden ratio.

Theory of the irrationale, Incommensurability, 2

Incommensurability of edge and diagonal:There is no common measue however small!

2a2 = c2 2 = =p q =

pq

2 = p/q, without common divisor

2 = p2/q2

2q2 = p2

p2 even p even

p = 2z p2 = 4z2

2q2 = 4z2

q2 = 2z2

q2 even q even

p,q common divisor 2

Theory of the irrational, incommensurability, 2

Theory of the irrational, incommensurability, 2

Euclid (325 - 275)

The elements, book 10: Theory of incommensurability

The sum of two incommensurables is incommensurable to its summands. Example 1 + 2.

Irrational numbers : alogos = unspeakable

According to Platon (427 - 348) Theodoros of Cyrene has proofs of irrationality for all non-integer roots up to 17.

His pupil Theaitetos (410 - 368) proved this for all natural numbers,

Early treatment of the Irrational by Eudoxos (408 - 355), Inventor of the method of exhaustion.

All fractions are periodic decimals.

Division by 3: At most 2 different remainders 0 possible. Length of period at most 2.

All periodical decimal numbers are fractions.

Example: 0,123123123... = 0,123

123 = 123,123 - 0,123 = (1000 - 1)0,123

0,123 = 123/999

Irrational numbers cannot be periodic. They have an infinite sequence of non-periodic decimals, defined by a formula.

All terminating decimals can be represented in a unique way by periodic decimals:

100,999 - 10,999 = 9,999 - 0,999 (10 - 1)0,999 = 9

0,999 = 1,000 = 1

All roots of natural numbers are integers or irrationals.

Squares of rationals are rationals again: (3/4)2 = 33 / 44

(Uniqueness of primefactorization)

7 = p/q with p,q having no common divisor 7qq = pp

9 = p/q 3q3q = pp p = 3q

And since p, q have no common divisor q = 1, p = 3

The fundamental theorem of number theory

Uniqueness of prime factorization

is not trivial because for even numbers

60 = 30 2 = 10 6

Proof of uniqueness for the integers:

4 = 22, 6 = 32,... have unique factorizations.

Let pP = qQ the smallest equivocal factorization.

Let p > q with no common factors.

(p-q)P = pP - qP = qQ - qP = q(Q-P) < qQ

(p-q) does not contain the factor q, because p and q prime.

10 = n/m 10m2 = n2

n2 ends with an even number of zeros.

Example n = 3 n2 = 9 n = 20 n2 = 400

10m2 ends with an odd number of zeros.

lg5 = n/m 5 = 10n/m 5m = 10n

LHS odd number, RHS even number

( 2 + 3 )2 = 2 + 2 6 + 3 is irrational because 6 is irrational.

Golden section

a/b = b/(a-b) > 0

= 1/(- 1) - 1 = 1/

2 - - 1 = 0 (- 1)(+ 1) =

= 1/2 + (1 + 1/4)

= (1 + 5)/2 = 1,618...

...1111 1 2 = 1 +

= 1 + 1/

11

1

11

11

11

11

Martin Ohm (1792 - 1872):Brother of Georg Simon Ohm

Goldener Schnitt (1835)

Luca Pacioli (1446 - 1517) ital. mathematician

Inventor of double entry book-keeping calculated log2, failed to find the solution of the cubic equ.Leonardo da Vinci (1452 - 1519) ital. painter, draftsman, sculpturist, architect, musician, natural scientist, engineer

Golden section

Devina proportione (1509)

- 1 = 0,618...

5/8 = 0,625

Parthenon

Johannes Campanus de Novarra (1220 -1296) Chaplain of pope Urban IV. (1261-1264)Euclid-translatorProof of irrationality of the golden section by descente infinie

Stetige Teilung

Let x1, x2 Multiply by common denominator to obtain: n1, n2

(n1 + n2)/n1 = n1/n2 with n2 < n1

1 + n2/n1 = n1/n2

n1/n2 = n2/(n1 - n2)

(n1 - n2) n3 with n3 < n2 < n1

(n2 + n3)/n2 = n2/n3

and so on ad infinitum

Golden section in the pentagon. Poor Pythagoras!

Golden section in the pentagon. Poor Pythagoras!

Golden section in the pentagon. Poor Pythagoras!

First active as an engineer From 1816 professor in France and ItalyExtraordinary prolific and versatile mathematician with 800 papers.

Cauchy-criterionQuotient-criterionRoot-criterion Series-compressionRadius of convergenceDiagonal-procedureDerivative and integral as limitsTheory of complex functions

|ak| < k = m

n

Augustin Louis Cauchy (1789 - 1857)

Potentiell understanding of limits: If the values taken by the variable are converging without end to a fixed number such that the difference becomes arbitrarily small, then we call it the limit of all other values.

x = a x2 = a 2x2 = x2 + a

Irrational numbers can be defined as limits of sequences of rational numbers.

x2

a2x

x

n

n1n x2

a2

xx

Richard Dedekind (1831 - 1916)

Doctorated in 1852 in Göttingen supervisor: C.F. Gauß

From 1862 until 1894 Professor at the Polytechnikumin Braunschweig (today Technical University)

Had an extended correspondence with G. CantorAnd took part in formulating set theory.

"... thus the negative and the rational numbershave been created by human spirit“

„We can call the numbers a free creation of human spirit.“

What are and what shall the numbers? (1887) "My main answer on this question is: The numbers are free creations of the human spirit, they serve as a means to recognize the differences of the things easier and sharper.“

Johann Heinrich Lambert (1728 - 1777)Son of a taylorAutodidact, universal scholar:Mathematican, physicist, astronomer, philosopher1759 member of Bavarian academy of sciences. 1765 member of Prussian academy of sciences.1027 manuskripts,190 published

Proof of Irrationality of e and (continued fraction development of tanx)

„Vorläufige Kenntnis für die, so die Quadratur des Circuls suchen“ (1770) Preliminary knowledge for those who want to square the circle.

Continued fractions

aa

a

a

11

1

11for a < 1:

2

11

1

231

3

2

Example:

32

3

1

3111

113

Finite continued fractions are rational numbers.

Infinite continued fractions are irrational numbers.

Infinite continued fractions are irrational numbers.

25

2

32

11

4

2

2

2

Lord Brouncker (1620 - 1684)

Continued fractions

Leonhard Euler (1707 - 1783)

1410

62

2

2

2

A = r U2

= 3,1415926...

Geometric attempts to square the circle

Anaxagoras (in prison) und Hippocrates of Chios were among the first who considered the problem. (5th century BC)

In 414 BC the problem has been so well known already that Aristophanes (445 - 385) in „The Birds“ talks about people trying to squre the circle as of people who try the impossible.

Allowed tools, according to Platon: (427 - 348):

Ruler and compasses

Squaring of circle by Dinostratus (400 BC) Using the Quadratrix of Hippias of Elis (420 BC)

a

r

r

K

a

r

r

r

2

a

r

2

Egypt, Ahmosis, 2nd millenium BC: p/4 = (8/9)2 = 3,16...

Babylon, 2nd millenium BC: = 3 + 1/8 = 3,125

Jews: 5th century BC: = 3

Greece, Archimedes (287 - 212): = 22/7 = 3,1428...

China, Tsu Ch’ung-Chih (430 - 501): = 355/113 = 3,1415929... China, Liu hwuy (7th century AC): = 157/50 = 3,14

India, Brahmagupta (7th century AC): = 10 = 3,16...

Middle Ages, relapse into barbarism:Michael Psellus, Byzantium, 11th century AC: = 8 = 2,828...Franco of Liège, 11th century AC = (9/5)2 = 3,24Adrian Metius (1585): = 355/113 = 3,1415929... re-discovered

Digits of and processing speed

Jamshid Masud Al-Kashi (Persia) had 16 digits in 1424Ludolph van Ceulen (1540-1610) had 20 digits in 1596, 35 by the end of his life Ludolphs numberIsaac Newton (1642 - 1727) calculated 15 digits in 1665 to pass the time (pestilence)Abraham Sharp, beginning of the 18 century, 71 digitsSherwin 72 digitsMachin (1680 - 1752), calculated 100 digits in 1706Leonhard Euler (1707 – 1783) 20 digits within few hoursJohn Dase (1824 - 1861), wizard of calculaion, multiplied within hours 100-digit-numbers by mental arithmetic, calculated 205 digitsWilliam Shanks (1812 - 1882) produced 607 digits, 527 of which were correct, later (1853) 707 digits, but wrong beyond 527.The mistake was recognized only in 1945, when D.F. Ferguson calculated 530 digits. Last calculation with pencil and paper.1947 Ferguson calculated 808 digits using a desk calculator.1949 ENIAC (Electronic Numerical Integrator And Computer): 2037 digits in 70 hours1957 10000 digits, but only 7480 correct (machine fault)1958 IBM 704: 10.000 digits within 100 minutes1961 IBM 7090: 100.000 digits within 9 hours1973 CDC 7600: 1 mio. digits in less than 1 day1985 Symbolics 3670: 17 mio. digits1986 CRAY-2: 29 mio digits in less than 28 hours1987 100 mio.1995 6.442.450.000 digits, Yasumasa Kanada, Univ. Tokyo, within 116 hours1999 206.158.430.000 digits, Takahashi and Kanada on Hitachi SR8000, Univ. TokyoThe digits appear to be normally distributed p seems to be a normal number.

Modular identities (infinite series)

S. Ramanujan (1914)

10

- = 410-88

D.V. Chudnovsky and G.V. Chudnovsky (1989)

10

- = -310-156

Iterative prozesses lim an = 1/ for n (infinite sequences)

Borwein (1989)

I1/an - I < 164ne-24n

I1/a3 - I < 10-171 I1/a10 - I < 10-2861297

Bailey (1996)

I1/an - I < 165ne-5n

I1/a3 - I < 10-167 I1/a10 - I < 10-13323979

quanciesIn 1897 the draft bill 246 passed the parliament of Indiana defining

:= 3

It failed only in senate on intervention of prof. C. Waldo.

Rajan Mahadevan, on 5. 7. 1981 recited

in 3 h 49 min 31812 digits of by heart.

Hideaki Tomoyori, in 1987 recited

in 17 h 21 min 40000 digits of by heart.

World records :

04 Jun 1979 - 11 Jul 1979 (15,151)02 Oct 1979 - 26 Jun 1980 (20,000)10 Mar 1987 - 17 Feb 1995 (40,000)

quanciesIn 1897 the draft bill 246 passed the parliament of Indiana defining

:= 3

It failed only in senate on intervention of prof. C. Waldo.

Rajan Mahadevan, on 5. 7. 1981 recited

in 3 h 49 min 31812 digits of by heart.

Hideaki Tomoyori, in 1987 recited

in 17 h 21 min 40000 digits of by heart.

The approximation of the Japanese Arima

= 3.141592653589793238462643383275

supplies 30 digits of .

701171363081215493044282245933

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989

Among the first 1000 digits we find

116 1

103 2

103 3

93 4

97 5

94 6

95 7

100 8

106 9

93 0