Numbers are man s work
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Numbers are man's work. Gerhard Post, DWMP Mathematisch Café, 17 juni 2013. Numbers are man's work. The dear God has made the whole numbers, all the rest is man's work . Leopold Kronecker (1823 - 1891). Two interwoven stories : The concept “ number ”

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Numbers are man s work

Numbers are man's work

Gerhard Post, DWMP

Mathematisch Café, 17 juni 2013


Numbers are man s work

Numbers are man's work

The dear God has made the whole numbers,

all the rest is man's work.

Leopold Kronecker (1823 - 1891)

  • Twointerwovenstories:

  • The concept “number”

  • The representation of a number.

Leopold Kronecker

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Numbers are man s work

Egyptian fractions

®

A Number is a sum of distinct unit fractions,

such as = + +

Rhind papyrus (1650 BC)

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Numbers are man s work

Egyptian fractions: construction

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Numbers are man s work

Egyptian fractions: why ?

A possiblereason is easier (physical) division:

= +

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Numbers are man s work

The Greek

A Number is a ratio of integers

®

or: a numberis a solution toanequation of the form:

c1 x + c0 = 0 (c1and c0 integers)

Hippasus (5thcentury BC) is believedto have discoveredthat is not a number

is not a number

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Numbers are man s work

The Greek (after Hippasus)

®

A Numberis a solution toanequation of the form:

cn xn + cn-1 xn-1 + … + c1x + c0 = 0

for integers cn,…,c0.

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Numbers are man s work

Orloj, Prague (15thcentury)

Orloj- AstronomicalClock - Prague

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Numbers are man s work

Orloj, Prague (15thcentury)

Toothedwheels

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Numbers are man s work

Orloj, Prague

A Number is a ratio of ‘small’ integers

®

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Numbers are man s work

Orloj, Prague

A Number is a ratio of ‘small’ integers

®

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Numbers are man s work

Orloj, Prague

How to construct these small integers ?

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Numbers are man s work

The Italians (Cardano’s “Ars Magna”, 1545)

A Numberis a solution toanequation of the form:

cn xn + cn-1 xn-1 + … + c1x + c0 = 0

®

GirolamoCardano

NiccolòTartaglia

Lodovico Ferrari

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Numbers are man s work

Solve: x3+ a x2+ b x + c = 0

Replace x by (xa) (drop the prime) getsrid of x2 :

2. Substituteu - v for x

3. Take 3uv = b:

4. Substitutev = 1/3b/u→ quadraticequation in u3.

x3 + b x + c = 0

(u33uv(uv)  v3) + b(u  v) + c = 0

u3 v3 + c = 0

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Numbers are man s work

Simon StevinBrugensis (1548 1620)

A Number is a decimalexpansion

®

Simon Stevin

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Numbers are man s work

Beginning of 19thcentury

A Number is analgebraicnumber (since 500 BC)

®

An algebraicnumberis a solution toanequation of the form:

cn xn + cn-1 xn-1 + … + c1x + c0 = 0

for integers cn,…,c0.

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Numbers are man s work

Joseph Liouville (1809 - 1882)

f(x) = cn x n + cn-1 x n-1 + … + c1x + c0 = 0

(integers cn,…, c0).

If is an irrational algebraic number satisfying f()=0 the equation above, then there exists a number A > 0 such that, for all integers p and qwithq > 0:

The key observation to prove this is: |f()| if f() ≠ 0,

and) )

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Numbers are man s work

Joseph Liouville (1809 - 1882)

A Number is analgebraic or a Liouville number

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A Liouville number is a number  with the property that, for every positive integer n, there exist integers p and q with q > 0 and such that

0 <

Joseph Liouville

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Numbers are man s work

Joseph Liouville (1809 - 1882)

Liouville’s constant: + …

= 0.11000100000000000000000100…

Q: How many Liouville numbers are there?

A: As many as alldecimalexpansions…

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Numbers are man s work

Georg Cantor (1845 –1918)

A Number is a decimalexpansion

®

Notallinfinities are the same

Georg Cantor

Leopold Kronecker: “I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there.”

David Hilbert: “No one will drive us from the paradise which Cantor created for us.”

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Numbers are man s work

Conclusions

A Number is …

®

Although the numbers are man’swork,

theybroughtustoparadise…

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