Numbers are man's work

Gerhard Post, DWMP

Mathematisch Café, 17 juni 2013

The dear God has made the whole numbers,

all the rest is man's work.

Leopold Kronecker (1823 - 1891)

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Numbers are man's work

Leopold Kronecker Two interwoven stories:• The concept “number”• The representation of a number.

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Egyptian fractions

A Number is a sum of distinct unit fractions,

such as = + +

®

Rhind papyrus (1650 BC)

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Egyptian fractions: construction

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Egyptian fractions: why ?

A possible reason is easier (physical) division:

= +

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The Greek

A Number is a ratio of integers

is not a number

or: a number is a solution to an equation of the form:

c1 x + c0 = 0 (c1 and c0 integers)

Hippasus (5th century BC) is believed

to have discovered that is not a

number

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The Greek (after Hippasus)

A Number is a solution to an equation of the form:

cn x n + cn-1 x

n-1 + … + c1 x + c0 = 0

for integers cn ,…,c0.

®

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Orloj, Prague (15th century)

Orloj - Astronomical Clock - Prague

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Orloj, Prague (15th century)

Toothed wheels

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Orloj, Prague

A Number is a ratio of ‘small’ integers®

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Orloj, Prague

A Number is a ratio of ‘small’ integers®

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Orloj, Prague

How to construct these small integers ?

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

A Number is a solution to an equation of the form:

cn x n + cn-1 x

n-1 + … + c1 x + c0 = 0

®

Girolamo Cardano Niccolò Tartaglia Lodovico Ferrari

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Solve: x 3 + a x

2 + b x + c = 0

1. Replace x by (x a) (drop the prime) gets rid of x 2 :

2. Substitute u - v for x

3. Take 3u v = b:

4. Substitute v = 1/3 b/u → quadratic equation in u3.

x 3 + b x + c = 0

(u 3 3uv (u v) v

3) + b (u v) + c = 0

u 3 v

3 + c = 0

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Simon Stevin Brugensis (1548 1620)

A Number is a decimal expansion

Simon Stevin

®

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Beginning of 19th century

A Number is an algebraic number (since 500 BC)®

An algebraic number is a solution to an equation of the form:

cn x n + cn-1 x

n-1 + … + c1 x + c0 = 0

for integers cn ,…,c0.

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Joseph Liouville (1809 - 1882)

f(x) = cn x n + cn-1 x n-1 + … + c1 x + 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 q with q > 0:

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

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Joseph Liouville (1809 - 1882)

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 <

A Number is an algebraic or a Liouville number®

Joseph Liouville

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Joseph Liouville (1809 - 1882)

Liouville’s constant: + … = 0.11000100000000000000000100…

Q: How many Liouville numbers are there?

A: As many as all decimal expansions…

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Georg Cantor (1845 –1918)

A Number is a decimal expansion®

Not all infinities are the same

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.”

Georg Cantor

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Conclusions

A Number is …®

Although the numbers are man’s work,they brought us to paradise…