numbers are man's work gerhard post, dwmp mathematisch café, 17 juni 2013

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)

0-500-1000 1500-2000 500 1000-1500 1900

Numbers are man's work

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

0-500-1000 1500-2000 500 1000-1500 1900

Egyptian fractions

A Number is a sum of distinct unit fractions,

such as = + +

®

Rhind papyrus (1650 BC)

0-500-1000 1500-2000 500 1000-1500 1900

Egyptian fractions: construction

0-500-1000 1500-2000 500 1000-1500 1900

Egyptian fractions: why ?

A possible reason is easier (physical) division:

= +

0-500-1000 1500-2000 500 1000-1500 1900

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

®

0-500-1000 1500-2000 500 1000-1500 1900

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.

®

0-500-1000 1500-2000 500 1000-1500 1900

Orloj, Prague (15th century)

Orloj - Astronomical Clock - Prague

0-500-1000 1500-2000 500 1000-1500 1900

Orloj, Prague (15th century)

Toothed wheels

0-500-1000 1500-2000 500 1000-1500 1900

Orloj, Prague

A Number is a ratio of ‘small’ integers®

0-500-1000 1500-2000 500 1000-1500 1900

Orloj, Prague

A Number is a ratio of ‘small’ integers®

0-500-1000 1500-2000 500 1000-1500 1900

Orloj, Prague

How to construct these small integers ?

0-500-1000 1500-2000 500 1000-1500 1900

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

0-500-1000 1500-2000 500 1000-1500 1900

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

0-500-1000 1500-2000 500 1000-1500 1900

Simon Stevin Brugensis (1548 1620)

A Number is a decimal expansion

Simon Stevin

®

0-500-1000 1500-2000 500 1000-1500 1900

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.

0-500-1000 1500-2000 500 1000-1500 1900

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 ) )

0-500-1000 1500-2000 500 1000-1500 1900

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

0-500-1000 1500-2000 500 1000-1500 1900

Joseph Liouville (1809 - 1882)

Liouville’s constant: + … = 0.11000100000000000000000100…

Q: How many Liouville numbers are there?

A: As many as all decimal expansions…

0-500-1000 1500-2000 500 1000-1500 1900

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

0-500-1000 1500-2000 500 1000-1500 1900

Conclusions

A Number is …®

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