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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)

- Twointerwovenstories:
- The concept “number”
- The representation of a number.

Leopold Kronecker

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Egyptian fractions

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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 possiblereason is easier (physical) division:

= +

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The Greek

A Number is a ratio of integers

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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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The Greek (after Hippasus)

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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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Orloj, Prague (15thcentury)

Orloj- AstronomicalClock - Prague

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Orloj, Prague (15thcentury)

Toothedwheels

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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 Numberis a solution toanequation of the form:

cn xn + cn-1 xn-1 + … + c1x + c0 = 0

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GirolamoCardano

NiccolòTartaglia

Lodovico Ferrari

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

Replace x by (xa) (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

(u33uv(uv) v3) + b(u v) + c = 0

u3 v3 + c = 0

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Simon StevinBrugensis (1548 1620)

A Number is a decimalexpansion

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Simon Stevin

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Beginning of 19thcentury

A Number is analgebraicnumber (since 500 BC)

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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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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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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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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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Georg Cantor (1845 –1918)

A Number is a decimalexpansion

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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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Conclusions

A Number is …

®

Although the numbers are man’swork,

theybroughtustoparadise…

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