Numbers are man s work
Sponsored Links
This presentation is the property of its rightful owner.
1 / 21

Numbers are man's work PowerPoint PPT Presentation


  • 52 Views
  • Uploaded on
  • Presentation posted in: General

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 ”

Download Presentation

Numbers are man's work

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Egyptian fractions

®

A Number is a sum of distinct unit fractions,

such as = + +

Rhind papyrus (1650 BC)

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Egyptian fractions: construction

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Egyptian fractions: why ?

A possiblereason is easier (physical) division:

= +

-1500

-500

500

-2000

0

1500

-1000

1000

1900


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

-1500

-500

500

-2000

0

1500

-1000

1000

1900


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.

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Orloj, Prague (15thcentury)

Orloj- AstronomicalClock - Prague

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Orloj, Prague (15thcentury)

Toothedwheels

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Orloj, Prague

A Number is a ratio of ‘small’ integers

®

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Orloj, Prague

A Number is a ratio of ‘small’ integers

®

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Orloj, Prague

How to construct these small integers ?

-1500

-500

500

-2000

0

1500

-1000

1000

1900


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

-1500

-500

500

-2000

0

1500

-1000

1000

1900


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

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Simon StevinBrugensis (1548 1620)

A Number is a decimalexpansion

®

Simon Stevin

-1500

-500

500

-2000

0

1500

-1000

1000

1900


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.

-1500

-500

500

-2000

0

1500

-1000

1000

1900


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) )

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Joseph Liouville (1809 - 1882)

A Number is analgebraic or a Liouville number

®

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

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Joseph Liouville (1809 - 1882)

Liouville’s constant: + …

= 0.11000100000000000000000100…

Q: How many Liouville numbers are there?

A: As many as alldecimalexpansions…

-1500

-500

500

-2000

0

1500

-1000

1000

1900


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.”

-1500

-500

500

-2000

0

1500

-1000

1000

1900


Conclusions

A Number is …

®

Although the numbers are man’swork,

theybroughtustoparadise…

-1500

-500

500

-2000

0

1500

-1000

1000

1900


  • Login