1. Numbers are man's work Gerhard Post, DWMP Mathematisch Café, 17 juni 2013

2. 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.

3. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Egyptian fractions ® Rhind papyrus (1650 BC)

4. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Egyptian fractions: construction

5. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Egyptian fractions: why ? A possible reason is easier (physical) division:

6. 0 -500 -1000 1500-2000 500 1000 -1500 1900 The Greek A Number is a ratio of integers or: a number is a solution to an equation of the form: c 1 x + c 0 = 0 (c 1 and c 0 integers) ®

7. 0 -500 -1000 1500-2000 500 1000 -1500 1900 The Greek (after Hippasus) A Number is a solution to an equation of the form: c n x n + c n-1 x n-1 + … + c 1 x + c 0 = 0 for integers c n,…,c 0. ®

8. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Orloj, Prague (15 th century) Orloj - Astronomical Clock - Prague

9. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Orloj, Prague (15 th century) Toothed wheels

10. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Orloj, Prague A Number is a ratio of ‘small’ integers ®

11. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Orloj, Prague A Number is a ratio of ‘small’ integers ®

12. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Orloj, Prague How to construct these small integers ?

13. 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: c n x n + c n-1 x n-1 + … + c 1 x + c 0 = 0 ® Girolamo CardanoNiccolò Tartaglia Lodovico Ferrari

14. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Solve: x 3 + a x 2 + b x + c = 0 x 3 + b x + c = 0 (u 3  3uv (u  v)  v 3 ) + b (u  v) + c = 0 u 3  v 3 + c = 0

15. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Simon Stevin Brugensis (1548  1620) A Number is a decimal expansion Simon Stevin ®

16. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Beginning of 19 th century A Number is an algebraic number (since 500 BC) ® An algebraic number is a solution to an equation of the form: c n x n + c n-1 x n-1 + … + c 1 x + c 0 = 0 for integers c n,…,c 0.

17. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Joseph Liouville (1809 - 1882) f(x) = c n x n + c n-1 x n-1 + … + c 1 x + c 0 = 0 (integers c n,…, c 0 ).

18. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Joseph Liouville (1809 - 1882) A Number is an algebraic or a Liouville number ® Joseph Liouville

19. 0 -500 -1000 1500-2000 500 1000 -1500 1900 Joseph Liouville (1809 - 1882) Q: How many Liouville numbers are there? A: As many as all decimal expansions…

20. 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

21. 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…