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Numbers are man's work

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 ”

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Numbers are man's work

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

  2. 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

  3. 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

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

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

  6. 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

  7. 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

  8. Orloj, Prague (15thcentury) Orloj- AstronomicalClock - Prague -1500 -500 500 -2000 0 1500 -1000 1000 1900

  9. Orloj, Prague (15thcentury) Toothedwheels -1500 -500 500 -2000 0 1500 -1000 1000 1900

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

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

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

  13. 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

  14. 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

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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. Conclusions A Number is … ® Although the numbers are man’swork, theybroughtustoparadise… -1500 -500 500 -2000 0 1500 -1000 1000 1900

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