Math 2.0

Number theory Diophantine approximation and transcendence Theorem

Irrationality of pi

Builds on piSine and cosine

Statement

The number π\pi is irrational.

Remarks

Ivan Niven gave a short proof by calculus. Assume π=a/b\pi = a/b, and let f(x)=xn(a−bx)nn!f(x) = \frac{x^n (a - b x)^n}{n!}. Then ∫0πf(x)sin⁡x dx\int_0^\pi f(x) \sin x \, dx is a positive integer for every nn. But it tends to 00 as n→∞n \to \infty, a contradiction. In 1882, Ferdinand von Lindemann proved the stronger fact that π\pi is transcendental. Mathlib states irrationality as irrational_pi.

History

Johann Heinrich Lambert gave the first proof in the 1760s. He used a continued fraction for tan⁡x\tan x.

Reading path

To understand this page from the ground up, read these 57 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Bounded sequence definition
  11. Cauchy sequence definition
  12. Geometric sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Series definition
  16. Continuous function definition
  17. Fermat's theorem on stationary points statement
  18. Limit superior definition
  19. Cauchy sequences are bounded statement
  20. Sum of a finite geometric series statement
  21. Monotone convergence theorem for sequences statement
  22. Every sequence has a monotone subsequence statement
  23. Convergent sequence definition
  24. Pointwise convergence definition
  25. Squeeze theorem statement
  26. Partial sum definition
  27. Power series definition
  28. Intermediate value theorem statement
  29. Bolzano–Weierstrass theorem statement
  30. The sequence r^n tends to zero for |r| < 1 example
  31. Radius of convergence definition
  32. Convergent series definition
  33. Uniform convergence definition
  34. Sine and cosine definition
  35. Boundedness theorem statement
  36. Cauchy criterion for sequences statement
  37. Uniform convergence on a set definition
  38. Absolute convergence definition
  39. A series of nonnegative terms converges iff its partial sums are bounded statement
  40. Series of functions definition
  41. Sum of a series definition
  42. Extreme value theorem statement
  43. Uniformly Cauchy sequence of functions definition
  44. Comparison test statement
  45. Geometric series example
  46. Rolle's theorem statement
  47. Root test statement
  48. Weierstrass M-test statement
  49. Mean value theorem statement
  50. Cauchy–Hadamard theorem statement
  51. Power series converge uniformly on smaller discs inside the radius of convergence statement
  52. Uniform convergence of derivatives allows term-by-term differentiation statement
  53. Term-by-term differentiation statement
  54. Power series are differentiable inside the radius of convergence statement
  55. Derivative of cosine statement
  56. pi object
  57. Irrationality of pi statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Proof_that_pi_is_irrational, Niven's proof
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