Proofs without Words

77d59004fef10003e155461c4c47e037?s=47 Dana Ernst
October 01, 2013

Proofs without Words

In this FAMUS talk, we'll explore several cool mathematical theorems from a visual perspective.

This talk was given at the Northern Arizona University Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) on Friday, September 20, 2013.

77d59004fef10003e155461c4c47e037?s=128

Dana Ernst

October 01, 2013
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  1. Proofs without Words Dana C. Ernst Northern Arizona University Mathematics

    & Statistics Department http://danaernst.com Friday Afternoon Mathematics Undergraduate Seminar September 20, 2013 D.C. Ernst Proofs without Words 1 / 17
  2. Warning! Pictures can be misleading! D.C. Ernst Proofs without Words

    2 / 17
  3. Warning! Pictures can be misleading! D.C. Ernst Proofs without Words

    2 / 17
  4. Warning! Pictures can be misleading! Theorem? Hmmm, it looks like

    32.5 = 31.5. D.C. Ernst Proofs without Words 2 / 17
  5. D.C. Ernst Proofs without Words 3 / 17

  6. Theorem For all n ∈ N, 1 + 2 +

    · · · + n = n(n + 1) 2 . D.C. Ernst Proofs without Words 3 / 17
  7. Theorem For all n ∈ N, 1 + 2 +

    · · · + n = n(n + 1) 2 . Note The numbers Tn := 1 + 2 + · · · + n are called triangular numbers. D.C. Ernst Proofs without Words 3 / 17
  8. D.C. Ernst Proofs without Words 4 / 17

  9. Theorem For all n ∈ N, 1 + 2 +

    · · · + n = C(n + 1, 2) := (n + 1)! 2!(n − 1)! . D.C. Ernst Proofs without Words 4 / 17
  10. Theorem For all n ∈ N, 1 + 2 +

    · · · + n = C(n + 1, 2) := (n + 1)! 2!(n − 1)! . Corollary For all n ∈ N, C(n + 1, 2) = n(n + 1) 2 . D.C. Ernst Proofs without Words 4 / 17
  11. D.C. Ernst Proofs without Words 5 / 17

  12. Theorem For all n ∈ N, 1 + 3 +

    5 + · · · + (2n − 1) = n2. D.C. Ernst Proofs without Words 5 / 17
  13. D.C. Ernst Proofs without Words 6 / 17

  14. Theorem For all n ∈ N, 1 + 3 +

    5 + · · · + (2n − 1) = n2. D.C. Ernst Proofs without Words 6 / 17
  15. Theorem For all n ∈ N, 1 + 3 +

    5 + · · · + (2n − 1) = n2. Note This the same as the previous theorem, but with a different visual proof. D.C. Ernst Proofs without Words 6 / 17
  16. D.C. Ernst Proofs without Words 7 / 17

  17. Theorem Let Pn be the nth pentagonal number. Then Pn

    = 3Tn−1 + n. D.C. Ernst Proofs without Words 7 / 17
  18. Theorem Let Pn be the nth pentagonal number. Then Pn

    = 3Tn−1 + n. Note The nth pentagonal number is given by Pn := 3n2 − n 2 . D.C. Ernst Proofs without Words 7 / 17
  19. D.C. Ernst Proofs without Words 8 / 17

  20. Theorem (Nicomachus’ Theorem) For all n ∈ N, 13 +

    23 + · · · + n3 = (1 + 2 + · · · + n)2. D.C. Ernst Proofs without Words 8 / 17
  21. Theorem (Nicomachus’ Theorem) For all n ∈ N, 13 +

    23 + · · · + n3 = (1 + 2 + · · · + n)2. Corollary For all n ∈ N, 13 + 23 + · · · + n3 = n(n + 1) 2 2 . D.C. Ernst Proofs without Words 8 / 17
  22. D.C. Ernst Proofs without Words 9 / 17

  23. Theorem The alternating sum of the first n odd natural

    numbers is n. In other words, for all n ∈ N, n k=1 (−1)n−k(2k − 1) = n. D.C. Ernst Proofs without Words 9 / 17
  24. D.C. Ernst Proofs without Words 10 / 17

  25. Theorem (Pythagorean Theorem) If a, b, c ∈ N are

    the lengths of the sides of a right triangle, where c the length of the hypotenuse, then a2 + b2 = c2. D.C. Ernst Proofs without Words 10 / 17
  26. D.C. Ernst Proofs without Words 11 / 17

  27. Theorem (Pythagorean Theorem) If a, b, c ∈ N are

    the lengths of the sides of a right triangle, where c the length of the hypotenuse, then a2 + b2 = c2. D.C. Ernst Proofs without Words 11 / 17
  28. D.C. Ernst Proofs without Words 12 / 17

  29. Theorem We have the following fact concerning integrals: π/2 0

    sin2(x) dx = π 4 = π/2 0 cos2(x) dx. D.C. Ernst Proofs without Words 12 / 17
  30. D.C. Ernst Proofs without Words 13 / 17

  31. Theorem We have the following summation formula: ∞ k=1 1

    2 k = 1. D.C. Ernst Proofs without Words 13 / 17
  32. D.C. Ernst Proofs without Words 14 / 17

  33. Theorem We have the following summation formula: ∞ k=1 1

    4 k = 1 3 . D.C. Ernst Proofs without Words 14 / 17
  34. D.C. Ernst Proofs without Words 15 / 17

  35. Theorem We have the following summation formula: ∞ k=1 1

    3 k = 1 2 . D.C. Ernst Proofs without Words 15 / 17
  36. D.C. Ernst Proofs without Words 16 / 17

  37. Theorem A circle of radius r has area πr2. D.C.

    Ernst Proofs without Words 16 / 17
  38. Sources All the figures in this talk came from the

    following locations: • MathOverflow: http://mathoverflow.net/questions/8846/ proofs-without-words • Art of Problem Solving: http://www.artofproblemsolving.com/Wiki/index. php/Proofs_without_words • Strogatz, NY Times: http://opinionator.blogs.nytimes.com/2010/04/04/ take-it-to-the-limit/ D.C. Ernst Proofs without Words 17 / 17