Proofs without Words

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

This talk was given as a guest lecture in MAT 123 at Northern Arizona University on October 23, 2017.

Transcript

1. Proofs Without Words
   Dana C. Ernst
   Northern Arizona University
   Mathematics & Statistics Department
   http://danaernst.com
   MAT 123
   October 23, 2017
   D.C. Ernst Proofs Without Words 1 / 19
   

   View Slide

2. Warning!
   Pictures can be misleading!
   D.C. Ernst Proofs Without Words 2 / 19
   

   View Slide

3. Warning!
   Pictures can be misleading!
   D.C. Ernst Proofs Without Words 2 / 19
   

   View Slide

4. Warning!
   Pictures can be misleading!
   Theorem?
   Hmmm, it looks like 32.5 = 31.5.
   D.C. Ernst Proofs Without Words 2 / 19
   

   View Slide

5. Play Time
   Let’s play a game.
   • I’ll show you a picture,
   • You see if you can figure out what mathematical fact it
   describes or proofs.
   D.C. Ernst Proofs Without Words 3 / 19
   

   View Slide

6. D.C. Ernst Proofs Without Words 4 / 19
   

   View Slide

7. Theorem
   For all n ∈ N, 1 + 3 + 5 + · · · + (2n − 1) = n2.
   D.C. Ernst Proofs Without Words 4 / 19
   

   View Slide

8. D.C. Ernst Proofs Without Words 5 / 19
   

   View Slide

9. This the same as the previous theorem, but with a different visual
   proof.
   D.C. Ernst Proofs Without Words 5 / 19
   

   View Slide

10. This the same as the previous theorem, but with a different visual
    proof.
    Theorem
    For all n ∈ N, 1 + 3 + 5 + · · · + (2n − 1) = n2.
    D.C. Ernst Proofs Without Words 5 / 19
    

    View Slide

11. D.C. Ernst Proofs Without Words 6 / 19
    

    View Slide

12. Theorem
    For all n ∈ N, 1 + 2 + · · · + n =
    n(n + 1)
    2
    .
    D.C. Ernst Proofs Without Words 6 / 19
    

    View Slide

13. 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 6 / 19
    

    View Slide

14. D.C. Ernst Proofs Without Words 7 / 19
    

    View Slide

15. Theorem
    For all n ∈ N, 1 + 2 + · · · + n = C(n + 1, 2) :=
    (n + 1)!
    2!(n − 1)!
    .
    D.C. Ernst Proofs Without Words 7 / 19
    

    View Slide

16. 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 7 / 19
    

    View Slide

17. The nth pentagonal number is defined to be Pn
    :=
    3n2 − n
    2
    .
    D.C. Ernst Proofs Without Words 8 / 19
    

    View Slide

18. The nth pentagonal number is defined to be Pn
    :=
    3n2 − n
    2
    .
    D.C. Ernst Proofs Without Words 8 / 19
    

    View Slide

19. The nth pentagonal number is defined to be Pn
    :=
    3n2 − n
    2
    .
    Theorem
    Pn
    = 3Tn−1
    + n.
    D.C. Ernst Proofs Without Words 8 / 19
    

    View Slide

20. D.C. Ernst Proofs Without Words 9 / 19
    

    View Slide

21. Theorem (Nicomachus’ Theorem)
    For all n ∈ N, 13 + 23 + · · · + n3 = (1 + 2 + · · · + n)2.
    D.C. Ernst Proofs Without Words 9 / 19
    

    View Slide

22. 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 9 / 19
    

    View Slide

23. D.C. Ernst Proofs Without Words 10 / 19
    

    View Slide

24. Theorem
    2π > 6
    D.C. Ernst Proofs Without Words 10 / 19
    

    View Slide

25. D.C. Ernst Proofs Without Words 11 / 19
    

    View Slide

26. 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 / 19
    

    View Slide

27. D.C. Ernst Proofs Without Words 12 / 19
    

    View Slide

28. 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 12 / 19
    

    View Slide

29. D.C. Ernst Proofs Without Words 13 / 19
    

    View Slide

30. Theorem
    We have the following summation formula:
    ∞
    ∑
    k=1
    (
    1
    2
    )
    k
    = 1.
    D.C. Ernst Proofs Without Words 13 / 19
    

    View Slide

31. D.C. Ernst Proofs Without Words 14 / 19
    

    View Slide

32. Hint
    Focus on a single color.
    D.C. Ernst Proofs Without Words 14 / 19
    

    View Slide

33. Hint
    Focus on a single color.
    Theorem
    We have the following summation formula:
    ∞
    ∑
    k=1
    (
    1
    4
    )
    k
    =
    1
    3
    .
    D.C. Ernst Proofs Without Words 14 / 19
    

    View Slide

34. D.C. Ernst Proofs Without Words 15 / 19
    

    View Slide

35. Theorem
    We have the following summation formula:
    ∞
    ∑
    k=1
    (
    1
    3
    )
    k
    =
    1
    2
    .
    D.C. Ernst Proofs Without Words 15 / 19
    

    View Slide

36. D.C. Ernst Proofs Without Words 16 / 19
    

    View Slide

37. Theorem
    A circle of radius r has area πr2.
    D.C. Ernst Proofs Without Words 16 / 19
    

    View Slide

38. D.C. Ernst Proofs Without Words 17 / 19
    

    View Slide

39. 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 17 / 19
    

    View Slide

40. D.C. Ernst Proofs Without Words 18 / 19
    

    View Slide

41. 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 18 / 19
    

    View Slide

42. Sources
    MathOverflow:
    mathoverflow.net/questions/8846/proofs-without-words
    Art of Problem Solving:
    artofproblemsolving.com/Wiki/index.php/Proofs_
    without_words
    Wikipedia:
    en.wikipedia.org/wiki/Squared_triangular_number
    Strogatz, NY Times:
    opinionator.blogs.nytimes.com/2010/04/04/
    take-it-to-the-limit/
    D.C. Ernst Proofs Without Words 19 / 19
    

    View Slide