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Proofs without Words

Dana Ernst
October 23, 2017

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.

Dana Ernst

October 23, 2017
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  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

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

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

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

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  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

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

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

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

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

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  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

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

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

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  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

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

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  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

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  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

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

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

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  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

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

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

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  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

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

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

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

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  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

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

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  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

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

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  30. Theorem
    We have the following summation formula:


    k=1
    (
    1
    2
    )
    k
    = 1.
    D.C. Ernst Proofs Without Words 13 / 19

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

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

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  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

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

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  35. Theorem
    We have the following summation formula:


    k=1
    (
    1
    3
    )
    k
    =
    1
    2
    .
    D.C. Ernst Proofs Without Words 15 / 19

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

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

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

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  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

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

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  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

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  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

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