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Proofs without Words

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.

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

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

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

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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 / 17

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  5. D.C. Ernst Proofs without Words 3 / 17

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

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  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

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  8. D.C. Ernst Proofs without Words 4 / 17

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  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

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  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

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  11. D.C. Ernst Proofs without Words 5 / 17

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

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  13. D.C. Ernst Proofs without Words 6 / 17

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

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  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

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  16. D.C. Ernst Proofs without Words 7 / 17

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

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  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

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  19. D.C. Ernst Proofs without Words 8 / 17

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

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  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

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  22. D.C. Ernst Proofs without Words 9 / 17

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  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

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  24. D.C. Ernst Proofs without Words 10 / 17

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  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

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  26. D.C. Ernst Proofs without Words 11 / 17

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  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

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  28. D.C. Ernst Proofs without Words 12 / 17

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  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

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  30. D.C. Ernst Proofs without Words 13 / 17

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  31. Theorem
    We have the following summation formula:

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

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  32. D.C. Ernst Proofs without Words 14 / 17

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  33. Theorem
    We have the following summation formula:

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

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  34. D.C. Ernst Proofs without Words 15 / 17

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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 / 17

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  36. D.C. Ernst Proofs without Words 16 / 17

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  37. Theorem
    A circle of radius r has area πr2.
    D.C. Ernst Proofs without Words 16 / 17

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  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

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