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.

Transcript

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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