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 15, 2015.

Transcript

1. Proofs Without Words
   Dana C. Ernst
   Northern Arizona University
   Mathematics & Statistics Department
   http://dcernst.github.io
   Friday Afternoon Mathematics Undergraduate Seminar
   September 18, 2015
   D.C. Ernst Proofs Without Words 1 / 17
   

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2. 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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3. 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 / 17
   

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4. Theorem
   2π > 6
   D.C. Ernst Proofs Without Words 4 / 17
   

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5. 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 5 / 17
   

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6. 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 6 / 17
   

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7. Theorem
   For all n ∈ N, 1 + 3 + 5 + · · · + (2n − 1) = n2.
   D.C. Ernst Proofs Without Words 7 / 17
   

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8. 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 8 / 17
   

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9. The nth pentagonal number is defined to be Pn
   :=
   3n2 − n
   2
   .
   Theorem
   Pn
   = 3Tn−1
   + n.
   D.C. Ernst Proofs Without Words 9 / 17
   

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10. 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 10 / 17
    

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11. Theorem
    We have the following summation formula:
    ∞
    ∑
    k=1
    (
    1
    2
    )
    k
    = 1.
    D.C. Ernst Proofs Without Words 11 / 17
    

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12. 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 12 / 17
    

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13. Theorem
    We have the following summation formula:
    ∞
    ∑
    k=1
    (
    1
    3
    )
    k
    =
    1
    2
    .
    D.C. Ernst Proofs Without Words 13 / 17
    

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14. Theorem
    A circle of radius r has area πr2.
    D.C. Ernst Proofs Without Words 14 / 17
    

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15. 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 15 / 17
    

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16. 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 16 / 17
    

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17. 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 17 / 17
    

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