Upgrade to Pro — share decks privately, control downloads, hide ads and more …

Proofs Without Words

Dana Ernst
September 18, 2015

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.

Dana Ernst

September 18, 2015
Tweet

More Decks by Dana Ernst

Other Decks in Science

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

    View Slide

  2. Warning!
    Pictures can be misleading!
    Theorem?
    Hmmm, it looks like 32.5 = 31.5.
    D.C. Ernst Proofs Without Words 2 / 17

    View Slide

  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

    View Slide

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

    View Slide

  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

    View Slide

  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

    View Slide

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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  11. Theorem
    We have the following summation formula:


    k=1
    (
    1
    2
    )
    k
    = 1.
    D.C. Ernst Proofs Without Words 11 / 17

    View Slide

  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

    View Slide

  13. Theorem
    We have the following summation formula:


    k=1
    (
    1
    3
    )
    k
    =
    1
    2
    .
    D.C. Ernst Proofs Without Words 13 / 17

    View Slide

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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide