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
2. Warning! Pictures can be misleading! Theorem? Hmmm, it looks like 32.5 = 31.5. D.C. Ernst Proofs Without Words 2 / 17
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
4. Theorem 2π > 6 D.C. Ernst Proofs Without Words 4 / 17
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
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
7. Theorem For all n ∈ N, 1 + 3 + 5 + · · · + (2n − 1) = n2. D.C. Ernst Proofs Without Words 7 / 17
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
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
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
11. Theorem We have the following summation formula: ∞ ∑ k=1 ( 1 2 ) k = 1. D.C. Ernst Proofs Without Words 11 / 17
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
13. Theorem We have the following summation formula: ∞ ∑ k=1 ( 1 3 ) k = 1 2 . D.C. Ernst Proofs Without Words 13 / 17
14. Theorem A circle of radius r has area πr2. D.C. Ernst Proofs Without Words 14 / 17
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
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
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