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.

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
2. Warning! Pictures can be misleading! D.C. Ernst Proofs without Words 2 / 17
3. Warning! Pictures can be misleading! D.C. Ernst Proofs without Words 2 / 17
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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6. Theorem For all n ∈ N, 1 + 2 + · · · + n = n(n + 1) 2 . D.C. Ernst Proofs without Words 3 / 17
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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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
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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12. Theorem For all n ∈ N, 1 + 3 + 5 + · · · + (2n − 1) = n2. D.C. Ernst Proofs without Words 5 / 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
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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17. Theorem Let Pn be the nth pentagonal number. Then Pn = 3Tn−1 + n. D.C. Ernst Proofs without Words 7 / 17
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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20. Theorem (Nicomachus’ Theorem) For all n ∈ N, 13 + 23 + · · · + n3 = (1 + 2 + · · · + n)2. D.C. Ernst Proofs without Words 8 / 17
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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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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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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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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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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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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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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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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37. Theorem A circle of radius r has area πr2. D.C. Ernst Proofs without Words 16 / 17
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