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# 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.

October 01, 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

2 / 17

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

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

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

12. ### Theorem For all n ∈ N, 1 + 3 +

5 + · · · + (2n − 1) = n2. D.C. Ernst Proofs without Words 5 / 17

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

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

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

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

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

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

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

31. ### Theorem We have the following summation formula: ∞ k=1 1

2 k = 1. D.C. Ernst Proofs without Words 13 / 17

33. ### Theorem We have the following summation formula: ∞ k=1 1

4 k = 1 3 . D.C. Ernst Proofs without Words 14 / 17

35. ### Theorem We have the following summation formula: ∞ k=1 1

3 k = 1 2 . D.C. Ernst Proofs without Words 15 / 17

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