Dana Ernst
October 23, 2017
420

Proofs without Words

In this talk, we'll explore several cool mathematical theorems from a visual perspective.

This talk was given as a guest lecture in MAT 123 at Northern Arizona University on October 23, 2017.

October 23, 2017

Transcript

1. Proofs Without Words Dana C. Ernst Northern Arizona University Mathematics

& Statistics Department http://danaernst.com MAT 123 October 23, 2017 D.C. Ernst Proofs Without Words 1 / 19

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2 / 19
4. Warning! Pictures can be misleading! Theorem? Hmmm, it looks like

32.5 = 31.5. D.C. Ernst Proofs Without Words 2 / 19
5. 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 / 19

7. Theorem For all n ∈ N, 1 + 3 +

5 + · · · + (2n − 1) = n2. D.C. Ernst Proofs Without Words 4 / 19

9. This the same as the previous theorem, but with a

diﬀerent visual proof. D.C. Ernst Proofs Without Words 5 / 19
10. 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 5 / 19

12. Theorem For all n ∈ N, 1 + 2 +

· · · + n = n(n + 1) 2 . D.C. Ernst Proofs Without Words 6 / 19
13. 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 6 / 19

15. Theorem For all n ∈ N, 1 + 2 +

· · · + n = C(n + 1, 2) := (n + 1)! 2!(n − 1)! . D.C. Ernst Proofs Without Words 7 / 19
16. 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 7 / 19
17. The nth pentagonal number is deﬁned to be Pn :=

3n2 − n 2 . D.C. Ernst Proofs Without Words 8 / 19
18. The nth pentagonal number is deﬁned to be Pn :=

3n2 − n 2 . D.C. Ernst Proofs Without Words 8 / 19
19. The nth pentagonal number is deﬁned to be Pn :=

3n2 − n 2 . Theorem Pn = 3Tn−1 + n. D.C. Ernst Proofs Without Words 8 / 19

21. Theorem (Nicomachus’ Theorem) For all n ∈ N, 13 +

23 + · · · + n3 = (1 + 2 + · · · + n)2. D.C. Ernst Proofs Without Words 9 / 19
22. 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 9 / 19

/ 19

26. 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 / 19

28. 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 12 / 19

30. Theorem We have the following summation formula: ∞ ∑ k=1

( 1 2 ) k = 1. D.C. Ernst Proofs Without Words 13 / 19

32. Hint Focus on a single color. D.C. Ernst Proofs Without

Words 14 / 19
33. 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 14 / 19

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

( 1 3 ) k = 1 2 . D.C. Ernst Proofs Without Words 15 / 19

37. Theorem A circle of radius r has area πr2. D.C.

Ernst Proofs Without Words 16 / 19

39. 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 17 / 19

41. 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 18 / 19
42. 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 19 / 19