340

# 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

2. Warning!
D.C. Ernst Proofs Without Words 2 / 19

3. Warning!
D.C. Ernst Proofs Without Words 2 / 19

4. Warning!
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

6. D.C. Ernst Proofs Without Words 4 / 19

7. Theorem
For all n ∈ N, 1 + 3 + 5 + · · · + (2n − 1) = n2.
D.C. Ernst Proofs Without Words 4 / 19

8. D.C. Ernst Proofs Without Words 5 / 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

11. D.C. Ernst Proofs Without Words 6 / 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

14. D.C. Ernst Proofs Without Words 7 / 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

20. D.C. Ernst Proofs Without Words 9 / 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

23. D.C. Ernst Proofs Without Words 10 / 19

24. Theorem
2π > 6
D.C. Ernst Proofs Without Words 10 / 19

25. D.C. Ernst Proofs Without Words 11 / 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

27. D.C. Ernst Proofs Without Words 12 / 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

29. D.C. Ernst Proofs Without Words 13 / 19

30. Theorem
We have the following summation formula:

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

31. D.C. Ernst Proofs Without Words 14 / 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

34. D.C. Ernst Proofs Without Words 15 / 19

35. Theorem
We have the following summation formula:

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

36. D.C. Ernst Proofs Without Words 16 / 19

37. Theorem
A circle of radius r has area πr2.
D.C. Ernst Proofs Without Words 16 / 19

38. D.C. Ernst Proofs Without Words 17 / 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

40. D.C. Ernst Proofs Without Words 18 / 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