Dana Ernst
October 01, 2013
3.3k

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
September 20, 2013
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2. Warning!
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3. Warning!
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4. Warning!
Theorem?
Hmmm, it looks like 32.5 = 31.5.
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5. D.C. Ernst Proofs without Words 3 / 17

6. Theorem
For all n ∈ N, 1 + 2 + · · · + n =
n(n + 1)
2
.
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7. Theorem
For all n ∈ N, 1 + 2 + · · · + n =
n(n + 1)
2
.
Note
The numbers Tn
:= 1 + 2 + · · · + n are called triangular numbers.
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8. D.C. Ernst Proofs without Words 4 / 17

9. Theorem
For all n ∈ N, 1 + 2 + · · · + n = C(n + 1, 2) :=
(n + 1)!
2!(n − 1)!
.
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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
.
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11. D.C. Ernst Proofs without Words 5 / 17

12. Theorem
For all n ∈ N, 1 + 3 + 5 + · · · + (2n − 1) = n2.
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13. D.C. Ernst Proofs without Words 6 / 17

14. Theorem
For all n ∈ N, 1 + 3 + 5 + · · · + (2n − 1) = n2.
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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.
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16. D.C. Ernst Proofs without Words 7 / 17

17. Theorem
Let Pn
be the nth pentagonal number. Then Pn
= 3Tn−1
+ n.
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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
.
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19. D.C. Ernst Proofs without Words 8 / 17

20. Theorem (Nicomachus’ Theorem)
For all n ∈ N, 13 + 23 + · · · + n3 = (1 + 2 + · · · + n)2.
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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
.
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22. D.C. Ernst Proofs without Words 9 / 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.
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24. D.C. Ernst Proofs without Words 10 / 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.
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26. D.C. Ernst Proofs without Words 11 / 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.
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28. D.C. Ernst Proofs without Words 12 / 17

29. Theorem
We have the following fact concerning integrals:
π/2
0
sin2(x) dx =
π
4
=
π/2
0
cos2(x) dx.
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31. Theorem
We have the following summation formula:

k=1
1
2
k
= 1.
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33. Theorem
We have the following summation formula:

k=1
1
4
k
=
1
3
.
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35. Theorem
We have the following summation formula:

k=1
1
3
k
=
1
2
.
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36. D.C. Ernst Proofs without Words 16 / 17

37. Theorem
A circle of radius r has area πr2.
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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/
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