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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