Proof by Picture
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Transcript of Proof by Picture
![Page 1: Proof by Picture](https://reader034.fdocuments.us/reader034/viewer/2022051121/577cc0121a28aba7118ebfc7/html5/thumbnails/1.jpg)
Proof By Picture
Robertson Bayer
University of California, Berkeley
Math 55, Summer 2009August 13, 2009
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Today
Integer Sums
Non-Proof
Fibonacci Identities
Geometry
Partition Numbers
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Triangular Numbers Revisited
DefinitionThe nth triangular number is denoted Tn and is defined by
Tn = 1 + 2 + · · ·+ n
Theorem
Tn =n(n + 1)
2
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Triangular Numbers Revisited
Traditional Proof.Induction on n:
I BC: (n=1) T1 = 1 = 1·22
I IH: Suppose 1 + 2 + · · ·+ k = k(k+1)2
I IS:1 + 2 + · · ·+ k + k + 1 IH
=k(k + 1)
2+ k + 1
=
(k2
+ 1)
(k + 1)
=(k + 2)(k + 1)
2
Yuk! This gives no info about why this should be true, and requiredthat we knew the formula in advance.
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Triangular Numbers RevisitedRepresenting Tn with dots
Let’s try a visual representation:
n n
Tn blue dots and Tn red dots for a grand total of 2Tn dots
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Triangular Numbers Revisited
Proof.
n+1
n
2Tn = Total Dots = n(n + 1)
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T2n
Theorem
T2n = 3Tn + Tn−1
Traditional Proof.
T2n =2n(2n + 1)
2=
4n2 + 2n2
=(3n2 + 3n) + (n2 − n)
2= 3
n(n + 1)
2+
(n − 1)n2
= 3Tn + Tn−1
Yuk! Not only did we have to know the formula in advance, but wehad to do some creative algebra in the middle.
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T2n = 3Tn + Tn−1Induction?
Induction Proof.I BC: T4 = 10 = 3 · 3 + 1 = 3T2 + T1
I IH: T2k = 3Tk + Tk−1
I IS: T2n = 1 + 2 + · · ·+ 2k + k + 1 + 2k + 2IH= 3Tk + Tk−1 + (2k + 1) + (2k + 2)
= 3Tk + Tk−1 + (k + k + 1) + 2(k + 1)
= 3(Tk + (k + 1)) + Tk−1 + k= 3Tk+1 + Tk
Yuk! Not only did we have to know the formula in advance, but the ISalso required some creative algebra.Plus, we still don’t really know why this should be true.
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T2nRepresenting 3Tn + Tn−1 with dots
Let’s try a visual representation:
n n−1n n
And if we cleverly rearrange these...
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T2n = 3Tn + Tn−1
Proof.
n
3Tn + Tn−1 = Total Dots = T2n
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Plus, while doing this proof we also showed:
Corollary (of the proof)Tn + Tn−1 = n2
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The Main PointNot only are visual proofs sometimes easier than traditional proofs,they can also help explain why a certain result is true.which is of course the hallmark of a good proof.
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Sum of the Odd Integers
Theorem
1 + 3 + 5 + · · ·+ (2k − 1) = k2
Proof.
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Sum of Squares
DefinitionLet Sn be the sum of the first n perfect squares. So
Sn = 12 + 22 + 32 + · · ·+ n2
Theorem
Sn =n(n + 1)(2n + 1)
6
Traditional Proof.Messy induction–see old quiz solutions.
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Sum of Squares
Elegant Proof.
n(n+1)/2
2n+1
2Sn Black Dots + Sn Color Dots = 3SnTotal Dots = n(n+1)(2n+1)2
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When a Picture Isn’t a Proof
ClaimThe curves y = x2 and y = 2x − 1 cross exactly once.
“Proof”.
(1,1)
This depends on the ability to draw these graphs perfectly!
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GuidelinesIn general, a picture is a proof only if:
1. The picture represents an abstract idea
n+1
n
(1 + 2 + · · · + n) + (n + (n − 1) + · · · + 1) =
(n + 1) + (n + 1) + · · · (n + 1) = n(n + 1)
2. The specific drawing of the picture isn’t actually important3. The picture can be “scaled up” to as big an n as necessary
Remember: it’s not the picture that’s the proof–it’s the idea that thepicture is representing that really counts
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The Fibonacci Numbers
DefinitionThe Fibonacci numbers are defined by
F0 = F1 = 1; Fn+2 = Fn+1 + Fn
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Sum of Squares of Fibonacci Numbers
TheoremF 2
0 + F 21 + F 2
2 + · · ·+ F 2n = FnFn+1
Traditional Proof.1. BC (n=0): F 2
0 = 1 = 1 · 1 = F0F1
2. IH: F0 + · · ·+ F 2k = Fk Fk+1
3. IS: F0 + · · ·+ F 2k + F 2
k+1IH= Fk Fk+1 + F 2
k+1
= Fk+1(Fk + Fk+1)
= Fk+1Fk+2
Not bad, but still not very enlightening. Stillrequired that we knew the formula in advance.
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Let’s try a visual representation:2
0 2 3 4F
2
F2
1F
2
F2
F
Now let’s see if we can arrange these cleverly
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F 20 + F 2
1 + · · ·+ F 2n = FnFn+1
Visual Proof.
F 20 + F 2
1 + · · ·+ F 2n = Total Area = Fn(Fn + Fn−1) = FnFn+1
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Theorem
Fn+m = FnFm + Fn−1Fm−1
Induction Proof.Induction on what? n? m?Yuk!
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Fibonacci Numbers as Tilings
Recall:
TheoremThere are Fn ways to tile a 1× n board with 1× 1 and 1× 2rectangles.
Proof.
n−1
−or−
n−2
n
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Theorem
Fn+m = FnFm + Fn−1Fm−1
Tiling Proof.
nm
m+n −or−
m−1 n−1
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TheoremF0 + F2 + F4 + · · ·+ F2n = F2n+1
Proof.
−or−
−or−
−or−
2
4
2n−2
2n
....
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What about things that aren’t integers?Can visual proofs possibly work for theorems involving real numbers?
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The Pythagorean Theorem
TheoremIf c is the hypotenuse of a right triangle with sides a and b, then
a2 + b2 = c2
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Proof.
a
b
a b
ab
a
bc
c
c
c
c2 = Area(red square) = (a + b)2 − 4(
ab2
)= a2 + 2ab + b2 − 2ab
= a2 + b2
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12 +
(12
)2+
(12
)3+ · · · = 1
Proof.
1/2
1/4
1/8
1/16
1/32
1/64
1 = Total Area =12
+
(12
)2
+
(12
)3
+ · · ·
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14 +
(14
)2+
(14
)3+ · · · = 1
3
Proof.
1/256
1/64
1/16
1/4
13
= Green Area =14
+
(14
)2
+
(14
)3
+ · · ·
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Partitions
DefinitionA partition of n is a non-increasing list of integers that sum to n.
Examples22 = 8 + 5 + 5 + 3 + 1 22 = 6 + 6 + 6 + 1 + 1 +1 + 1In pictures:8
1
3
5
5
6
6
1
1
1
1
6
FYI, pictures like this are called Ferrers diagrams
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TheoremThe number of partitions of n into k parts is the same as the numberof partitions of n whose largest part has size k
ExamplesWith n = 6 and k = 3 we have:
3 parts2 + 2 + 23 + 2 + 14 + 1 + 1
Biggest piece is 33 + 33 + 2 + 13 + 1 + 1 + 1
With n = 7, k = 2 we have:2 parts6 + 15 + 24 + 3
Biggest piece is 22 + 1 + 1 + 1 + 1 + 12 + 2 + 1 + 1 + 12 + 2 + 2 + 1
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TheoremThe number of partitions of n into k parts is the same as the numberof partitions of n whose largest part has size k
Traditional Proof.Where would you even start?
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TheoremThe number of partitions of n into k parts is the same as the numberof partitions of n whose largest part has size k
Proof.
k
k
So there is a bijection between these two types of partitions and thusthere are the same number of each.
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References
Roger B. Nelson.Proofs Without Words, volume 1.Mathematical Association of America, 1993.
Roger B. Nelson.Proofs Without Words, volume 2.Mathematical Association of America, 2001.
Arthur T. Benjamin & Jennifer J. Quin.Proofs That Really Count.Mathematical Association of America, 2003.