continued fractions in combinatorial game theory mary a. cox
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Continued Fractions in Combinatorial Game Theory
Mary A. Cox
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Overview of talk Define general and simple continued fraction
Representations of rational and irrational numbers as continued fractions
Example of use in number theory: Pell’s Equation
Cominatorial Game Theory:The Game of Contorted Fractions
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What Is a Continued Fraction?
A general continued fraction representation of a real number x is one of the form
where ai and bi are integers for all i.
10
21
32
3 ...
bx a
ba
ba
a
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What Is a Continued Fraction?
A simple continued fraction representation of a real number x is one of the form
where
0
1
23
11
1...
x aa
aa
0ia
iaZ
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Notation
Simple continued fractions can be written as
or
0 1 2; , ,...x a a a
01 2
1 1...x a
a a
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Representations of Rational Numbers
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Finite Simple Continued Fraction0ia
01 2
1 1 1...
n
x aa a a
0 1 2; , ,..., nx a a a a
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Finite Simple Continued Fraction0ia
1 1 1 13
4 1 4 2x
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Finite Simple Continued Fraction0ia
13
14
11
14
2
x
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Finite Simple Continued Fraction0ia
13
14
11
9 / 2
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Finite Simple Continued Fraction0ia
13
14
21
9
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Finite Simple Continued Fraction0ia
13
94
11
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Finite Simple Continued Fraction0ia
113
53
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Finite Simple Continued Fraction0ia
170
53
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Theorem
The representation of a rational number as a finite simple continued fraction is unique (up to a fiddle).
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170 13
153 41
11
42
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1
11 1 1
1
n
n n n
a
a a a
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170 13
153 41
11
42
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170 13
153 41
11
41
11
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1703;4,1,4,2 3;4,1,4,1,1
53
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Finding The Continued Fraction
19
51x
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Finding The Continued Fraction
We use the Euclidean Algorithm!!
51 2 19 13
19 1 13 6
13 2 6 1
6 6 1 0
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Finding The Continued Fraction
We use the Euclidean Algorithm!!
51 1351 2 19 13 2
19 1919 6
19 1 13 6 113 13
13 113 2 6 1 2
6 66
6 6 1 0 16
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51 2 19 13
19 1 13 6
13 2 6 1
6 6 1 0
Finding The Continued Fraction
We use the Euclidean Algorithm!!
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19 10
151 21
11
26
Finding The Continued Fraction
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Finding The Continued Fraction
190;2,1,2,6
51
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Representations of Irrational Numbers
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Infinite Simple Continued Fraction0ia
01 2
1 1...x a
a a
0 1 2; , ,...x a a a
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Theorems
The value of any infinite simple continued fraction is an irrational number.
Two distinct infinite simple continued fractions represent two distinct irrational numbers.
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Infinite Simple Continued Fraction
3;7,15,1,292,...
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Infinite Simple Continued Fraction
23 ?
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Infinite Simple Continued Fraction
Let
and
1 20 0 1 1
1 1, ,...x x
x x x x
0 0 1 1 2 2, , ,...a x a x a x
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Infinite Simple Continued Fraction
23 4.8
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Infinite Simple Continued Fraction
0
1
2
3
4
23 4 23 4
1 23 4 23 31
7 723 4
23 33
2
23 41
7
23 4 8 23 4
x
x
x
x
x
0
1
2
3
4
4
1
3
1
8
a
a
a
a
a
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Infinite Simple Continued Fraction
23 4;1,3,1,8
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Theorem
If d is a positive integer that is not a perfect square, then the continued fraction expansion of necessarily has the form:d
0 1 2 2 1 0; , ,..., , ,2d a a a a a a
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Solving Pell’s Equation
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Pell’s Equation
2 2 1x dy
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Definition
The continued fraction made from
by cutting off the expansion after the kth partial denominator is called the kth convergent of the given continued fraction.
0 1 2; , ,...x a a a
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Definition
In symbols:
0 1 2; , ,... ,1k kC a a a a k n
0 0C a
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Theorem
If p, q is a positive solution of
then is a convergent of the continued
fraction expansion of
2 2 1x dy p
q
d
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Notice
The converse is not necessarily true.
In other words, not all of the convergents of supply solutions to Pell’s Equation.d
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Example
2 27 1x y
7 2;1,1,1,4
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Example
2
11
2 311 1 5
21 1 21 1 1 8
21 1 1 3
2 2
2 2
2 2
2 2
2 7 1 3
3 7 1 2
5 7 2 3
8 7 3 1