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Continued Fractions in Combinatorial Game Theory
Mary A. Cox
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
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
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
Notation
Simple continued fractions can be written as
or
0 1 2; , ,...x a a a
01 2
1 1...x a
a a
Representations of Rational Numbers
Finite Simple Continued Fraction0ia
01 2
1 1 1...
n
x aa a a
0 1 2; , ,..., nx a a a a
Finite Simple Continued Fraction0ia
1 1 1 13
4 1 4 2x
Finite Simple Continued Fraction0ia
13
14
11
14
2
x
Finite Simple Continued Fraction0ia
13
14
11
9 / 2
Finite Simple Continued Fraction0ia
13
14
21
9
Finite Simple Continued Fraction0ia
13
94
11
Finite Simple Continued Fraction0ia
113
53
Finite Simple Continued Fraction0ia
170
53
Theorem
The representation of a rational number as a finite simple continued fraction is unique (up to a fiddle).
170 13
153 41
11
42
1
11 1 1
1
n
n n n
a
a a a
170 13
153 41
11
42
170 13
153 41
11
41
11
1703;4,1,4,2 3;4,1,4,1,1
53
Finding The Continued Fraction
19
51x
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
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
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!!
19 10
151 21
11
26
Finding The Continued Fraction
Finding The Continued Fraction
190;2,1,2,6
51
Representations of Irrational Numbers
Infinite Simple Continued Fraction0ia
01 2
1 1...x a
a a
0 1 2; , ,...x a a a
Theorems
The value of any infinite simple continued fraction is an irrational number.
Two distinct infinite simple continued fractions represent two distinct irrational numbers.
Infinite Simple Continued Fraction
3;7,15,1,292,...
Infinite Simple Continued Fraction
23 ?
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
Infinite Simple Continued Fraction
23 4.8
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
Infinite Simple Continued Fraction
23 4;1,3,1,8
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
Solving Pell’s Equation
Pell’s Equation
2 2 1x dy
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
Definition
In symbols:
0 1 2; , ,... ,1k kC a a a a k n
0 0C a
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
Notice
The converse is not necessarily true.
In other words, not all of the convergents of supply solutions to Pell’s Equation.d
Example
2 27 1x y
7 2;1,1,1,4
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