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