1. Continued Fractions in Combinatorial Game Theory Mary A. Cox

2. 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

3. What Is a Continued Fraction? A general continued fraction representation of a real number x is one of the form where a i and b i are integers for all i.

4. What Is a Continued Fraction? A simple continued fraction representation of a real number x is one of the form where

5. Notation Simple continued fractions can be written as or

6. Representations of Rational Numbers

7. Finite Simple Continued Fraction

15. Theorem The representation of a rational number as a finite simple continued fraction is unique (up to a fiddle).

21. Finding The Continued Fraction

22. We use the Euclidean Algorithm!!

23. Finding The Continued Fraction We use the Euclidean Algorithm!!

24. Finding The Continued Fraction We use the Euclidean Algorithm!!

25. Finding The Continued Fraction

27. Representations of Irrational Numbers

28. Infinite Simple Continued Fraction

29. Theorems The value of any infinite simple continued fraction is an irrational number. Two distinct infinite simple continued fractions represent two distinct irrational numbers.

30. Infinite Simple Continued Fraction

32. Let and

33. Infinite Simple Continued Fraction

36. Theorem If d is a positive integer that is not a perfect square, then the continued fraction expansion of necessarily has the form:

37. Solving Pell’s Equation

38. Pell’s Equation

39. 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.

40. Definition In symbols:

41. Theorem If p, q is a positive solution of then is a convergent of the continued fraction expansion of

42. Notice The converse is not necessarily true. In other words, not all of the convergents of supply solutions to Pell’s Equation.

43. Example