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Continued Fractions in Combinatorial Game Theory

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

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Continued Fractions in Combinatorial Game Theory

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  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 ai and bi 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

  8. Finite Simple Continued Fraction

  9. Finite Simple Continued Fraction

  10. Finite Simple Continued Fraction

  11. Finite Simple Continued Fraction

  12. Finite Simple Continued Fraction

  13. Finite Simple Continued Fraction

  14. Finite Simple Continued Fraction

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

  16. Finding The Continued Fraction

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

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

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

  20. Finding The Continued Fraction

  21. Finding The Continued Fraction

  22. Representations of Irrational Numbers

  23. Infinite Simple Continued Fraction

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

  25. Infinite Simple Continued Fraction

  26. Infinite Simple Continued Fraction

  27. Infinite Simple Continued Fraction • Let • and

  28. Infinite Simple Continued Fraction

  29. Infinite Simple Continued Fraction

  30. Infinite Simple Continued Fraction

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

  32. Solving Pell’s Equation

  33. Pell’s Equation

  34. 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.

  35. Definition • In symbols:

  36. Theorem • If p, q is a positive solution of • then is a convergent of the continuedfraction expansion of

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

  38. Example

  39. Example

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