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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

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Continued Fractions in Combinatorial Game Theory

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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.


What Is a Continued Fraction?

  • A simple continued fraction representation of a real number x is one of the form

  • where


Notation

  • Simple continued fractions can be written as

  • or


Representations of Rational Numbers


Finite Simple Continued Fraction


Finite Simple Continued Fraction


Finite Simple Continued Fraction


Finite Simple Continued Fraction


Finite Simple Continued Fraction


Finite Simple Continued Fraction


Finite Simple Continued Fraction


Finite Simple Continued Fraction


Theorem

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


Finding The Continued Fraction


Finding The Continued Fraction

We use the Euclidean Algorithm!!


Finding The Continued Fraction

We use the Euclidean Algorithm!!


Finding The Continued Fraction

We use the Euclidean Algorithm!!


Finding The Continued Fraction


Finding The Continued Fraction


Representations of Irrational Numbers


Infinite Simple Continued Fraction


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


Infinite Simple Continued Fraction


Infinite Simple Continued Fraction

  • Let

  • and


Infinite Simple Continued Fraction


Infinite Simple Continued Fraction


Infinite Simple Continued Fraction


Theorem

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


Solving Pell’s Equation


Pell’s Equation


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.


Definition

  • In symbols:


Theorem

  • If p, q is a positive solution of

  • then is a convergent of the continuedfraction expansion of


Notice

  • The converse is not necessarily true.

  • In other words, not all of the convergents of supply solutions to Pell’s Equation.


Example


Example


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