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

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Presentation Transcript
overview of talk
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
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 fraction4
What Is a Continued Fraction?
  • A simple continued fraction representation of a real number x is one of the form
  • where
notation
Notation
  • Simple continued fractions can be written as
  • or
theorem
Theorem
  • The representation of a rational number as a finite simple continued fraction is unique (up to a fiddle).
finding the continued fraction22
Finding The Continued Fraction

We use the Euclidean Algorithm!!

finding the continued fraction23
Finding The Continued Fraction

We use the Euclidean Algorithm!!

finding the continued fraction24
Finding The Continued Fraction

We use the Euclidean Algorithm!!

theorems
Theorems
  • The value of any infinite simple continued fraction is an irrational number.
  • Two distinct infinite simple continued fractions represent two distinct irrational numbers.
theorem36
Theorem
  • If d is a positive integer that is not a perfect square, then the continued fraction expansion of necessarily has the form:
definition
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.
definition40
Definition
  • In symbols:
theorem41
Theorem
  • If p, q is a positive solution of
  • then is a convergent of the continuedfraction expansion of
notice
Notice
  • The converse is not necessarily true.
  • In other words, not all of the convergents of supply solutions to Pell’s Equation.
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