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Continued Fractions in Combinatorial Game TheoryPowerPoint Presentation

Continued Fractions in Combinatorial Game Theory

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Continued Fractions in Combinatorial Game Theory

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Continued Fractions in Combinatorial Game Theory

Mary A. Cox

- 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

- A general continued fraction representation of a real number x is one of the form
- where ai and bi are integers for all i.

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

- Simple continued fractions can be written as
- or

Representations of Rational Numbers

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

We use the Euclidean Algorithm!!

We use the Euclidean Algorithm!!

We use the Euclidean Algorithm!!

Finding The Continued Fraction

Representations of Irrational Numbers

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

- Let
- and

- 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

- 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.

- In symbols:

- If p, q is a positive solution of
- then is a convergent of the continuedfraction expansion of

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