1 / 44

# Continued Fractions in Combinatorial Game Theory - PowerPoint PPT Presentation

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

Related searches for Continued Fractions in Combinatorial Game Theory

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Continued Fractions in Combinatorial Game Theory' - lew

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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!!

### 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.

• 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.