Continued Fractions in Combinatorial Game Theory

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

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

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

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

### Representations of Irrational Numbers

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

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.