Continued Fractions in Combinatorial Game Theory Mary A. Cox.

Slides:

Advertisements

Similar presentations

Goal When you have finished this presentation you should understand:

Advertisements

Rational and Irrational Numbers

Mrs.Volynskaya Real Numbers

EXAMPLE 5 Rewrite a conditional statement in if-then form

Organizing Numbers into Number Sets. Definitions and Symbols for Number Sets Counting numbers ( maybe 0, 1, 2, 3, 4, and so on) Natural Numbers: Positive.

Classification of Numbers

Real Numbers Week 1 Topic 1. Real Numbers  Irrational Numbers  Numbers that cannot be written as a fraction  √2, π  Rational Numbers  Numbers that.

The Real Number System. Real Numbers The set of all rational and the set of all irrational numbers together make up the set of real numbers. Any and all.

Imaginary & Complex Numbers 5-3 English Casbarro Unit 5: Polynomials.

Lesson 9-2 Pages The Real Number System Lesson Check 9-1.

MATHS PROJECT NUMBER SYSTEMS BY BINDIYA GURUNG,CLASS IX.

How do we know what’s what?.  A rational number is a number that can be written as a simple fraction ◦ Ex. 5 5/1  ◦ Ex …1/3  ◦ Ex …?/?

Roots of Real Numbers and Radical Expressions. Definition of n th Root ** For a square root the value of n is 2. For any real numbers a and b and any.

2-5: Imaginary & Complex Numbers Unit 2 English Casbarro.

Obj: To solve equations using the Rational Root Theorem.

Integrated Mathematics Real Numbers. Rational Numbers Examples of Rational Numbers.

Lesson 2-7 Square Roots and Real Numbers. Definitions: Square Root- one of two equal factors of a number. Perfect Square- A number, like 64, whose square.

Mathletes Fab Five Divide ) 122 2) ) 0.5.

Isosceles Right Angled Triangle. The three sides cannot be all integers 1 1.

The Real Numbers 4-7 Learn to determine if a number is rational or irrational.

Chapter V Characteristics of quadratic number and square root form and it uses in solving simple problem.

11/10/2015.  Copy these definitions into your notes: 1. Rational number: Any number that can be put into the form of a fraction. 2. Irrational number:

Review for Unit 2 Test Directions: Show your work for each slide on the next 2-3 blank pages of your INB. Write the title above on each page you use in.

Fractional Equations and Extraneous Solutions

Table of Contents Rational Functions and Domains where P(x) and Q(x) are polynomials, Q(x) ≠ 0. A rational expression is given by.

Rational Functions and Domains where P(x) and Q(x) are polynomials, Q(x) ≠ 0. A rational expression is given by.

Section 1-2 Classifying Numbers and the Number Line.

Copyright © Ed2Net Learning, Inc.1 Pythagorean Theorem Grade 7 Pre-Algebra.

Perfect squares: 1, 4, 9, __, __, __, __, __, __, __,…

Number Sets. Symbols for Number Set Counting numbers ( maybe 0, 1, 2, 3, 4, and so on) Natural Numbers: Positive and negative counting numbers (-2, -1,

Irrational Numbers (4.2). If the definition of a rational number is a number that can be written in the form m/n, where m and n are integers with n not.

Square Roots All positive real numbers have two square roots, a positive and negative square root. All positive real numbers have two square roots, a positive.

Module XIII, Lesson 3 Online Algebra

(as opposed to fake numbers?)

Expressions and Equations Part 2

(as opposed to fake numbers?)

(as opposed to fake numbers?)

(as opposed to fake numbers?)

Real Zeros Intro - Chapter 4.2.

Properties of Real Numbers

(as opposed to fake numbers?)

Algebra Review Radical Expressions page 280

The Quadratic Formula..

All numbers that can be represented on a number line are called real numbers and can be classified according to their characteristics.

UNIT 1 VOCABULARY The Real number system.

(as opposed to fake numbers?)

(as opposed to fake numbers?)

Warm-Up #12 (Monday, 9/28) 3 ∙ ∙ (2 5 − 125)

Algebra 1 Section 1.1.

Lesson 9.1 How do you find and approximate the square root of a number? You need to find a number b such that

Rational and Irrational

(as opposed to fake numbers?)

(as opposed to fake numbers?)

Lesson 7.4 Real Numbers.

(as opposed to fake numbers?)

(as opposed to fake numbers?)

NUMBER SYSTEMS.

(as opposed to fake numbers?)

(as opposed to fake numbers?)

(as opposed to fake numbers?)

Rational Numbers.

(as opposed to fake numbers?)

(as opposed to fake numbers?)

Irrational Numbers.

(as opposed to fake numbers?)

(as opposed to fake numbers?)

(as opposed to fake numbers?)

(as opposed to fake numbers?)

(as opposed to fake numbers?)

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 a i and b i 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

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!!

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

Let and

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 continued fraction 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