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## Chapter 9

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**Chapter 9**Rational Numbers and Real Numbers, with an Introduction to Algebra**9.1 The Rational Numbers**The setof rational numbers is the set**Definition: Equality or Rational Numbers**if and only if Theorem: Let be any rational number and n any nonzero integer. Then**Simplest Form**A rational number is said to be in simplest form (lowest terms) when its numerator and denominator have no common prime factors, and its denominator is positive.**Addition of Rational Numbers**Definition: Let and be any rational numbers. Then Theorem: Let be any rational number. Then**Properties of Rational Numbers**• Closure Property of Addition. • Commutative Property of Addition • Associative Property of Addition • Additive Identity Property • Additive Inverse Property**Theorem: Let be any rational numbers.**If then Theorem: Let be any rational number. Then**Subtraction**Definition: Let and be any rational numbers. Then**Multiplication**Definition:Let and be any rational numbers. Then**Properties of Rational Number Multiplication**• Closure Property of Multiplication. • Commutative Property of Multiplication. • Associative Property of Multiplication • Multiplicative Identity Property • Multiplicative Inverse Property**6. Distributive Property of Fraction Multiplication over**Addition:**Division**Definition: Let and be any rational numbers where Then**Three Methods of Rational-Number Division**Let and be any rational numbers where Then the following are equivalent.**Cross Multiplication of Rational Number Inequality**Let and be any rational numbers, where and Then**9.2 The Real Numbers**Theorem:There is no rational number whose square is 2.**Definition: The set of real numbers, R, is the set of all**numbers that have an infinite decimal representation. Real Numbers (Decimals) Irrational Numbers (nonterminating, nonrepeating decimals) Rational Numbers Terminating Decimals Nonterminating Decimals**Definition: Let a be a nonnegative real number. Then the**principal square root of a is defined as Definition: Let a be a real number and n a positive integer. • If then if and only if • If and n is odd, then if and only if where**Rational Exponents**Let a be any real number and n any positive integer. Then Let a be a nonnegative number and be a rational number in simplest form. Then**Introduction to Algebra**• An equation is a mathematical statement. • A solution is a number that makes the mathematical statement true. • The set of all solutions is called a solution set. • An inequality is a mathematical statement using the phrase “less than” or “greater than” or “less than or equal to”, etc.**9.3 Functions and Their Graphs**The Cartesian Coordinate System y Quadrant I Quadrant II x Quadrant III Quadrant IV**Linear Functions**A linear function has the form The y intercept is (0,b) and the slope is m.**Quadratic Functions**A quadratic function has the form The x coordinate of the vertex is To find the y coordinate of the vertex, evaluate the function at To find x-intercepts, solve**Exponential Functions**An exponential function has the form**The Vertical Line Test**• To determine if a graph represents a function, we use the Vertical Line Test. • If it is possible for a vertical line to cross a graph more than once, then the graph is not the graph of a function.