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
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.