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Chapter 8 Real Numbers and Introduction to Algebra Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 8.1 Symbols and Sets of Numbers Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Set of Numbers • Natural Numbers: {1, 2, 3, 4, 5, 6 . . .} • Whole Numbers: {0, 1, 2, 3, 4 . . .} • Integers: {. . . –3, –2, –1, 0, 1, 2, 3 . . .} • Rational Numbers: the set of all numbers that can be expressed as a quotient of integers, with denominator 0. • Irrational Numbers: nonrational numbers that correspond to points on a number line. • Real Numbers: all numbers that correspond to pints on a number line. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 3 Equality and Inequality Symbols Symbol a=b ab a<b a>b ab ab Meaning a is equal to b. a is not equal to b. a is less than b. a is greater than b. a is less then or equal to b. a is greater than or equal to b. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 4 The Number Line A number line is a line on which each point is associated with a number. –5 –4 –3 –2 –1 0 – 4.8 1 2 3 4 5 1.5 Negative numbers Positive numbers Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 5 Order Property for Real Numbers For any two real numbers a and b, a is less than b if a is to the left of b on the number line. • a < b means a is to the left of b on a number line. • a > b means a is to the right of b on a number line. a b a < b or also b > a Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 6 Example Determine whether each mathematical statement is true. a. 4 < 5 True b. 27 ≥ 27 True c. 0>5 d. 16 ≤ 9 False False Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 7 Example Translate each sentence into a mathematical statement. a. Thirteen is less than or equal to nineteen. 13 ≤ 19 b. Five is greater than two. 5 > 2 c. Seven is not equal to eight. 7 ≠ 8 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 8 Example Graph the numbers on a number line. 1 3 1 ,2.5, , 3.75 4 2 3.75 1 1 4 3 2.5 2 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 9 Example Insert <, >, or = between the pairs of numbers to form true statements. a. 4.7 > 4.697 b. 32.61 = 32.61 c. – 4 > –7 d. 1 4 < 2 3 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 10 Absolute Value The absolute value of a real number a, denoted by |a|, is the distance between a and 0 on the number line. | –4| = 4 Symbol for absolute value |5| = 5 Distance of 4 –5 –4 –3 –2 –1 Distance of 5 0 1 2 3 4 5 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 11 Example Find the absolute value of each number. a. 9 9 b. 6 6 c. 4 4 5 5 d. 0 0 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 12 Example Insert <, >, or = between the pairs of numbers to form true statements. a. 4 > 0 b. 5 < 9 2 1 c. 6 > 6 3 3 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 13