Chapter 1 Section 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Real Numbers and the Number Line Classify numbers and graph them on number lines. Tell which of two real numbers is less than the other. Find additive inverses and absolute values of real numbers. Interpret the meanings of real numbers from a table of data.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Slide Classify numbers and graph them on number lines.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Classify numbers and graph them on a number line. The natural numbers and the whole numbers, along with many others, can be represented on a number line like the one below. We draw a number line by choosing any point on the line and labeling it 0. Then we choose any point to the right of 0 and label it 1. The distance between 0 and 1 gives a unit of measure used to locate other points. The “arrowhead” is used to indicate the positive direction on a number line. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Classify numbers and graph them on a number line. (cont’d) The natural numbers are located to the right of 0 on the number line. For each natural number, we can place a corresponding number to the left of 0. Each is the opposite, or negative, of a natural number. Positive numbers and negative numbers are called signed numbers. The natural numbers, their opposites, and 0 form a new set of numbers called the integers. {..., −3, −2, −1, 0, 1, 2, 3,...} Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use an integer to express the number in boldface italics in each application. Erin discovers that she has spent $53 more than she has in her checking account. The record-high Fahrenheit temperature in the United States was 134° in Death Valley, California, on July 10, (Source: World Almanac and Book of Facts 2006.) EXAMPLE 1 Using Negative Numbers in Applications Solution: −53 Solution: 134 Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Classify numbers and graph them on a number line. (cont’d) { is a quotient of two integers, with denominator not 0} is the set of rational numbers. (Read as “the set of all numbers x such that x is a quotient of two integers, with denominator not 0.”) A decimal number that comes to an end (terminates), such as 0.23 is a rational number. For example, Since any integer can be written as a quotient of itself and 1, all integers are also rational numbers. Example: This is called set-builder notation. This notation is convenient to use when it is not possible to list all the elements of a set. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Classify numbers and graph them on a number line. (cont’d) To graph a number, we place a dot on the number line at the point that corresponds to the number. The number is called the coordinate of the point. { is a nonrational number represented by a point on the number} is the set of irrational numbers. The decimal form of an irrational number neither terminates nor repeats. { is a rational or an irrational number} is the set of real numbers. Decimals numbers that repeat in a fixed block of digits, such as …, are also rational numbers. Example: Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Identify each real number in the set as rational or irrational. Solution: Determining whether a Number Belongs to a Set and are rational; and π are irrational Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Tell which of two real numbers is less than the other. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Ordering of Real Numbers This means that any negative number is less than 0, and any negative number is less than any positive number. Also, 0 is less than any positive number. We can also say that, for any two real numbers a and b, a is greater than b, if a is to the right of b on the number line. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: False Determining the Order of Real Numbers Determine whether the statement is true or false. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Find additive inverses and absolute values of real numbers. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley By a property of the real numbers, for any real number x (except 0), there is exactly one number, called the additive inverse, on the number line the same distance from 0 as x, but on the opposite side of 0. Finding additive inverses and absolute values of real numbers. The additive inverse of a number can be indicated by writing the − symbol in front of the number. The additive inverse of −7 is written −(−7) and can be read “the opposite of −7” or “the negative of −7” The Double Negative Rule, states that for any real number x, −(−x) = x Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Distance is a physical measurement, which is never negative. Therefore, the absolute value of a number is never negative. Finding additive inverses and absolute values of real numbers. (cont’d) The absolute value of a real number can be defined as the distance between 0 and the number on the number line. The symbol for the absolute value of the number x is |x|, read “the absolute value of x.” The rule of Absolute Value says that, for any real number x,. The “ − x ” in the second part of the definition does not represent a negative number. Since x is negative in the second part, −x represents the opposite of a negative number—that is, a positive number. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Finding the Absolute Value Simplify by finding the absolute value. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Interpret the meanings of real numbers from a table of data. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solution: Iron and steal, from 2003 to 2004 Interpreting Data In the table, which commodity in which year represents the greatest percent increase? Slide