1 Copyright © Cengage Learning. All rights reserved.

2 1.1 Real Numbers

3 What You Will Learn  Classify numbers and plot them on a real number line  Use the real number line and inequality symbols to order real numbers  Find the absolute value of a number.

4 Classifying Real Numbers

5 The numbers you use in everyday life are called real numbers. They are classified into different categories, as shown below. Classifying Real Numbers

6 Example 1 – Classifying Real Numbers Which of the numbers in the following set are (a) natural numbers, (b) integers, (c) rational numbers, and (d) irrational numbers? Solution a. Natural numbers: b. Integers: c. Rational numbers: d. Irrational number:

7 The Real Number Line

8 The diagram used to represent the real numbers is called the real number line. It consists of a horizontal line with a point (the origin) labeled 0. Numbers to the left of 0 are negative and numbers to the right of 0 are positive, as shown below.

9 Example 2 – Plotting Real Numbers a. The point corresponds to the real number. b. The point corresponds to the real number 2.

10 Example 2 – Plotting Real Numbers c. The point corresponds to the real number. d. The point corresponds to the real number 1. cont’d

11 Ordering Real Numbers

12 Ordering Real Numbers The real number line provides you with a way of comparing any two real numbers. If a is to the left of b, then a is less than b, which is written as a < b. You can also describe this relationship by saying that b is greater than a, or b > a.

13 Example 3 – Ordering Real Numbers a. Ordering Integers 3 < 5 (3 is less than 5.) -5 < -3 (-5 is less than -3.)

14 Example 3 – Ordering Real Numbers cont’d b. Ordering Decimals -3.1 < 2.8 (-3.1 is less than 2.8.) < (-1.90 is less than )

15 Example 3 – Ordering Real Numbers cont’d c. Ordering Fractions To order fractions, you can write both fractions with the same denominator, or you can rewrite both fractions in decimal form. Here are two examples: The symbol ≈ means “is approximately equal to.”

16 Example 4 – Comparing Profits The bar graph shows the profits for a company from 2008 through 2013 a.Compare the profit for 2008 with the profit for b.Write a statement that summarizes the trend in the company’s profits.

17 Example 4 – Comparing Profits Solution a.The profit for 2008 was greater than the profit for b.Here is one possible summary statement: “The company’s profit had a big fall from 2008 to After that, profits have been increasing, but as of 2013, they have still not reached the level of profit in 2008.” cont’d

18 Absolute Value

19 Absolute Values For any real number, the distance between the number and 0 on the real number line is its absolute value. A pair of vertical bars, | |, is used to denote absolute value. For instance, the absolute value of -8 is 8.

20 Example 5 – Evaluating Absolute Values a.|-10| = 10, because the distance between -10 and 0 is 10. b. =, because the distance between and 0 is. c.|-3.2| = 3.2, because the distance between -3.2 and 0 is 3.2. d.-|-6| = -(6) = -6

21 Example 6 – Comparing Real Numbers Place the correct symbol (, or =) between the real numbers.

22 Example 6 – Comparing Real Numbers Solution a.|-9| = |9|, because |-9| = 0 and |9| = 9. b.|-3| < |5|, because |-3| = 3 and 3 is less than 5. c.0 < |-7| because |-7| = 7 and 0 is less than 7. d.-4 = -|-4|, because -|-4| = -4 and -4 is equal to -4 e.|12| < |-15|, because |12| = 12, |-15| = 15, and 12 is less than 15. f.2 > -|-2|, because -|-2| = -2 and 2 is greater than -2. cont’d

23 Example 7 – Explaining Elevation You are visiting Death Valley, California. Your GPS receiver lists your elevation at -178 feet. However, an outdoor sign states that the elevation is +178 feet below sea level. How can you explain the difference in signs?

24 Example 7 – Explaining Elevation Solution GPS receivers usually display elevation above sea level as positive numbers and elevation below sea level as negative numbers. If you want to talk about the elevation “below sea level”, the absolute value of the actual elevation. cont’d