Understanding Real Numbers and Number Lines

1. Chapter 1 Section 4

2. 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 the additive inverse of a real number. Find the absolute value of a real number. Interpret the meanings of real numbers from a table of data. 1.4 2 3 4 5

3. Classify numbers and graph them on number lines. Objective 1 Slide 1.4-3

4. Natural numbers (or counting numbers) and whole numbers, along with many others, can be represented on a number line like the one below. Classify numbers and graph them on a number line. 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, and then label other points. The “arrowhead” is used to indicate the positive direction on a number line. Slide 1.4-4

5. 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. Classify numbers and graph them on a number line. (cont’d) The natural numbers, their opposites, and 0 form a new set of numbers called the integers. { . . . , −3, −2, −1, 0, 1, 2, 3, . . . } Positive numbersand negative numbersare called signed numbers. The three dots ( … ) show that the list of numbers continues in the same way indefinitely. Slide 1.4-5

6. EXAMPLE 1 Using Negative Numbers in Applications 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, 1913. (Source: World Almanac and Book of Facts.) Solution: −53 Solution: 134 Slide 1.4-6

7. { 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.”) Classify numbers and graph them on a number line. (cont’d) 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. Since any number that can be written as the quotient of two integers is a rational number, all integers, mixed numbers, terminating (or ending decimals), and repeating decimals are rational. Slide 1.4-7

8. To grapha 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 line} 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. Classify numbers and graph them on a number line. (cont’d) Slide 1.4-8

9. Identify each real number in the set as rational or irrational. and −π are irrational EXAMPLE 2 Determining Whether a Number Belongs to a Set Solution: are rational; Slide 1.4-9

10. Tell which of two real numbers is less than the other. Objective 2 Slide 1.4-10

11. Tell which of two real numbers is less than the other. Ordering of Real NumbersFor any two real numbers a and b, a is less than b if a is to the left of b on the number line. 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 1.4-11

12. EXAMPLE 3 Determining the Order of Real Numbers Determine whether the statement is true or false. Solution: False Slide 1.4-12

13. Find the additive inverse of a real number. Objective 3 Slide 1.4-13

14. Find the additive inverse of a real number. By a property of the real numbers, for any real number x (except 0), there is exactly one number on the number line the same distance from 0 as x, but on the opposite side of 0. Such pairs are called additive inverses, or opposites, of each other. Additive Inverse The additive inverse of a number x is the number that is the same distance from 0 on the number line as x, but on the opposite side of 0. The additive inverse of −7 is written −(−7) and can be read “the opposite of −7” or “the negative of −7” or 7. Double Negative Rule For any real number x, −(−x) = x. Slide 1.4-14

15. Find the absolute value of a real number. Objective 4 Slide 1.4-15

16. Find the absolute value of a real number. The absolute valueof 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.” Distance is a physical measurement, which is never negative. Therefore, the absolute value of a number is never negative. Absolute Value For any real number x, . The “−x ” in the second part of the definition doesNOTrepresent a negative number. Since x is negative in the second part, −x represents the opposite of a negative number—that is, a positive number. The absolute value of a number is never negative. Slide 1.4-16

17. EXAMPLE 4 Finding the Absolute Value Simplify by finding the absolute value. Solution: 32 32 −32 −30 Slide 1.4-17

18. Interpret the meanings of real numbers from a table of data. Objective 5 Slide 1.4-18

19. In the table, which category in which year represents the greatest percent increase? EXAMPLE 5 Interpreting Data Solution: gasoline, from 2005 to 2006 Slide 1.4-19