1. Chapter 1 Section 2 Copyright © 2011 Pearson Education, Inc.

2. 1 1 2 2 3 3 Operations on Real Numbers Add real numbers. Subtract real numbers. Find the distance between two points on a number line. Multiply real numbers. Divide real numbers. 4 4 5 5 1.21.2

3. Slide 1.1- 3 Copyright © 2011 Pearson Education, Inc. Objective 1 Add real numbers. Slide 1.1- 3

4. Slide 1.1- 4 Copyright © 2011 Pearson Education, Inc. Adding Real Numbers Same Sign To add two numbers with the same sign, add their absolute values. The sum has the same sign as the given numbers. Different Signs To add two numbers with different signs, find the absolute values of the numbers, and subtract the smaller absolute value from the larger. The sum has the same sign as the number with the larger absolute value. Slide 1.1- 4

5. Slide 1.1- 5 Copyright © 2011 Pearson Education, Inc. EXAMPLE 1 Find each sum. a.  6 + (  15) Find the absolute values. |  6| = 6|  15| = 15 Because they have the same sign, add their absolute values.  6 + (  15) =  (6 + 15) Add the absolute values. =  (21) =  21 Slide 1.1- 5 Both numbers are negative, so the answer will be negative.

6. Slide 1.1- 6 Copyright © 2011 Pearson Education, Inc. continued b.  1.1 + (  1.2) =  (1.1 +1.2) =  2.3 c. Slide 1.1- 6 Add the absolute values. Both numbers are negative, so the answer will be negative. The least common denominator is 8. Add numerators; keep the same denominator.

7. Slide 1.1- 7 Copyright © 2011 Pearson Education, Inc. EXAMPLE 2 Find each sum. a. 3 + (  7) Find the absolute values. | 3| = 3|  7| = 7 Because they have different signs, subtract their absolute values. The number  7 has the larger absolute value, so the answer is negative. 3 + (  7) =  4 Slide 1.1- 7 The sum is negative because |  7| > |3|

8. Slide 1.1- 8 Copyright © 2011 Pearson Education, Inc. continued b.  3 + 7 = = 4 d. Slide 1.1- 8 c.  3.8 + 4.6 = = 0.8 Subtract the absolute values. -3/8 has the larger absolute value, so the answer will be negative. Subtract numerators; keep the same denominator. 7 – 3 4.6 – 3.8

9. Slide 1.1- 9 Copyright © 2011 Pearson Education, Inc. Objective 2 Subtract real numbers. Slide 1.1- 9

10. Slide 1.1- 10 Copyright © 2011 Pearson Education, Inc. Subtraction For all real numbers a and b, a – b = a + (–b). That is, to subtract b from a, add the additive inverse (or opposite) of b to a. Slide 1.1- 10

11. Slide 1.1- 11 Copyright © 2011 Pearson Education, Inc. EXAMPLE 3 Find each difference. a. 12  (–5) = 12 + 5 = 17 b.  11.5 – (  6.3) =  11.5 + 6.3 =  5.2 Slide 1.1- 11 Change to addition. The additive inverse of  5 is 5. Change to addition. The additive inverse of  6.3 is 6.3.

12. Slide 1.1- 12 Copyright © 2011 Pearson Education, Inc. continued c. Slide 1.1- 12 Write each fraction with the least common denominator, 12. Add numerators; keep the same denominator. To subtract, add the additive inverse (opposite).

13. Slide 1.1- 13 Copyright © 2011 Pearson Education, Inc. EXAMPLE 4 Perform the indicated operations. a.  6 – (–2) – 8 – 1 Work from left to right. = (  6 + 2) – 8 – 1 =  4 – 8 – 1 =  12 – 1 =  13 Slide 1.1- 13

14. Slide 1.1- 14 Copyright © 2011 Pearson Education, Inc. continued b.  3 – [(  7) + 15] – 6 Work inside brackets. =  3 – [8] – 6 =  11 – 6 =  17 Slide 1.1- 14

15. Slide 1.1- 15 Copyright © 2011 Pearson Education, Inc. Objective 3 Find the distance between two points on a number line. Slide 1.1- 15

16. Slide 1.1- 16 Copyright © 2011 Pearson Education, Inc. To find the distance between the points 2 and 8, we subtract 8 – 2 = 6. Since distance is always positive, we must be careful to subtract in such a way that the answer is positive. Or, to avoid this problem altogether, we can find the absolute value of the difference. Then the distance is either |8 – 2| = 6 or |2 – 8| = 6.

17. Slide 1.1- 17 Copyright © 2011 Pearson Education, Inc. Distance The distance between two points on a number line is the absolute value of the difference between their coordinates. Slide 1.1- 17

18. Slide 1.1- 18 Copyright © 2011 Pearson Education, Inc. EXAMPLE 5 Find the distance between the points  12 and  1. Find the absolute value of the difference of the numbers, taken in either order. |  12 – (  1)| = 11 or |  1 – (  12)| = 11 Slide 1.1- 18

19. Slide 1.1- 19 Copyright © 2011 Pearson Education, Inc. Objective 4 Multiply real numbers. Slide 1.1- 19

20. Slide 1.1- 20 Copyright © 2011 Pearson Education, Inc. Multiplying Real Numbers Same Sign The product of two numbers with the same sign is positive. Different Signs The product of two numbers with different signs is negative. Slide 1.1- 20

21. Slide 1.1- 21 Copyright © 2011 Pearson Education, Inc. EXAMPLE 6 Find each product. a. 7(–2) b. –0.9(–15) c. Slide 1.1- 21 Different signs; product is negative. Same signs; product is positive. Different signs; product is negative. =  14 = 13.5 =  10

22. Slide 1.1- 22 Copyright © 2011 Pearson Education, Inc. Objective 5 Divide real numbers. Slide 1.1- 22

23. Slide 1.1- 23 Copyright © 2011 Pearson Education, Inc. CAUTION Division by 0 is undefined. However, dividing 0 by a nonzero number gives the quotient 0. For example, Be careful when 0 is involved in a division problem.

24. Slide 1.1- 24 Copyright © 2011 Pearson Education, Inc. Recall that Thus, dividing by b is the same a multiplying by 1/b. If b ≠0, then 1/b is the reciprocal, or multiplicative inverse, of b. When multiplied, reciprocals have a product of 1. NumberReciprocal 66 0.0520 0None

25. Slide 1.1- 25 Copyright © 2011 Pearson Education, Inc. Division For all real numbers a and b (where b ≠ 0), a  b = That is, multiply the first number by the reciprocal of the second number. Slide 1.1- 25 Dividing Real Numbers Same Sign The quotient of two nonzero real numbers with the same sign is positive. Different Signs The product of two nonzero real numbers with different signs is negative.

26. Slide 1.1- 26 Copyright © 2011 Pearson Education, Inc. EXAMPLE 7 Find each quotient. a. b. c. Slide 1.1- 26 Same signs; quotient is positive. The reciprocal of 11/16 is 16/11. Multiply by the reciprocal.

27. Slide 1.1- 27 Copyright © 2011 Pearson Education, Inc. Equivalent Forms of a Fraction The fractions Example: The fractions Example: Slide 1.1- 27

28. Slide 1.1- 28 Copyright © 2011 Pearson Education, Inc. EXAMPLE Identify which fractions equal a. b. c. d. Slide 1.1- 28 Equivalent fraction Not Equal