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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1. 2 - 1

Chapter 1 Review of the Real Number System Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1. 2 - 2

1. 2 Operations on Real Numbers Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1. 2 - 3

1. 2 Operations on Real Numbers Objectives 1. 2. 3. 4. 5. Add real numbers. Subtract real numbers. Multiply real numbers. Find the reciprocal of a number. Divide real numbers. Copyright © 2010 Pearson Education, Inc. All rights reserved. 4

1. 2 Operations on Real Numbers Adding Real Numbers Number lines can be used to illustrate addition and subtraction of real numbers. Move left (the negative direction) to add negative numbers. – 1 + (– 3) = – 4 – 1 – 3 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 Sum Copyright © 2010 Pearson Education, Inc. All rights reserved. 5

1. 2 Operations on Real Numbers Adding Real Numbers Number lines can be used to illustrate addition and subtraction of real numbers. Move right (the positive direction) to add positive numbers or left (the negative direction) to add negative numbers. – 4 + 6 = 2 6 – 4 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 Sum Copyright © 2010 Pearson Education, Inc. All rights reserved. 6

1. 2 Operations on Real Numbers Rules for Adding Real Numbers Same signs To add two numbers with same signs, 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 lesser absolution value from the greater. The sum has the same sign as the number with the greater absolute value. Copyright © 2010 Pearson Education, Inc. All rights reserved. 7

1. 2 Operations on Real Numbers Adding Two Negative Real Numbers – 25 + (– 6) = First find the absolute values. | – 25| = 25 and | – 6| = 6 Because – 25 and – 6 have the same sign, add their absolute values. Both numbers are negative, so the answer is negative. – 25 + (– 6) = –(25 + 6) = –(31) = – 31 Copyright © 2010 Pearson Education, Inc. All rights reserved. 8

1. 2 Operations on Real Numbers Adding Two Negative Real Numbers – 2. 4 + (– 4. 8) = First find the absolute values. | – 2. 4| = 2. 4 and | – 4. 8| = 4. 8 Because – 2. 4 and – 4. 8 have the same sign, add their absolute values; the answer will be negative. – 2. 4 + (– 4. 8) = –(2. 4 + 4. 8) = –(7. 2) = – 7. 2 Copyright © 2010 Pearson Education, Inc. All rights reserved. 9

1. 2 Operations on Real Numbers Adding Two Negative Real Numbers Copyright © 2010 Pearson Education, Inc. All rights reserved. 10

1. 2 Operations on Real Numbers Adding Real Numbers with Different Signs – 23 + 9 = First find the absolute values. | – 23| = 23 and | 9| = 9 Because – 23 and 9 have different signs, subtract their absolute values. 23 – 9 = 14 The number – 23 has the larger absolute value, so the answer is negative. – 23 + 9 = – 13 Copyright © 2010 Pearson Education, Inc. All rights reserved. 11

1. 2 Operations on Real Numbers Adding Real Numbers with Different Signs 19 + (– 11) = First find the absolute values. | 19| = 19 and | – 11| = 11 Because 19 and – 11 have different signs, subtract their absolute values. 19 – 11 = 8 The number 19 has the larger absolute value, so the answer is positive. 19 + (– 11) = 8 Copyright © 2010 Pearson Education, Inc. All rights reserved. 12

1. 2 Operations on Real Numbers Adding Real Numbers with Different Signs Copyright © 2010 Pearson Education, Inc. All rights reserved. 13

1. 2 Operations on Real Numbers Adding Real Numbers with Different Signs Copyright © 2010 Pearson Education, Inc. All rights reserved. 14

1. 2 Operations on Real Numbers Subtracting Real Numbers Subtraction For all real numbers a and b, a – b = a + (– b). In words, to subtract b from a, add the additive inverse (or opposite) of b to a. Subtraction changes to addition Change the sign 7 – 5 = 7 + (– 5) = 2 Difference Subtrahend = 5 Copyright © 2010 Pearson Education, Inc. All rights reserved. 15

1. 2 Operations on Real Numbers Subtracting Real Numbers Change to addition Sign changed 5 – (– 19) = 5 + [–(– 19)] = 5 + 19 = 24 Change to addition Sign changed – 17 – (– 6) = – 17 + [–(– 6)] = – 17 + 6 = – 11 Copyright © 2010 Pearson Education, Inc. All rights reserved. 16

1. 2 Operations on Real Numbers Subtracting Real Numbers Find the differences: (a) – 6. 3 – 3. 2 = (b) 5. 1 – ( – 4. 7) = – 9. 5 9. 8 (c) Copyright © 2010 Pearson Education, Inc. All rights reserved. 17

1. 2 Operations on Real Numbers Adding and Subtracting Real Numbers For problems involving both addition and subtraction: 1. Add and subtract in order from left to right. 2. Work inside brackets or parentheses first – 2 + 7 – 12 = (– 2 + 7) – 12 = 5 + (– 12) = – 7 Copyright © 2010 Pearson Education, Inc. All rights reserved. 18

1. 2 Operations on Real Numbers Adding and Subtracting Real Numbers 19 – 11 – (– 8) = (19 – 11) – (– 8) = (19 + (– 11)) – (– 8) = 8 – (– 8) =8+8 = 16 Copyright © 2010 Pearson Education, Inc. All rights reserved. 19

1. 2 Operations on Real Numbers Adding and Subtracting Real Numbers 7 – [(– 12) – (– 5)] – 2 = 7 – [(– 12) + 5] – 2 = 7 – [– 7] – 2 =7+7– 2 = 14 + (– 2) = 12 Copyright © 2010 Pearson Education, Inc. All rights reserved. 20

1. 2 Operations on Real Numbers Multiplying Real Numbers Same signs The product of two numbers with the same sign is positive. Different signs The product of two numbers with different signs is negative. Copyright © 2010 Pearson Education, Inc. All rights reserved. 21

1. 2 Operations on Real Numbers Multiplying Real Numbers Same Sign (+) Different Sign (–) – 7(– 4) = 28 – 9(11) = – 99 – 0. 2(– 0. 6) = 0. 12 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1. 5(– 6) = – 9 22

1. 2 Operations on Real Numbers Finding the Reciprocal of a Number Reciprocal The reciprocal of a nonzero number a is Number . Reciprocal – 9 0. 02 50 – 4 0 Copyright © 2010 Pearson Education, Inc. All rights reserved. None 23

1. 2 Operations on Real Numbers Finding the Reciprocal of a Number CAUTION A number and its additive inverse have opposite signs; however, a number and its reciprocal have the same sign. Opposite signs – 8 and 8 are additive inverses of each other Same signs are reciprocals of each other Copyright © 2010 Pearson Education, Inc. All rights reserved. 24

1. 2 Operations on Real Numbers Dividing Real Numbers Division For all real numbers a and b (where b ≠ 0), a÷b= . In words, multiplying the first number by the reciprocal of the second number. CAUTION Division by 0 is undefined. Dividing 0 by a nonzero number gives 0. Copyright © 2010 Pearson Education, Inc. All rights reserved. 25

1. 2 Operations on Real Numbers Dividing Real Numbers Same signs The quotient of two nonzero real numbers with the same sign is positive. Different signs The quotient of two nonzero real numbers with different signs is negative. Copyright © 2010 Pearson Education, Inc. All rights reserved. 26

1. 2 Operations on Real Numbers Dividing Real Numbers Copyright © 2010 Pearson Education, Inc. All rights reserved. 27

1. 2 Operations on Real Numbers Equivalent (Equal) Fraction Copyright © 2010 Pearson Education, Inc. All rights reserved. 28