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Free Pre-Algebra Lesson 22 page 1 Lesson 22 More Operations with Negative Numbers Now that we can add and subtract with negative numbers, we fill in the rest of the arithmetic operations: multiplying, dividing, and raising negative numbers to a power. Multiplying with Negative Numbers If you think of multiplication as repeated addition, a problem like 3(4) means 4 + 4 + 4 = 12. To show multiplication graphically we could line up three columns of four. The graphic model is helpful because it makes it clear that multiplication is commutative, since 3 columns of 4 is the same as 4 rows of 3. To show 3(–4), then, we could line up 3 columns with –4 in each column: Adding –4 three times we get –4 + –4 +–4 = –12. And since multiplication is commutative, this should be the same as the multiplication (–4)(3). We now have: 4 • 3 = 12 3 • –4 = –12 –4 • 3 = –12 You can see from the way we did the above examples that the problems below will work the same way. (If you take a minute to verify this yourself by drawing a picture as above, you are destined for mathematical greatness.) 4 • 3 = 12 4 • –3 = –12 –3 • 4 = –12 The only difficulty remaining is the problem (–3)(–4). I can’t make –3 columns, or –4 rows, those don’t make sense. It’s probably some form of 12, but is it positive or negative? A Negative Times A Negative There are two ways to think about a double negative in grammar. In some languages or sentence constructions a double negative intensifies the negative effect. “I will never, never go there again,” means that the person feels very strongly about never going to that awful place again. Sometimes a double negative is a kind of reversal, the opposite of the opposite, that turns around to a positive. “I do not disagree,” means the speaker agrees with you. Let’s look at a simple situation. Each number is multiplied by –1. At first we know what to do, and then we hope our momentum will carry us along: It’s easy to see a pattern. Multiplying a positive number by –1 changes the number to its opposite, which is negative. The products are moving along the number line in the positive direction. To fill in the missing numbers, we go with the pattern. © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 22 page 2 The pattern on the number line suggests that multiplying a number by –1 changes it to its opposite. If the number is positive and multiplied by –1 it will become negative and if the number is negative and is multiplied by –1 it will become positive. The product of two negative numbers is positive. The Four Types of Multiplication Problems SAME SIGN DIFFERENT SIGNS 3 • 4 = 12 (–3) • (–4) = 12 3 • (–4) = –12 (–3) • 4 = –12 positive • positive negative • negative positive • negative negative • positive ANSWER is POSITIVE ANSWER is NEGATIVE Example: Multiply. 8•2 8 • –2 16 –8 • –2 –16 –8 • 2 16 –16 When we multiply strings of factors, remember that pairs of negatives turn positive. If there are three negatives, (negative • negative) • negative = (positive) • negative = negative When multiplying several numbers, you multiply them in order, two by two. Or you can figure out the sign of the answer first. Then just multiply the numbers. Example: Multiply. (2)(–3)(5) (–2)(–3)(5) (–6)(5) = –30 (6)(5) = 30 (–2)(–3)(–5) (2)(–3)(–5) (6)(–5) = –30 (–6)(–5) = 30 Example: Write a multiplication to find the answer. Travis owed $10 each to 5 different friends. How much did he owe in all? The submarine started at the surface and made 4 dives, each 200 feet down. How far down was the submarine? (–10)(5) = –50 4(–200) = –800 He owed $50. It was 800 feet below the surface. © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 22 page 3 Using the Distributive Property with Negative Numbers Everything works exactly the same way it did before. Really. Example: Simplify using the distributive property. 3( 5x 8) 3(2x 7) 3(2x ) –3(7) 6x 3( 5x ) 3(4x 6) 15x 15x –21 6x 21 Remember to use the efficient notation and change +–21 to – 21. 3(4x 3(8) 3(4x ) 24 24 6) 3( 6) 12x 18 Same here, change adding a negative back to subtraction. Here change the subtraction to addition before you start, so you know it’s a –6. Dividing with Negative Numbers The related multiplication/division equations allow us to figure out the rules for division from the multiplication rules. 3 • 4 = 12 is related to the equation 12 ÷ 4 = 3 –3 • –4 = 12 is related to the equation 12 ÷ –4 = –3 3 • –4 = –12 is related to the equation –12 ÷ –4 = 3 –3 • 4 = –12 is related to the equation –12 ÷ 4 = –3 positive ÷ positive POSITIVE positive ÷ negative NEGATIVE negative ÷ negative POSITIVE negative ÷ positive NEGATIVE Just as in multiplication, if the signs are the same, the quotient is positive. If the signs are different, the quotient is negative. Example: Divide. 8÷2 8 ÷ –2 4 –8 ÷ –2 –4 –8 ÷ 2 4 –4 Example: Write a division to find the answer. The 25 negative team members were split evenly between 5 houses. How many in each house? Jake wants to lose 16 pounds over the next 8 weeks. How much should he lose per week? –25 ÷ 5 = –5 –16 ÷ 8 = –2 5 negative team members in each house. He should lose 2 lb per week. © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 22 page 4 Negative Fractions Since fraction bars are a kind of division symbol, the rules about division with negatives help with dealing with negative fractions. Think of the fraction 1/2 as the answer to the division problem 1 ÷ 2. Then the fraction –(1/2) is the answer to the division problem –1 ÷ 2 or to the division problem 1 ÷ –2. In fraction notation, if the fraction is negative, we can write the negative sign in front, in the numerator, or in the denominator. Since negative ÷ negative = positive, when numerator and denominator are both negative the fraction should be written as positive. Negative Fraction Notation The negative sign can go in front of the fraction, or in the numerator, or in the denominator (least preferred). 1 2 1 2 1 2 If both numerator and denominator are negative, the fraction is positive. 1 2 1 2 Example: Simplify. 5 3 • 6 10 1 1 5 3 6 2 • 10 6 1 4 18 • 2 positive • negative = negative 5 3 18 5 • 1 3 30 negative • negative = positive Exponents on Negative Numbers Now that we know how to multiply with negative numbers, we can figure out what a negative number to a power should be. Example: Write each expression as a multiplication, then find the product. 35 (3)(3)(3)(3)(3) 243 ( 3)5 ( 3)( 3)( 3)( 3)( 3) 243 (3)4 (3)(3)(3)(3) 81 ( 3)4 ( 3)( 3)( 3)( 3) 81 Notice the signs of the answers. If a negative number is raised to a power, the result may be either positive or negative. It depends on how many negatives there are in the product. If there are an even number of negatives, they will form positives in pairs, and the result will be positive. If there are odd numbers of negatives, the pairs will make positives and there will be one more negative left over, making the result negative. © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 22 page 5 Exponentiation comes first in the order of operations, so unless the parentheses hold the negative sign onto the number, the exponent doesn’t apply to the negative sign. It works like this: ( 3)4 ( 3)( 3)( 3)( 3) but 34 (3)(3)(3)(3) Every math student I have ever met has found this confusing at first, so don’t get discouraged if you don’t remember it every time right away. It seems a bit arbitrary and finicky at first, but there are good reasons for it when working with variables, so invest the time to understand and remember the difference between the two expressions. Example: Write each expression as a multiplication, then find the product. ( 5)2 52 (5)(5) 25 (52 ) ( 5)( 5) 25 52 (5 • 5) 25 (5)(5) 25 A Lot of Rules Negative numbers really altered the way we look at addition and subtraction, because a negative is opposite a positive in terms of addition. But once we established how the new numbers behave in addition and subtraction, the rest of the operations follow. Multiplication is repeated addition, and our understanding of the negatives from addition gave us the two most important rules of multiplication: The product of two numbers with the same sign is positive. The product of two numbers with different signs is negative. Everything else is a consequence of this. Understanding division comes from the related multiplication/division equations, and the same rules apply there. Understanding exponents with negatives comes from writing them as multiplications, and being finicky about notation and the order of operations. Now that we can do all the arithmetic operations, the negatives are full-fledged real numbers at our disposal. Now we’ll start using them in formulas and equations. © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 22 page 6 Lesson 22: More Operations with Negative Numbers Worksheet Name _________________________________________ 1. Fill in the answers to the problems and the sign of the answer in the chart: SAME SIGN DIFFERENT SIGNS 3 • 4 = 12 (–3) • (–4) = 12 3 • (–4) = –12 (–3) • 4 = –12 positive • positive negative • negative positive • negative negative • positive ANSWER is POSITIVE ANSWER is NEGATIVE 2. Simplify. 7( 3) 2•5 35 ( 7) 6 3 1• 9 ( 4)( 6) 54 ( 6) 32 ( 8) 3. Write the exponential expression as a multiplication, then evaluate. 24 ( 2)4 24 ( 2)5 25 \ 25 4. Use the distributive property to simplify. 5(2x 1) © 2010 Cheryl Wilcox 5( 2x 1) 5(2x 1) Free Pre-Algebra Lesson 22 page 7 5. Simplify 1 5 15 2 1 5 15 2 1 5 15 2 6. Mixed Practice 3 4 3 4 3( 4) 3 4 3 3 9 3( 9) 3 ( 9) 9 18 9 18 9 18 ( 9) 18( 9) 23 20 5 © 2010 Cheryl Wilcox ( 2)3 20 5 ( 2)(3) 20 5 ( 2)( 3) 20 5 Free Pre-Algebra Lesson 22 page 8 Lesson 22: More Operations with Negative Numbers Homework 22A Name _____________________________________ 1. Simplify. 2. Simplify. a. 10 16 b. 8 2 c. 17 1 d. a. 6 2•3 5 1 b. 2 7•2 2•9 3•6 c. 102 510 2a 5b • b2 6 g. 6 33 • 13 35 42 23 d. |4 5| 2 e. –9 4 2•5 5 189a 2 e. 819a f. 43 3. Solve the equations. a. 5x h. Build four fractions equivalent to 2 . 5 b. 3(x i. Change 8 22 1) 27 111 to a mixed number. 5 j. Change 17 2 to an improper fraction. 9 © 2010 Cheryl Wilcox c. 2x 3 6 Free Pre-Algebra Lesson 22 page 9 4. Simplify each expression, then compare with >, <, =. 5. Change any subtractions to additions, then find the sum. a. 5 a. 6 3 8 8 2 b. 7 b. 5 8 3 c. 8 ( 7) d. 9 c. 5 (–3)2 2 8 10 9 e. 8 ( 8) 6. Using the formula |a – b|, find the distance between 7. Simplify. a. 10 and 3 a. 4 • 9 b. 10 and –3 b. ( 4)( 9) c. 36 ( 4) c. Sam on Floor 87 and Holly on Floor –22. d. 36 ( 9) e. 36 4 8. Write the expression as a multiplication, then evaluate. 9. Simplify. a. 53 a. 6x 3x b. ( 5)3 b. 6(x 3) c. 53 c. 6(x 3) d. 6(x 3) 2( 3x 9) d. ( 5)4 © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 22 page 10 Lesson 22: More Operations with Negative Numbers Homework 22A Answers 1. Simplify. a. 10 16 b. 8 2 2 •5 2 •2•2•2 2 •2•2 17 c. 1 d. 2. Simplify. 4 1 2 5 8 a. 6 2•3 5 1 4 b. 2 7•2 2•9 3•6 17 c. 2 • 3 • 17 102 510 2 • 3 • 5 • 17 189a 2 e. 819a 1 5 3 • 3 •3• 7 • a •a 3 • 3 • 7 • 13 • a 2a 5b • b2 6 2 •a 5• b • b• b 2 •3 5a 3b 6 33 • g. 13 35 2 • 3 3 • 11 • 13 5 • 7 198 455 f. h. Build four fractions equivalent to 2 5 i. Change 3a 13 4 10 6 15 43 42 2 d. |4 5| 2 e. –9 4 2•5 5 j. Change 17 a. 5x 2 14 18 18 | 1| 2 16 undefined 0 48 8 6 5 5 1 1 2 5 10 5 8 22 5x 8 22 5x 30 x b. 3(x 1 5 3x c. 2x 3 5x 8 8 22 8 5x / 5 30 / 5 6 1) 27 3 27 3x 24 x 2 to an improper fraction. 9 2 153 2 155 17 9 9 9 9 © 2010 Cheryl Wilcox 0 3. Solve the equations. 2 . 5 8 10 20 25 22 0 6 64 16 8 3 111 to a mixed number. 5 111 5 22r 1 6 6 6 3x 3 3 27 3 3x / 3 24 / 3 8 6 2x 3 2x x 6 18 9 2x •3 6•3 3 2x / 2 18 / 2 Free Pre-Algebra Lesson 22 page 11 4. Simplify each expression, then compare with >, <, =. 5. Change any subtractions to additions, then find the sum. a. 5 a. 6 3 8 5 5 8 2 b. 7 6 2 5 9 8 7 15 5 d. 9 8 10 9 (9 8) (10 5 < 5 5 2 c. 8 ( 7) 8 3 5 c. 0 7 5 < 5 b. 8 2 9) 1 1 2 e. 8 ( 8) (–3)2 8 8 0 5 < 9 6. Using the formula |a – b|, find the distance between 7. Simplify. a. 10 and 3 a. 4 • 9 10 3 7 36 36 b. ( 4)( 9) b. 10 and –3 10 ( 3) 10 3 13 c. Sam on Floor 87 and Holly on Floor –22. 87 ( 22) 87 22 c. 36 ( 4) 9 d. 36 ( 9) 4 109 floors e. 36 4 9 8. Write the expression as a multiplication, then evaluate. 9. Simplify. a. 53 a. 6x 3x (5)(5)(5) 125 9x b. 6(x b. ( 5)3 ( 5)( 5)( 5) 125 c. 6(x c. 53 (5)(5)(5) d. ( 5) d. 6(x ( 5)( 5)( 5)( 5) 625 6x 18 6x 18 3) 125 4 © 2010 Cheryl Wilcox 3) 3) 2( 3x 9) 6x 18 6x 18 0 Free Pre-Algebra Lesson 22 page 12 Lesson 22: More Operations with Negative Numbers Homework 22B Name _____________________________________ 1. Simplify. 2. Simplify. a. 12 16 b. 9 3 b. 15 1 c. c. d. e. a. 260 910 f. 52 3 2 5 4 5(3 1) 32 12 e. –6 4 5(8 6) 3. Solve the equations. 8 14 g. • 15 3 a. 9 h. Build four fractions equivalent to 4x 3) 14 218 to a mixed number. 7 j. Change 23 4 to an improper fraction. 5 © 2010 Cheryl Wilcox 7 3 . 10 b. 2(x i. Change 22 • 32 | 8 10 | 2 285x 3 7x 2 y • y 2 21x 6 d. 190x 2 6 2•3 2 c. 2x 7 6 Free Pre-Algebra Lesson 22 page 13 4. Simplify each expression, then compare with >, <, =. a. 5 5. Change any subtractions to additions, then find the sum. a. 3 ( 5) 2 b. 3 b. 5 1 c. 5 1 5 1 5 1 5 c. 2 ( 3 5) d. 2 ( 2) e. 4 5 6 6. Using the formula |a – b|, find the distance between 7. Simplify. a. 15 and 8 a. 3 • 5 b. –15 and 8 b. ( 3)( 5) c. 15 ( 3) c. Beatrice on Floor –55 and Ted on Floor 22. d. 15 ( 5) e. 15 5 8. Write the expression as a multiplication, then evaluate. 9. Simplify. a. 63 a. 7x 2x b. ( 6)3 b. 7(x 2) c. 63 c. 7(x 2) d. 7(x 2) 7(x 2) d. ( 6)4 © 2010 Cheryl Wilcox