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10.6 Multiplication and division of signed numbers

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to multiply and divide signed numbers. By the end of the module students should be able to multiply and divide signed numbers and be able to multiply and divide signed numbers using a calculator.

Section overview

  • Multiplication of Signed Numbers
  • Division of Signed Numbers
  • Calculators

Multiplication of signed numbers

Let us consider first, the product of two positive numbers. Multiply: 3 ⋅ 5 size 12{"3 " cdot " 5"} {} .

3 ⋅ 5 size 12{"3 " cdot " 5"} {} means 5 + 5 + 5 = 15 size 12{5+5+5="15"} {}

This suggests In later mathematics courses, the word "suggests" turns into the word "proof." One example does not prove a claim. Mathematical proofs are constructed to validate a claim for all possible cases. that

( positive number ) ⋅ ( positive number ) = ( positive number ) size 12{ \( "positive number" \) cdot \( "positive number" \) = \( "positive number" \) } {}

More briefly,

( + ) ( + ) = ( + ) size 12{ \( + \) \( + \) = \( + \) } {}

Now consider the product of a positive number and a negative number. Multiply: ( 3 ) ( − 5 ) size 12{ \( 3 \) \( - 5 \) } {} .

( 3 ) ( − 5 ) size 12{ \( 3 \) \( - 5 \) } {} means ( − 5 ) + ( − 5 ) + ( − 5 ) = − 15 size 12{ \( - 5 \) + \( - 5 \) + \( - 5 \) = - "15"} {}

This suggests that

( positive number ) ⋅ ( negative number ) = ( negative number ) size 12{ \( "positive number" \) cdot \( "negative number" \) = \( "negative number" \) } {}

More briefly,

( + ) ( - ) = ( - ) size 12{ \( + \) \( - \) = \( - \) } {}

By the commutative property of multiplication, we get

( negative number ) ⋅ ( positive number ) = ( negative number ) size 12{ \( "negative number" \) cdot \( "positive number" \) = \( "negative number" \) } {}

More briefly,

( − ) ( + ) = ( − ) size 12{ \( - \) \( + \) = \( - \) } {}

The sign of the product of two negative numbers can be suggested after observing the following illustration.

Multiply -2 by, respectively, 4, 3, 2, 1, 0, -1, -2, -3, -4.

A list of equations. 4 times negative 2 equals negative 8. 3 times negative 2 equals negative 6. 2 times negative 2 equals negative 4. 1 times negative 2 equals negative 2. For all this, the following label is listed on the side: As we know, a negative times a positive equals a negative. The list continues. 0 times negative 2 equals 0. The following label is listed to the side: As we know, 0 times any number equals 0. The list continues further. Negative 1 times negative 2 equals 2. Negative 2 times negative 2 equals 4. Negative 3 times negative 2 equals 6. Negative 4 times negative 2 equals 8. The following label is listed to the side: The pattern suggested is a negative times a negative equals a positive. For the entire list, the label at the top says: when this number decreases by 1, the first factor in each multiplication problem, the product increases by 2.

We have the following rules for multiplying signed numbers.

Rules for multiplying signed numbers

Multiplying signed numbers:
  1. To multiply two real numbers that have the same sign , multiply their absolute values. The product is positive.
    ( + ) ( + ) = ( + ) size 12{ \( + \) \( + \) = \( + \) } {}
    ( − ) ( − ) = ( + ) size 12{ \( - \) \( - \) = \( + \) } {}
  2. To multiply two real numbers that have opposite signs , multiply their abso­lute values. The product is negative.
    ( + ) ( − ) = ( − ) size 12{ \( + \) \( - \) = \( - \) } {}
    ( − ) ( + ) = ( − ) size 12{ \( - \) \( + \) = \( - \) } {}

Sample set a

Find the following products.

8 ⋅ 6 size 12{"8 " cdot " 6"} {}

| 8 | = 8 | 6 | = 6 Multiply these absolute values.

8 ⋅ 6 = 48 size 12{8 cdot 6="48"} {}

Since the numbers have the same sign, the product is positive.

Thus, 8 ⋅ 6 =+ 48 size 12{8 cdot 6"=+""48"} {} , or 8 ⋅ 6 = 48 size 12{8 cdot 6="48"} {} .

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( − 8 ) ( − 6 ) size 12{ \( - 8 \) \( - 6 \) } {}

| - 8 | = 8 | - 6 | = 6 Multiply these absolute values.

8 ⋅ 6 = 48 size 12{8 cdot 6="48"} {}

Since the numbers have the same sign, the product is positive.

Thus, ( − 8 ) ( − 6 ) =+ 48 size 12{ \( - 8 \) \( - 6 \) "=+""48"} {} , or ( − 8 ) ( − 6 ) = 48 size 12{ \( - 8 \) \( - 6 \) ="48"} {} .

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( − 4 ) ( 7 ) size 12{ \( - 4 \) \( 7 \) } {}

| - 4 | = 4 | 7 | = 7 Multiply these absolute values.

4 ⋅ 7 = 28 size 12{4 cdot 7="28"} {}

Since the numbers have opposite signs, the product is negative.

Thus, ( − 4 ) ( 7 ) = − 28 size 12{ \( - 4 \) \( 7 \) = - "28"} {} .

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6 ( − 3 ) size 12{6 \( - 3 \) } {}

| 6 | = 6 | - 3 | = 3 Multiply these absolute values.

6 ⋅ 3 = 18 size 12{6 cdot 3="18"} {}

Since the numbers have opposite signs, the product is negative.

Thus, 6 ( − 3 ) = − 18 size 12{6 \( - 3 \) = - "18"} {} .

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Practice set a

Find the following products.

3 ( − 8 ) size 12{3 \( - 8 \) } {}

-24

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4 ( 16 ) size 12{4 \( "16" \) } {}

64

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( − 6 ) ( − 5 ) size 12{ \( - 6 \) \( - 5 \) } {}

30

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( − 7 ) ( − 2 ) size 12{ \( - 7 \) \( - 2 \) } {}

14

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( − 1 ) ( 4 ) size 12{ \( - 1 \) \( 4 \) } {}

-4

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( − 7 ) 7 size 12{ \( - 7 \) \( 7 \) } {}

-49

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Division of signed numbers

To determine the signs in a division problem, recall that

12 3 = 4 size 12{ { {"12"} over {3} } =4} {} since 12 = 3 ⋅ 4 size 12{"12"=3 cdot 4} {}

This suggests that

( + ) ( + ) = ( + ) size 12{ { { \( + \) } over { \( + \) } } = \( + \) } {}

( + ) ( + ) = ( + ) size 12{ { { \( + \) } over { \( + \) } } = \( + \) } {} since ( + ) = ( + ) ( + ) size 12{ \( + \) = \( + \) \( + \) } {}

What is 12 − 3 size 12{ { {"12"} over { - 3} } } {} ?

− 12 = ( − 3 ) ( − 4 ) size 12{ - "12"= \( - 3 \) \( - 4 \) } {} suggests that 12 − 3 = − 4 size 12{ { {"12"} over { - 3} } = - 4} {} . That is,

( + ) ( − ) = ( − ) size 12{ { { \( + \) } over { \( - \) } } = \( - \) } {}

( + ) = ( − ) ( − ) size 12{ \( + \) = \( - \) \( - \) } {} suggests that ( + ) ( − ) = ( − ) size 12{ { { \( + \) } over { \( - \) } } = \( - \) } {}

What is − 12 3 size 12{ { { - "12"} over {3} } } {} ?

− 12 = ( 3 ) ( − 4 ) size 12{ - "12"= \( 3 \) \( - 4 \) } {} suggests that − 12 3 = − 4 size 12{ { { - "12"} over {3} } = - 4} {} . That is,

( − ) ( + ) = ( − ) size 12{ { { \( - \) } over { \( + \) } } = \( - \) } {}

( − ) = ( + ) ( − ) size 12{ \( - \) = \( + \) \( - \) } {} suggests that ( − ) ( + ) = ( − ) size 12{ { { \( - \) } over { \( + \) } } = \( - \) } {}

What is − 12 − 3 size 12{ { { - "12"} over { - 3} } } {} ?

− 12 = ( − 3 ) ( 4 ) size 12{ - "12"= \( - 3 \) \( 4 \) } {} suggests that − 12 − 3 = 4 size 12{ { { - "12"} over { - 3} } =4} {} . That is,

( − ) ( − ) = ( + ) size 12{ { { \( - \) } over { \( - \) } } = \( + \) } {}

( − ) = ( − ) ( + ) size 12{ \( - \) = \( - \) \( + \) } {} suggests that ( − ) ( − ) = ( + ) size 12{ { { \( - \) } over { \( - \) } } = \( + \) } {}

We have the following rules for dividing signed numbers.

Rules for dividing signed numbers

Dividing signed numbers:
  1. To divide two real numbers that have the same sign , divide their absolute values. The quotient is positive.
    ( + ) ( + ) = ( + ) size 12{ { { \( + \) } over { \( + \) } } = \( + \) } {} ( − ) ( − ) = ( + ) size 12{ { { \( - \) } over { \( - \) } } = \( + \) } {}
  2. To divide two real numbers that have opposite signs , divide their absolute values. The quotient is negative.
    ( − ) ( + ) = ( − ) size 12{ { { \( - \) } over { \( + \) } } = \( - \) } {} ( + ) ( − ) = ( − ) size 12{ { { \( + \) } over { \( - \) } } = \( - \) } {}

Sample set b

Find the following quotients.

− 10 2 size 12{ { { - "10"} over {2} } } {}

| - 10 | = 10 | 2 | = 2 Divide these absolute values.

10 2 = 5 size 12{ { {"10"} over {2} } =5} {}

Since the numbers have opposite signs, the quotient is negative.

Thus − 10 2 = − 5 size 12{ { { - "10"} over {2} } = - 5} {} .

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− 35 − 7 size 12{ { { - "35"} over { - 7} } } {}

| - 35 | = 35 | - 7 | = 7 Divide these absolute values.

35 7 = 5 size 12{ { {"35"} over {7} } =5} {}

Since the numbers have the same signs, the quotient is positive.

Thus, − 35 − 7 = 5 size 12{ { { - "35"} over { - 7} } =5} {} .

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18 − 9 size 12{ { {"18"} over { - 9} } } {}

| 18 | = 18 | - 9 | = 9 Divide these absolute values.

18 9 = 2 size 12{ { {"18"} over {9} } =2} {}

Since the numbers have opposite signs, the quotient is negative.

Thus, 18 − 9 = 2 size 12{ { {"18"} over { - 9} } =2} {} .

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Practice set b

Find the following quotients.

− 24 − 6 size 12{ { { - "24"} over { - 6} } } {}

4

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30 − 5 size 12{ { {"30"} over { - 5} } } {}

-6

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− 54 27 size 12{ { { - "54"} over {"27"} } } {}

-2

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51 17 size 12{ { {"51"} over {"17"} } } {}

3

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Sample set c

Find the value of − 6 ( 4 − 7 ) − 2 ( 8 − 9 ) − ( 4 + 1 ) + 1 size 12{ { { - 6 \( 4 - 7 \) - 2 \( 8 - 9 \) } over { - \( 4+1 \) +1} } } {} .

Using the order of operations and what we know about signed numbers, we get,

- 6 ( 4 - 7 ) - 2 ( 8 - 9 ) - ( 4 + 1 ) + 1 = - 6 ( - 3 ) - 2 ( - 1 ) - ( 5 ) + 1 = 18 + 2 - 5 + 1 = 20 - 4 = - 5

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Practice set c

Find the value of − 5 ( 2 − 6 ) − 4 ( − 8 − 1 ) 2 ( 3 − 10 ) − 9 ( − 2 ) size 12{ { { - 5 \( 2 - 6 \) - 4 \( - 8 - 1 \) } over {2 \( 3 - "10" \) - 9 \( - 2 \) } } } {} .

14

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Calculators

Calculators with the A box with a plus and minus sign. key can be used for multiplying and dividing signed numbers.

Sample set d

Use a calculator to find each quotient or product.

( − 186 ) ⋅ ( − 43 ) size 12{ \( - "186" \) cdot \( - "43" \) } {}

Since this product involves a ( negative ) ⋅ ( negative ) size 12{ \( "negative" \) cdot \( "negative" \) } {} , we know the result should be a positive number. We'll illustrate this on the calculator.

Display Reads
Type 186 186
Press A box with a plus and minus sign. -186
Press × -186
Type 43 43
Press A box with a plus and minus sign. -43
Press = 7998

Thus, ( − 186 ) ⋅ ( − 43 ) = 7, 998 size 12{ \( - "186" \) cdot \( - "43" \) =7,"998"} {} .

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158 . 64 − 54 . 3 size 12{ { {"158" "." "64"} over { - "54" "." 3} } } {} . Round to one decimal place.

Display Reads
Type 158.64 158.64
Press ÷ 158.64
Type 54.3 54.3
Press A box with a plus and minus sign. -54.3
Press = -2.921546961

Rounding to one decimal place we get -2.9.

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Practice set d

Use a calculator to find each value.

( - 51 . 3 ) ⋅ ( - 21 . 6 ) size 12{ \( "–51" "." 3 \) cdot \( "–21" "." 6 \) } {}

1,108.08

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- 2 . 5746 ÷ - 2 . 1 size 12{"–2" "." "5746" div " –2" "." 1} {}

1.226

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( 0 . 006 ) ⋅ ( - 0 . 241 ) size 12{ \( 0 "." "006" \) cdot \( – 0 "." "241" \) } {} . Round to three decimal places.

-0.001

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Exercises

Find the value of each of the following. Use a calculator to check each result.

− 2 − 8 size 12{ left (-2 right ) left (-8 right )} {}

16

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− 3 − 9 size 12{ left (-3 right ) left (-9 right )} {}

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− 4 − 8 size 12{ left (-4 right ) left (-8 right )} {}

32

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− 5 − 2 size 12{ left (-5 right ) left (-2 right )} {}

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3 − 12 size 12{ left (3 right ) left (-"12" right )} {}

-36

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4 − 18 size 12{ left (4 right ) left (-"18" right )} {}

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10 − 6 size 12{ left ("10" right ) left (-6 right )} {}

-60

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− 6 4 size 12{ left (-6 right ) left (4 right )} {}

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− 2 6 size 12{ left (-2 right ) left (6 right )} {}

-12

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− 8 7 size 12{ left (-8 right ) left (7 right )} {}

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21 7 size 12{ { {"21"} over {7} } } {}

3

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42 6 size 12{ { {"42"} over {6} } } {}

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− 39 3 size 12{ { {-"39"} over {3} } } {}

-13

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− 20 10 size 12{ { {-"20"} over {"10"} } } {}

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− 45 − 5 size 12{ { {-"45"} over {-5} } } {}

9

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− 16 − 8 size 12{ { {-"16"} over {-8} } } {}

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25 − 5 size 12{ { {"25"} over {-5} } } {}

-5

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36 − 4 size 12{ { {"36"} over {-4} } } {}

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8 − − 3 size 12{8- left (-3 right )} {}

11

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14 − − 20 size 12{"14"- left (-"20" right )} {}

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20 − − 8 size 12{"20"- left (-8 right )} {}

28

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− 4 − − 1 size 12{-4- left (-1 right )} {}

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0 − 4 size 12{0-4} {}

-4

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0 − − 1 size 12{0- left (-1 right )} {}

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− 6 + 1 − 7 size 12{-6+1-7} {}

-12

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15 − 12 − 20 size 12{"15"-"12"-"20"} {}

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1 − 6 − 7 + 8 size 12{1-6-7+8} {}

-4

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2 + 7 − 10 + 2 size 12{2+7-"10"+2} {}

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3 4 − 6 size 12{3 left (4-6 right )} {}

-6

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8 5 − 12 size 12{8 left (5-"12" right )} {}

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− 3 1 − 6 size 12{-3 left (1-6 right )} {}

15

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− 8 4 − 12 + 2 size 12{-8 left (4-"12" right )+2} {}

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− 4 1 − 8 + 3 10 − 3 size 12{-4 left (1-8 right )+3 left ("10"-3 right )} {}

49

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− 9 0 − 2 + 4 8 − 9 + 0 − 3 size 12{-9 left (0-2 right )+4 left (8-9 right )+0 left (-3 right )} {}

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6 − 2 − 9 − 6 2 + 9 + 4 − 1 − 1 size 12{6 left (-2-9 right )-6 left (2+9 right )+4 left (-1-1 right )} {}

-140

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3 4 + 1 − 2 5 − 2 size 12{ { {3 left (4+1 right )-2 left (5 right )} over {-2} } } {}

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4 8 + 1 − 3 − 2 − 4 − 2 size 12{ { {4 left (8+1 right )-3 left (-2 right )} over {-4-2} } } {}

-7

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− 1 3 + 2 + 5 − 1 size 12{ { {-1 left (3+2 right )+5} over {-1} } } {}

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− 3 4 − 2 + − 3 − 6 − 4 size 12{ { {-3 left (4-2 right )+ left (-3 right ) left (-6 right )} over {-4} } } {}

-3

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− 1 4 + 2 size 12{-1 left (4+2 right )} {}

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− 1 6 − 1 size 12{-1 left (6-1 right )} {}

-5

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− 8 + 21 size 12{- left (8+"21" right )} {}

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− 8 − 21 size 12{- left (8-"21" right )} {}

13

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Exercises for review

( [link] ) Use the order of operations to simplify 5 2 + 3 2 + 2 ÷ 2 2 size 12{ left (5 rSup { size 8{2} } +3 rSup { size 8{2} } +2 right )¸2 rSup { size 8{2} } } {} .

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( [link] ) Find 3 8 of 32 9 size 12{ { {3} over {8} } " of " { {"32"} over {9} } } {} .

4 3 = 1 1 3 size 12{ { {4} over {3} } =1 { {1} over {3} } } {}

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( [link] ) Write this number in decimal form using digits: “fifty-two three-thousandths”

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( [link] ) The ratio of chlorine to water in a solution is 2 to 7. How many mL of water are in a solution that contains 15 mL of chlorine?

52 1 2 size 12{"52" { {1} over {2} } } {}

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( [link] ) Perform the subtraction − 8 − − 20 size 12{-8- left (-"20" right )} {}

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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