# Multiplication and division of real 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: $\text{3}\cdot \text{5}$ .

$\text{3}\cdot \text{5}$ means $5+5+5=\text{15}$

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

$\left(\text{positive number}\right)\cdot \left(\text{positive number}\right)=\left(\text{positive number}\right)$

More briefly,

$\left(+\right)\left(+\right)=\left(+\right)$

Now consider the product of a positive number and a negative number. Multiply: $\left(3\right)\left(-5\right)$ .

$\left(3\right)\left(-5\right)$ means $\left(-5\right)+\left(-5\right)+\left(-5\right)=-\text{15}$

This suggests that

$\left(\text{positive number}\right)\cdot \left(\text{negative number}\right)=\left(\text{negative number}\right)$

More briefly,

$\left(+\right)\left(-\right)=\left(-\right)$

By the commutative property of multiplication, we get

$\left(\text{negative number}\right)\cdot \left(\text{positive number}\right)=\left(\text{negative number}\right)$

More briefly,

$\left(-\right)\left(+\right)=\left(-\right)$

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.

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.
$\left(+\right)\left(+\right)=\left(+\right)$
$\left(-\right)\left(-\right)=\left(+\right)$
2. To multiply two real numbers that have opposite signs , multiply their abso­lute values. The product is negative.
$\left(+\right)\left(-\right)=\left(-\right)$
$\left(-\right)\left(+\right)=\left(-\right)$

## Sample set a

Find the following products.

$\text{8}\cdot \text{6}$

$\left(\begin{array}{ccc}|8|& =& 8\\ |6|& =& 6\end{array}}$ Multiply these absolute values.

$8\cdot 6=\text{48}$

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

Thus, $8\cdot 6\text{=+}\text{48}$ , or $8\cdot 6=\text{48}$ .

$\left(-8\right)\left(-6\right)$

$\left(\begin{array}{ccc}|-8|& =& 8\\ |-6|& =& 6\end{array}}$ Multiply these absolute values.

$8\cdot 6=\text{48}$

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

Thus, $\left(-8\right)\left(-6\right)\text{=+}\text{48}$ , or $\left(-8\right)\left(-6\right)=\text{48}$ .

$\left(-4\right)\left(7\right)$

$\left(\begin{array}{ccc}|-4|& =& 4\\ |7|& =& 7\end{array}}$ Multiply these absolute values.

$4\cdot 7=\text{28}$

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

Thus, $\left(-4\right)\left(7\right)=-\text{28}$ .

$6\left(-3\right)$

$\left(\begin{array}{ccc}|6|& =& 6\\ |-3|& =& 3\end{array}}$ Multiply these absolute values.

$6\cdot 3=\text{18}$

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

Thus, $6\left(-3\right)=-\text{18}$ .

## Practice set a

Find the following products.

$3\left(-8\right)$

-24

$4\left(\text{16}\right)$

64

$\left(-6\right)\left(-5\right)$

30

$\left(-7\right)\left(-2\right)$

14

$\left(-1\right)\left(4\right)$

-4

$\left(-7\right)7$

-49

## Division of signed numbers

To determine the signs in a division problem, recall that

$\frac{\text{12}}{3}=4$ since $\text{12}=3\cdot 4$

This suggests that

## $\frac{\left(+\right)}{\left(+\right)}=\left(+\right)$

$\frac{\left(+\right)}{\left(+\right)}=\left(+\right)$ since $\left(+\right)=\left(+\right)\left(+\right)$

What is $\frac{\text{12}}{-3}$ ?

$-\text{12}=\left(-3\right)\left(-4\right)$ suggests that $\frac{\text{12}}{-3}=-4$ . That is,

## $\frac{\left(+\right)}{\left(-\right)}=\left(-\right)$

$\left(+\right)=\left(-\right)\left(-\right)$ suggests that $\frac{\left(+\right)}{\left(-\right)}=\left(-\right)$

What is $\frac{-\text{12}}{3}$ ?

$-\text{12}=\left(3\right)\left(-4\right)$ suggests that $\frac{-\text{12}}{3}=-4$ . That is,

## $\frac{\left(-\right)}{\left(+\right)}=\left(-\right)$

$\left(-\right)=\left(+\right)\left(-\right)$ suggests that $\frac{\left(-\right)}{\left(+\right)}=\left(-\right)$

What is $\frac{-\text{12}}{-3}$ ?

$-\text{12}=\left(-3\right)\left(4\right)$ suggests that $\frac{-\text{12}}{-3}=4$ . That is,

## $\frac{\left(-\right)}{\left(-\right)}=\left(+\right)$

$\left(-\right)=\left(-\right)\left(+\right)$ suggests that $\frac{\left(-\right)}{\left(-\right)}=\left(+\right)$

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.
$\frac{\left(+\right)}{\left(+\right)}=\left(+\right)$ $\frac{\left(-\right)}{\left(-\right)}=\left(+\right)$
2. To divide two real numbers that have opposite signs , divide their absolute values. The quotient is negative.
$\frac{\left(-\right)}{\left(+\right)}=\left(-\right)$ $\frac{\left(+\right)}{\left(-\right)}=\left(-\right)$

## Sample set b

Find the following quotients.

$\frac{-\text{10}}{2}$

$\left(\begin{array}{ccc}|-10|& =& 10\\ |2|& =& 2\end{array}}$ Divide these absolute values.

$\frac{\text{10}}{2}=5$

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

Thus $\frac{-\text{10}}{2}=-5$ .

$\frac{-\text{35}}{-7}$

$\left(\begin{array}{ccc}|-35|& =& 35\\ |-7|& =& 7\end{array}}$ Divide these absolute values.

$\frac{\text{35}}{7}=5$

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

Thus, $\frac{-\text{35}}{-7}=5$ .

$\frac{\text{18}}{-9}$

$\left(\begin{array}{ccc}|18|& =& 18\\ |-9|& =& 9\end{array}}$ Divide these absolute values.

$\frac{\text{18}}{9}=2$

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

Thus, $\frac{\text{18}}{-9}=2$ .

## Practice set b

Find the following quotients.

$\frac{-\text{24}}{-6}$

4

$\frac{\text{30}}{-5}$

-6

$\frac{-\text{54}}{\text{27}}$

-2

$\frac{\text{51}}{\text{17}}$

3

## Sample set c

Find the value of $\frac{-6\left(4-7\right)-2\left(8-9\right)}{-\left(4+1\right)+1}$ .

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

$\begin{array}{ccc}\hfill \frac{-6\left(4-7\right)-2\left(8-9\right)}{-\left(4+1\right)+1}& =& \frac{-6\left(-3\right)-2\left(-1\right)}{-\left(5\right)+1}\hfill \\ & =& \frac{18+2}{-5+1}\hfill \\ & =& \frac{20}{-}\hfill \\ & =& -5\hfill \end{array}$

## Practice set c

Find the value of $\frac{-5\left(2-6\right)-4\left(-8-1\right)}{2\left(3-\text{10}\right)-9\left(-2\right)}$ .

14

## Calculators

Calculators with the key can be used for multiplying and dividing signed numbers.

## Sample set d

Use a calculator to find each quotient or product.

$\left(-\text{186}\right)\cdot \left(-\text{43}\right)$

Since this product involves a $\left(\text{negative}\right)\cdot \left(\text{negative}\right)$ , we know the result should be a positive number. We'll illustrate this on the calculator.

 Display Reads Type 186 186 Press -186 Press × -186 Type 43 43 Press -43 Press = 7998

Thus, $\left(-\text{186}\right)\cdot \left(-\text{43}\right)=7,\text{998}$ .

$\frac{\text{158}\text{.}\text{64}}{-\text{54}\text{.}3}$ . Round to one decimal place.

 Display Reads Type 158.64 158.64 Press ÷ 158.64 Type 54.3 54.3 Press -54.3 Press = -2.921546961

Rounding to one decimal place we get -2.9.

## Practice set d

Use a calculator to find each value.

$\left(-51\text{.}3\right)\cdot \left(-21\text{.}6\right)$

1,108.08

$-2\text{.}\text{5746}÷-2\text{.}1$

1.226

$\left(0\text{.}\text{006}\right)\cdot \left(-0\text{.}\text{241}\right)$ . Round to three decimal places.

-0.001

## Exercises

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

$\left(-2\right)\left(-8\right)$

16

$\left(-3\right)\left(-9\right)$

$\left(-4\right)\left(-8\right)$

32

$\left(-5\right)\left(-2\right)$

$\left(3\right)\left(-\text{12}\right)$

-36

$\left(4\right)\left(-\text{18}\right)$

$\left(\text{10}\right)\left(-6\right)$

-60

$\left(-6\right)\left(4\right)$

$\left(-2\right)\left(6\right)$

-12

$\left(-8\right)\left(7\right)$

$\frac{\text{21}}{7}$

3

$\frac{\text{42}}{6}$

$\frac{-\text{39}}{3}$

-13

$\frac{-\text{20}}{\text{10}}$

$\frac{-\text{45}}{-5}$

9

$\frac{-\text{16}}{-8}$

$\frac{\text{25}}{-5}$

-5

$\frac{\text{36}}{-4}$

$8-\left(-3\right)$

11

$\text{14}-\left(-\text{20}\right)$

$\text{20}-\left(-8\right)$

28

$-4-\left(-1\right)$

$0-4$

-4

$0-\left(-1\right)$

$-6+1-7$

-12

$\text{15}-\text{12}-\text{20}$

$1-6-7+8$

-4

$2+7-\text{10}+2$

$3\left(4-6\right)$

-6

$8\left(5-\text{12}\right)$

$-3\left(1-6\right)$

15

$-8\left(4-\text{12}\right)+2$

$-4\left(1-8\right)+3\left(\text{10}-3\right)$

49

$-9\left(0-2\right)+4\left(8-9\right)+0\left(-3\right)$

$6\left(-2-9\right)-6\left(2+9\right)+4\left(-1-1\right)$

-140

$\frac{3\left(4+1\right)-2\left(5\right)}{-2}$

$\frac{4\left(8+1\right)-3\left(-2\right)}{-4-2}$

-7

$\frac{-1\left(3+2\right)+5}{-1}$

$\frac{-3\left(4-2\right)+\left(-3\right)\left(-6\right)}{-4}$

-3

$-1\left(4+2\right)$

$-1\left(6-1\right)$

-5

$-\left(8+\text{21}\right)$

$-\left(8-\text{21}\right)$

13

## Exercises for review

( [link] ) Use the order of operations to simplify $\left({5}^{2}+{3}^{2}+2\right)÷{2}^{2}$ .

( [link] ) Find $\frac{3}{8}\text{of}\frac{\text{32}}{9}$ .

$\frac{4}{3}=1\frac{1}{3}$

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

( [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?

$\text{52}\frac{1}{2}$

( [link] ) Perform the subtraction $-8-\left(-\text{20}\right)$

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