10.6: Multiplication and Division of Signed Numbers
- be able to multiply and divide signed numbers
- be able to multiply and divide signed numbers using a calculator
Multiplication of Signed Numbers
Let us consider first, the product of two positive numbers. Multiply: \(3 \cdot 5\).
\(3 \cdot 5\) means \(5 + 5 + 5 = 15\)
This suggests that (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.)
\(\text{(positive number)} \cdot \text{(positive number)} = \text{(positive number)}\)
More briefly,
(+) (+) = (+)
Now consider the product of a positive number and a negative number. Multiply: (3)(-5)
(3)(-5) means (-5) + (-5) + (-5) = -15
This suggests that
\(\text{(positive number)} \cdot \text{(negative number)} = \text{(negative number)}\)
More briefly,
(+) (-) = (-)
By the commutative property of multiplication, we get
\(\text{(negative number)} \cdot \text{(positive number)} = \text{(negative number)}\)
More briefly,
(-) (+) = (-)
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:
-
To multiply two real numbers that have the
same sign
, multiply their absolute values. The product is positive.
(+) (+) = (+)
(-) (-) = (+) -
To multiply two real numbers that have
opposite signs
, multiply their absolute values. The product is negative.
(+) (-) = (-)
(-) (+) = (-)
Find the following products.
\(8 \cdot 6\)
Solution
\(\begin{array} {ccl} {|8|} & = & {8} \\ {|6|} & = & {6} \end{array} \big \}\) Multiply these absolute values.
\(8 \cdot 6 = 48\)
Since the numbers have the same sign, the product is positive.
Thus, \(8 \cdot 6 = +48\), or \(8 \cdot 6 = 48\).
(-8)(-6)
Solution
\(\begin{array} {ccl} {|-8|} & = & {8} \\ {|-6|} & = & {6} \end{array} \big \}\) Multiply these absolute values.
\(8 \cdot 6 = 48\)
Since the numbers have the same sign, the product is positive.
Thus, \((-8)(-6) = +48\), or \((-8)(-6) = 48\).
(-4)(7)
Solution
\(\begin{array} {ccl} {|-4|} & = & {4} \\ {|7|} & = & {7} \end{array} \big \}\) Multiply these absolute values.
\(4 \cdot 7 = 28\)
Since the numbers have opposite signs, the product is negative.
Thus, (-4)(7) = -28.
6(-3)
Solution
\(\begin{array} {ccl} {|6|} & = & {6} \\ {|-3|} & = & {3} \end{array} \big \}\) Multiply these absolute values.
\(6 \cdot 3 = 18\)
Since the numbers have opposite signs, the product is negative.
Thus, 6(-3) = -18.
Practice Set A
Find the following products.
3(-8)
- Answer
-
-24
Practice Set A
4(16)
- Answer
-
64
Practice Set A
(-6)(-5)
- Answer
-
30
Practice Set A
(-7)(-2)
- Answer
-
14
Practice Set A
(-1)(4)
- Answer
-
-4
Practice Set A
(-7)7
- Answer
-
-49
Division of Signed Numbers
To determine the signs in a division problem, recall that
\(\dfrac{12}{3} = 4\) since \(12 = 3 \cdot 4\)
This suggests that
\(\dfrac{(+)}{(+)} = (+)\)
\(\dfrac{(+)}{(+)} = (+)\) since (+) = (+) (+)
What is \(\dfrac{12}{-3}\)?
12 = (-3)(-4) suggets that \(\dfrac{12}{-3} = -4\). That is,
\(\dfrac{(+)}{(-)} = (-)\)
(+) = (-) (-) suggets that \(\dfrac{(+)}{(-)} = (-)\)
What is \(\dfrac{-12}{3}\)?
-12 = (3)(-4) suggests that \(\dfrac{-12}{3} = -4\). That is,
\(\dfrac{(-)}{(+)} = (-)\)
(-) = (+) (-) suggets that \(\dfrac{(-)}{(+)} = (-)\)
What is \(\dfrac{-12}{-3}\)?
-12 = (-3)(4) suggets that \(\dfrac{-12}{-3} = 4\). That is,
\(\dfrac{(-)}{(-)} = (+)\)
(-) = (-)(+) suggests that \(\dfrac{(-)}{(-)} = (+)\)
We have the following rules for dividing signed numbers.
Rules for Dividing Signed Numbers
Dividing signed numbers:
-
To divide two real numbers that have the
same sign
, divide their absolute values. The quotient is positive.
\(\dfrac{(+)}{(+)} = (+)\dfrac{(-)}{(-)} = (+)\) -
To divide two real numbers that have
opposite signs
, divide their absolute values. The quotient is negative.
\(\dfrac{(-)}{(+)} = (-)\dfrac{(+)}{(-)} = (-)\)
Find the following quotients.
\(\dfrac{-10}{2}\)
Solution
\(\begin{array} {ccc} {|-10|} & = & {10} \\ {|2|} & = & {2} \end{array} \big \}\) Divide these absolute values.
\(\dfrac{10}{2} = 5\)
Since the numbers have opposite signs, the quotient is negative.
Thus \(\dfrac{-10}{2} = -5\).
\(\dfrac{-35}{-7}\)
Solution
\(\begin{array} {ccc} {|-35|} & = & {35} \\ {|-7|} & = & {7} \end{array} \big \}\) Divide these absolute values.
\(\dfrac{35}{7} = 5\)
Since the numbers have the same signs, the quotient is positive.
Thus \(\dfrac{-35}{-7} = 5\).
\(\dfrac{18}{-9}\)
Solution
\(\begin{array} {ccc} {|18|} & = & {18} \\ {|-9|} & = & {9} \end{array} \big \}\) Divide these absolute values.
\(\dfrac{18}{9} = 2\)
Since the numbers have opposite signs, the quotient is negative.
Thus \(\dfrac{18}{-9} = -2\).
Practice Set B
Find the following quotients.
\(\dfrac{-24}{-6}\)
- Answer
-
4
Practice Set B
\(\dfrac{30}{-5}\)
- Answer
-
-6
Practice Set B
\(\dfrac{-54}{27}\)
- Answer
-
-2
Practice Set B
\(\dfrac{51}{17}\)
- Answer
-
3
Find the value of \(\dfrac{-6(4 - 7) - 2(8 - 9)}{-(4 + 1) + 1}\).
Solution
Using the order of operations and what we know about signed numbers, we get,
\(\begin{array} {rcl} {\dfrac{-6(4 - 7) - 2(8 - 9)}{-(4 + 1) + 1}} & = & {\dfrac{-6(-3) - 2(-1)}{-(5) + 1}} \\ {} & = & {\dfrac{18 + 2}{-5 + 1}} \\ {} & = & {\dfrac{20}{-4}} \\ {} & = & {-5} \end{array}\)
Practice Set C
Find the value of \(\dfrac{-5(2 - 6) - 4(-8 - 1)}{2(3 - 10) - 9(-2)}\).
- Answer
-
14
Calculators
Calculators with the
key can be used for multiplying and dividing signed numbers.
Use a calculator to find each quotient or product.
\((-186) \cdot (-43)\)
Solution
Since this product involves a \(\text{(negative)} \cdot \text{(negative)}\), we know the result should be a positive number. We'll illustrate this on the calculator.
| Display Reads | ||
| Type | 186 | 186 |
| Press | -186 | |
| Press | \(\times\) | -186 |
| Type | 43 | 43 |
| Press | -43 | |
| Press | = | 7998 |
Thus, \((-186) \cdot (-43) = 7,998\)
\(\dfrac{158.64}{-54.3}\). Round to one decimal place.
Solution
Since this product involves a \(\text{(negative)} \cdot \text{(negative)}\), we know the result should be a positive number. We'll illustrate this on the calculator.
| Display Reads | ||
| Type | 158.64 | 158.64 |
| Press | \(\div\) | 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.
\((-51.3) \cdot (-21.6)\)
- Answer
-
1,108.08
Practice Set D
\(-2.5746 \div -2.1\)
- Answer
-
1.226
Practice Set D
\((0.006) \cdot (-0.241)\). Round to three decimal places.
- Answer
-
-0.001
Exercises
Find the value of each of the following. Use a calculator to check each result.
Exercise \(\PageIndex{1}\)
(-2)(-8)
- Answer
-
16
Exercise \(\PageIndex{2}\)
(-3)(-9)
Exercise \(\PageIndex{3}\)
(-4)(-8)
- Answer
-
32
Exercise \(\PageIndex{4}\)
(-5)(-2)
Exercise \(\PageIndex{5}\)
(3)(-12)
- Answer
-
-36
Exercise \(\PageIndex{6}\)
(4)(-18)
Exercise \(\PageIndex{7}\)
(10)(-6)
- Answer
-
-60
Exercise \(\PageIndex{8}\)
(-6)(4)
Exercise \(\PageIndex{9}\)
(-2)(6)
- Answer
-
-12
Exercise \(\PageIndex{10}\)
(-8)(7)
Exercise \(\PageIndex{11}\)
\(\dfrac{21}{7}\)
- Answer
-
3
Exercise \(\PageIndex{12}\)
\(\dfrac{42}{6}\)
Exercise \(\PageIndex{13}\)
\(\dfrac{-39}{3}\)
- Answer
-
-13
Exercise \(\PageIndex{14}\)
\(\dfrac{-20}{10}\)
Exercise \(\PageIndex{15}\)
\(\dfrac{-45}{-5}\)
- Answer
-
9
Exercise \(\PageIndex{16}\)
\(\dfrac{-16}{-8}\)
Exercise \(\PageIndex{17}\)
\(\dfrac{25}{-5}\)
- Answer
-
-5
Exercise \(\PageIndex{18}\)
\(\dfrac{36}{-4}\)
Exercise \(\PageIndex{19}\)
8 - (-3)
- Answer
-
11
Exercise \(\PageIndex{20}\)
14 - (-20)
Exercise \(\PageIndex{21}\)
20 - (-8)
- Answer
-
28
Exercise \(\PageIndex{22}\)
(-4) - (-1)
Exercise \(\PageIndex{23}\)
0 - 4
- Answer
-
-4
Exercise \(\PageIndex{24}\)
0 - (-1)
Exercise \(\PageIndex{25}\)
-6 + 1 - 7
- Answer
-
-12
Exercise \(\PageIndex{26}\)
15 - 12 - 20
Exercise \(\PageIndex{27}\)
1 - 6 - 7 + 8
- Answer
-
-4
Exercise \(\PageIndex{28}\)
2 + 7 - 10 + 2
Exercise \(\PageIndex{29}\)
3(4 - 6)
- Answer
-
-6
Exercise \(\PageIndex{30}\)
8(5 - 12)
Exercise \(\PageIndex{31}\)
-3(1 - 6)
- Answer
-
15
Exercise \(\PageIndex{32}\)
-8(4 - 12) + 2
Exercise \(\PageIndex{33}\)
-4(1 - 8) + 3(10 - 3)
- Answer
-
49
Exercise \(\PageIndex{34}\)
-9(0 - 2) + 4(8 - 9) + 0(-3)
Exercise \(\PageIndex{35}\)
6(-2 - 9) - 6(2 + 9) + 4(-1 - 1)
- Answer
-
-140
Exercise \(\PageIndex{36}\)
\(\dfrac{3(4 + 1) - 2 (5)}{-2}\)
Exercise \(\PageIndex{37}\)
\(\dfrac{4(8 + 1) - 3 (-2)}{-4 - 2}\)
- Answer
-
-7
Exercise \(\PageIndex{38}\)
\(\dfrac{-1(3 + 2) + 5}{-1}\)
Exercise \(\PageIndex{39}\)
\(\dfrac{-3(4 - 2) + (-3)(-6)}{-4}\)
- Answer
-
-3
Exercise \(\PageIndex{40}\)
-1(4 + 2)
Exercise \(\PageIndex{41}\)
-1(6 - 1)
- Answer
-
-5
Exercise \(\PageIndex{42}\)
-(8 + 21)
Exercise \(\PageIndex{43}\)
-(8 - 21)
- Answer
-
13
Exercises for Review
Exercise \(\PageIndex{44}\)
Use the order of operations to simplify \((5^2 + 3^2 + 2) \div 2^2\).
Exercise \(\PageIndex{45}\)
Find \(\dfrac{3}{8}\) of \(\dfrac{32}{9}\).
- Answer
-
\(\dfrac{4}{3} = 1 \dfrac{1}{3}\)
Exercise \(\PageIndex{46}\)
Write this number in decimal form using digits: “fifty-two three-thousandths”
Exercise \(\PageIndex{47}\)
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?
- Answer
-
\(52 \dfrac{1}{2}\)
Exercise \(\PageIndex{48}\)
Perform the subtraction -8 - (-20)