Multiply and Divide Integers

Multiply integers

Since multiplication is mathematical shorthand for repeated addition, our model for adding integers can also show multiplication. Remember that $a \cdot b$ means add $a$, $b$ times.

Using that idea, $5 \cdot 3$ means add $5$ three times, giving $15$ positives, so $5 \cdot 3 = 15$. And $-5(3)$ means add $-5$ three times, giving $15$ negatives, so $-5(3) = -15$.

What does it mean to multiply $5$ by $-3$? It means subtract $5$, three times. Looking at subtraction as “taking away,” it means to take away $5$, three times. But there is nothing to take away, so we start by adding neutral pairs, then take away $5$ three times — leaving $15$ negatives, so $5(-3) = -15$. To find $(-5)(-3)$, we instead take away $-5$ three times, which leaves $15$ positives, so $(-5)(-3) = 15$.

In summary:

Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative. We’ll put this all together in the chart below.

Example. Multiply: (a) $-9 \cdot 3$ (b) $-2(-5)$ (c) $4(-8)$ (d) $7 \cdot 6$.

(a) The signs are different, so the product is negative: $-9 \cdot 3 = -27$.

(b) The signs are the same, so the product is positive: $-2(-5) = 10$.

(c) The signs are different: $4(-8) = -32$.

(d) The signs are the same: $7 \cdot 6 = 42$.

When we multiply a number by $1$, the result is the same number. What happens when we multiply a number by $-1$? Let’s multiply a positive number and then a negative number by $-1$ to see what we get.

Here $-4$ is the opposite of $4$, and $3$ is the opposite of $-3$. Each time we multiply a number by $-1$, we get its opposite!

Example. Multiply: (a) $-1 \cdot 7$ (b) $-1(-11)$.

(a) The signs are different, so the product is negative, and $-7$ is the opposite of $7$: $-1 \cdot 7 = -7$.

(b) The signs are the same, so the product is positive, and $11$ is the opposite of $-11$: $-1(-11) = 11$.

Divide integers

Division is the inverse operation of multiplication. So $15 \div 3 = 5$ because $5 \cdot 3 = 15$. In words, this expression says that $15$ can be divided into three groups of five each, because adding five three times gives $15$. Look at some examples of multiplying integers to figure out the rules for dividing integers.

Division follows the same rules as multiplication! For division of two signed numbers, when the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative. And remember that we can always check the answer of a division problem by multiplying.

Example. Divide: (a) $-27 \div 3$ (b) $-100 \div (-4)$.

(a) The signs are different, so the quotient is negative: $-27 \div 3 = -9$.

(b) The signs are the same, so the quotient is positive: $-100 \div (-4) = 25$.

Simplify expressions with integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally? Let’s try some examples. We’ll simplify expressions that use all four operations with integers — addition, subtraction, multiplication, and division. Remember to follow the order of operations.

Example. Simplify: $7(-2) + 4(-7) - 6$.

Multiply first: $7(-2) + 4(-7) - 6 = -14 + (-28) - 6$. Then add: $-42 - 6$. Then subtract: $-48$.

Example. Simplify: (a) $(-2)^4$ (b) $-2^4$.

(a) Write in expanded form and multiply: $(-2)^4 = (-2)(-2)(-2)(-2) = 4(-2)(-2) = -8(-2) = 16$.

(b) Write in expanded form — we are asked to find the opposite of $2^4$: $-2^4 = -(2 \cdot 2 \cdot 2 \cdot 2) = -(4 \cdot 2 \cdot 2) = -(8 \cdot 2) = -16$.

Notice the difference between parts (a) and (b). In part (a), the exponent means to raise what is in the parentheses, the $(-2)$, to the $4$th power. In part (b), the exponent means to raise just the $2$ to the $4$th power and then take the opposite.

The next example reminds us to simplify inside parentheses first.

Example. Simplify: $12 - 3(9 - 12)$.

Subtract in parentheses first: $12 - 3(9 - 12) = 12 - 3(-3)$. Then multiply: $12 - (-9)$. Then subtract: $21$.

Example. Simplify: $8(-9) \div (-2)^3$.

Exponents first: $8(-9) \div (-2)^3 = 8(-9) \div (-8)$. Then multiply: $-72 \div (-8)$. Then divide: $9$.

Example. Simplify: $-30 \div 2 + (-3)(-7)$.

Multiply and divide left to right, so divide first: $-30 \div 2 + (-3)(-7) = -15 + (-3)(-7)$. Then multiply: $-15 + 21$. Then add: $6$.

Evaluate variable expressions with integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

Example. When $n = -5$, evaluate: (a) $n + 1$ (b) $-n + 1$.

(a) Substitute $-5$ for $n$: $n + 1 = -5 + 1 = -4$.

(b) Substitute $-5$ for $n$: $-n + 1 = -(-5) + 1 = 5 + 1 = 6$.

Example. Evaluate $(x + y)^2$ when $x = -18$ and $y = 24$.

Substitute $-18$ for $x$ and $24$ for $y$: $(x + y)^2 = (-18 + 24)^2$. Add inside the parenthesis: $(6)^2$. Simplify: $36$.

Example. Evaluate $20 - z$ when (a) $z = 12$ and (b) $z = -12$.

(a) Substitute $12$ for $z$ and subtract: $20 - z = 20 - 12 = 8$.

(b) Substitute $-12$ for $z$ and subtract: $20 - z = 20 - (-12) = 32$.

Example. Evaluate $2x^2 + 3x + 8$ when $x = 4$.

Substitute $4$ for $x$, using parentheses to show multiplication: $2x^2 + 3x + 8 = 2(4)^2 + 3(4) + 8$. Evaluate exponents first: $2(16) + 3(4) + 8$. Multiply: $32 + 12 + 8$. Add: $52$.

Translate phrases to expressions with integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

Example. Translate and simplify: the sum of $8$ and $-12$, increased by $3$.

Translate “the sum of $8$ and $-12$, increased by $3$”: $[8 + (-12)] + 3$. Simplify inside the brackets — be careful not to confuse the brackets with an absolute value sign: $(-4) + 3$. Add: $-1$.

When we first introduced the operation symbols, we saw that the expression $a - b$ may be read in several ways: $a$ minus $b$, the difference of $a$ and $b$, $b$ subtracted from $a$, or $b$ less than $a$. Be careful to get $a$ and $b$ in the right order!

Example. Translate and then simplify: (a) the difference of $13$ and $-21$ (b) subtract $24$ from $-19$.

(a) Translate and simplify: $13 - (-21) = 34$.

(b) Translate — remember “subtract $b$ from $a$” means $a - b$ — and simplify: $-19 - 24 = -43$.

Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

Example. Translate to an algebraic expression and simplify if possible: the product of $-2$ and $14$.

Translate “the product of $-2$ and $14$”: $(-2)(14)$. Simplify: $-28$.

Example. Translate to an algebraic expression and simplify if possible: the quotient of $-56$ and $-7$.

Translate “the quotient of $-56$ and $-7$”: $-56 \div (-7)$. Simplify: $8$.

Use integers in applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

Example. In the morning, the temperature in Urbana, Illinois was $11$ degrees. By mid-afternoon, the temperature had dropped to $-9$ degrees. What was the difference of the morning and afternoon temperatures?

We are asked to find the difference of the morning and afternoon temperatures. The phrase that gives the information to find it is “the difference of $11$ and $-9$.” Translate the phrase to an expression: $11 - (-9)$. Simplify: $20$. The difference in temperatures was $20$ degrees.

Example. The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

We are asked to find the number of yards lost. The phrase that gives the information to find it is “three times a $15$-yard penalty.” Translate the phrase to an expression: $3(-15)$. Simplify: $-45$. The team lost $45$ yards.

Key terms

opposite — the number that is the same distance from zero on the number line but on the opposite side of zero. integer — a number in the set $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. order of operations — the sequence in which operations should be simplified: parentheses first, then exponents, then multiplication and division left to right, then addition and subtraction left to right (Please Excuse My Dear Aunt Sally).

Practice

Multiply integers

Divide integers

Simplify expressions with integers

Evaluate variable expressions with integers

Translate English phrases to algebraic expressions

Use integers in applications

This section is adapted from Elementary Algebra 2e, Section 1.4: Multiply and Divide Integers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the colored-counter multiplication model as narrated prose instead of a hotlinked figure; omitted the Be Prepared callout and the Self Check checklist; adapted selected Section 1.4 Exercises (“Practice Makes Perfect”) into the section-final interactive Practice block and omitted the remaining Section 1.4 Exercises (Everyday Math, Writing Exercises); and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.