Multiply and Divide Integers

Multiply integers

Since multiplication is mathematical shorthand for repeated addition, the counter model that worked for addition and subtraction can show multiplication too. Remember that $a \cdot b$ means add $a$, $b$ times.

$5 \cdot 3$ means add $5$ three times: $5+5+5 = 15$ positives, so $5 \cdot 3 = 15$. And $-5(3)$ means add $-5$ three times: $-15$, so $-5(3) = -15$.

Now consider $5(-3)$. This means subtract $5$, three times. Thinking of subtraction as “taking away,” this means take away $5$ three times — but there’s nothing to take away, so we start by adding neutral pairs. Taking away $5$ positives three times (after adding enough neutral pairs) leaves $15$ negatives: $5(-3) = -15$. Likewise, $(-5)(-3)$ means take away $-5$ three times; after adding neutral pairs and taking away $-5$ three times, $15$ positives are left: $(-5)(-3) = 15$.

In both cases we started with $15$ neutral pairs; taking away $5$ positives three times left $-15$, while taking away $-5$ three times left $15$. So:

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.

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

(a) Different signs, so the product is negative: $-27$. (b) Same signs, so the product is positive: $10$. (c) Different signs, so the product is negative: $-32$. (d) Same signs, so the product is positive: $42$.

When we multiply a number by $1$, the result is the same number. What happens when we multiply by $-1$? Try $-1 \cdot 4 = -4$ (the opposite of $4$) and $-1(-3) = 3$ (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) Different signs, so the product is negative — and indeed $-7$ is the opposite of $7$. (b) Same signs, so the product is positive — and $11$ is the opposite of $-11$.

Divide integers

Division is the inverse operation of multiplication: $15 \div 3 = 5$ because $5 \cdot 3 = 15$. If we look at the multiplication facts above, we can figure out the rules for dividing integers:

Division of signed numbers follows the same rules as multiplication: when the signs are the same the quotient is positive, and when the signs are different the quotient is negative.

Remember that you can always check a division answer by multiplying.

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

(a) Different signs, so the quotient is negative: $-9$. (b) Same signs, so the quotient is positive: $25$.

Just as with multiplication, dividing a number by $1$ leaves it unchanged. What about dividing by $-1$? Try $8 \div (-1) = -8$ (the opposite of $8$) and $-9 \div (-1) = 9$ (the opposite of $-9$). Dividing a number by $-1$ gives its opposite.

Example. Divide: (a) $16 \div (-1)$, (b) $-20 \div (-1)$.

(a) The dividend $16$ is divided by $-1$: the signs are different, so the result is negative, $-16$. (b) The dividend $-20$ is divided by $-1$: the signs are the same, so the result is positive, $20$.

Simplify expressions with integers

Now let’s simplify expressions that use all four operations — addition, subtraction, multiplication, and division — with integers. Remember to follow the order of operations.

Example. Simplify $7(-2) + 4(-7) - 6$. Multiply first: $-14 + (-28) - 6$. Add: $-42-6$. Subtract: $-48$.

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

(a) The exponent is $4$ and the base is $-2$: raise $-2$ to the fourth power. In expanded form, $(-2)(-2)(-2)(-2) = 16$. (b) The exponent is $4$ and the base is $2$: raise $2$ to the fourth power, then take the opposite. In expanded form, $-(2 \cdot 2 \cdot 2 \cdot 2) = -16$.

Example. Simplify $12 - 3(9-12)$. Simplify inside the parentheses first: $12-3(-3)$. Multiply: $12-(-9)$. Subtract: $21$.

Example. Simplify $8(-9) \div (-2)^3$. Simplify the exponent first: $8(-9) \div (-8)$. Multiply: $-72 \div (-8)$. Divide: $9$.

Example. Simplify $-30 \div 2 + (-3)(-7)$. Divide first (left to right with multiplication): $-15 + (-3)(-7)$. Multiply: $-15+21$. Add: $6$.

Evaluate variable expressions with integers

Now we can evaluate expressions that include multiplication and division with integers — substitute the given numbers for the variables, then simplify.

Example. Evaluate $2x^2 - 3x + 8$ when $x = -4$. Substitute $-4$ for $x$: $2(-4)^2 - 3(-4) + 8$. Simplify the exponent: $2(16) - 3(-4) + 8$. Multiply: $32 - (-12) + 8$. Subtract: $44 + 8$. Add: $52$. (Keep in mind that we use parentheses to show the multiplication when substituting a negative value — without them, $2 \cdot -4^2 - 3 \cdot -4 + 8$ would be ambiguous.)

Example. Evaluate $3x + 4y - 6$ when $x = -1$ and $y = 2$. Substitute: $3(-1) + 4(2) - 6$. Multiply: $-3 + 8 - 6$. Simplify: $-1$.

Translate word phrases to algebraic expressions

All our prior work translating words to algebra transfers to phrases that include multiplying and dividing integers. 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: $(-2)(14)$. Simplify: $-28$.

Example. Translate to an algebraic expression and simplify if possible: the quotient of $-56$ and $-7$. Translate: $-56 \div (-7)$. Simplify: $8$.

Key terms

multiplication/division sign rule — for both multiplying and dividing signed numbers: same signs give a positive result; different signs give a negative result. multiplication and division by $-1$ — multiplying or dividing a number by $-1$ gives its opposite.

Practice

Multiply integers

Divide integers

Simplify expressions with integers

Evaluate variable expressions with integers

Translate word phrases to algebraic expressions

This section is adapted from Prealgebra 2e, Section 3.4: 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: condensed the counter-model derivation of the multiplication sign rule into prose and recreated the sign-rule summaries as callouts; omitted the Be Prepared quiz and media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block, with each multipart exercise expanded into one question per part.