Dividing Integers

Learning Outcomes

• Divide integers with the same sign
• Divide integers with different signs
• Divide integers by -1

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

$\begin{array}{ccccc}5\cdot 3=15\text{ so }15\div 3=5\hfill & & & & -5\left(3\right)=-15\text{ so }-15\div 3=-5\hfill \\ \left(-5\right)\left(-3\right)=15\text{ so }15\div \left(-3\right)=-5\hfill & & & & 5\left(-3\right)=-15\text{ so }-15\div -3=5\hfill \end{array}$

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.

Division of Signed Numbers

The sign of the quotient of two numbers depends on their signs.

Same signs Quotient
•Two positives

•Two negatives

Positive

Positive

Different signs Quotient
•Positive & negative

•Negative & positive

Negative

Negative

Remember, you can always check the answer to a division problem by multiplying.

example

Divide each of the following:

1. $-27\div 3$
2. $-100\div \left(-4\right)$

Solution

 1. $-27\div 3$ Divide, noting that the signs are different and so the quotient is negative. $-9$
 2. $-100\div \left(-4\right)$ Divide, noting that the signs are the same and so the quotient is positive. $25$

try it

Just as we saw with multiplication, when we divide a number by $1$, the result is the same number. What happens when we divide a number by $-1?$ Let’s divide a positive number and then a negative number by $-1$ to see what we get.

$\begin{array}{cccc}8\div \left(-1\right)\hfill & & & -9\div \left(-1\right)\hfill \\ -8\hfill & & & 9\hfill \\ \hfill \text{-8 is the opposite of 8}\hfill & & & \hfill \text{9 is the opposite of -9}\hfill \end{array}$

When we divide a number by $-1$ we get its opposite.

Division by $-1$

Dividing a number by $-1$ gives its opposite.

$a\div \left(-1\right)=-a$

example

Divide each of the following:

1. $16\div \left(-1\right)$
2. $-20\div \left(-1\right)$

try it

Watch the following video for more examples of how to divide integers with the same and different signs.