Mathematics » Introducing Integers » Multiply and Divide Integers

Dividing Integers

Dividing Integers

Division is the inverse operation of multiplication. So, \(15÷3=5\) because \(5·3=15\) 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.

\(\begin{array}{ccccc}5·3=15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15÷3=5\hfill & & & & -5\left(3\right)=-15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15÷3=-5\hfill \\ \left(-5\right)\left(-3\right)=15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15÷\left(-3\right)=-5\hfill & & & & 5\left(-3\right)=-15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15÷-3=5\hfill \end{array}\)

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 signsQuotient
•Two positives

 

•Two negatives

Positive

 

Positive

Different signsQuotient
•Positive & negative

 

•Negative & positive

Negative

 

Negative

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

Optional Video: Division of Integers – The Basics

Example

Divide each of the following:

  1. \(−27÷3\)
  2. \(\phantom{\rule{0.2em}{0ex}}-100÷\left(-4\right)\)

Solution

  
 \(–27÷3\)
Divide, noting that the signs are different and so the quotient is negative.\(–9\)
  
 \(–100÷\left(–4\right)\)
Divide, noting that the signs are the same and so the quotient is positive.\(25\)

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.

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

When we divide a number by, \(-1\) we get its opposite.

Division by -1

Dividing a number by \(-1\) gives its opposite.

\(a÷\left(-1\right)=\mathit{\text{−a}}\)

 

Example

Divide each of the following:

  1. \(\phantom{\rule{0.2em}{0ex}}16÷\left(-1\right)\phantom{\rule{1em}{0ex}}\)
  2. \(\phantom{\rule{0.2em}{0ex}}-20÷\left(-1\right)\)

Solution

  
 \(16÷\left(–1\right)\)
The dividend, 16, is being divided by –1.\(–16\)
Dividing a number by –1 gives its opposite. 
Notice that the signs were different, so the result was negative. 
  
 \(–20÷\left(–1\right)\)
The dividend, –20, is being divided by –1.\(20\)
Dividing a number by –1 gives its opposite. 

Notice that the signs were the same, so the quotient was positive.

Optional Video: Dividing Integers

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