Multiplying and dividing integers

Multiplying and dividing integers or whole numbers are done the same way as multiplying and dividing natural numbers, with one exception – we have to watch out for the minuses.

As we already know, if we’re multiplying or dividing two positive numbers, we’ll get a positive number as a result. Also, if we’re multiplying or dividing two negative numbers, we’ll get a positive number as well. But, if we’re multiplying or dividing a negative and a positive number, we’ll get a negative number as a result.

$”+”\cdot “+” = “+”$

$”+”\cdot “-” = “-“$

$”-“\cdot “+” = “-“$

$”-“\cdot “-” = “+”$

What does that mean for doing calculations?

Well, when it comes to multiplying and dividing positive integers, nothing changes. There is no difference between natural numbers and positive integers, so we can multiply and divide them the same way.

One positive and one negative integer

$-2 \cdot 4 = ?$ so $2 \cdot 4 = 8$, but if we multiply a positive by a negative number or vice versa, the result will be a negative number. This means that $-2 \cdot 4= – 8$.

Also, if we divide a positive by a negative number or a negative with a positive number, we’ll get a negative number as a result:

$-8 : 4 = – \frac{8}{4} = -2$

Every division can also be represented as a fraction. So, if you decide to keep it in the form of a fraction, you can put the minus anywhere you want:

$\frac{-1}{2} = – \frac{1}{2} = \frac{1}{-2}$

Although all of these ways are correct, the preferred way to deal with the minus is to leave it in front of the whole fraction.

Two negative numbers

When multiplying or dividing two negative numbers, we’ll get a positive number:

$-8 \cdot (-4) = 32$

$\frac{-8}{-4} = 2$

If this confuses you a bit, remember rule number one from the lesson on the addition and subtraction of integers. A minus in front of a number changes the sign of the number, so two of them cancel each other out. Think of it as doubling back – you basically end up where you started.

More than two negative integers

What if you don’t have only one or two negative numbers to multiply or divide, and don’t want to take it step by step? How will you keep track of the minuses? Here is a simple trick you can do to make things easier on yourself: count them. If their number is odd, the result will be negative, if their number is even, the result will be positive.

$(-(-(-(-2)))) \cdot (-(-5)) = ?$

For example, the number of minuses is this task is $6$, which means that the final result will be positive and the result of the multiplication is $10$.

$– \frac{-8}{-4} = ?$

What about this example? Well, the number of minuses here is number $3$, which is an odd number, and that means that the result will be negative and the final result of the division is number $-2$.

Just take special care when you see a negative number multiplying an expression that’s in parentheses. Remember that, in that case, every number inside the parentheses will change its sign!

Examples of solving complicated expressions with negative numbers

When a complicated expression is given, the first thing you do is to reduce the number of parenthesis. If you can, solve the expressions in parenthesis first. Then you solve those numbers that are bounded by a multiplication or division then you can go on solving numbers bounded by addition or subtraction. After that you multiply or divide what you can. When you have only whole numbers you can add or subtract them.

Example:

$[1 + (5 \cdot 3 – 4) – (25 + 3) ] + 5 (4 + 8) – 5 \cdot 2$

$= [1 + (15 – 4) – 28] + 5 \cdot 12 – 10$

$= 1 + 11 – 28 + 60 – 10 = 34$

Then there are unknowns which can also complicate things.

Example: Reduce the expression

$$-8 \cdot (a + b) + a + 2 \cdot (4 + 3b) – (b + a)$$

$$= -8a – 8b + a + 8 + 6b – b – a$$

$$= -8a-3b+8$$

Well, that’s that about the multiplication and division of integers! If you wish to practice a bit to really master this subject, we’ve prepared some worksheets for you and you can find them below.