Multiplying and dividing integers | Pre-algebra

The sign rules

To multiply or divide integers, work with the absolute values as usual, then decide the sign.

Notice that parentheses placed side by side mean multiplication: $$(-6)(-4)$$ is $(-6) \cdot (-4)$ .

A negative times a positive is easy to picture. If you spend $$4$$ dollars a day for $$6$$ days, your money changes by $6 \cdot (-4) = -24$ dollars.

Why is a negative times a negative positive?

Watch what happens when you keep multiplying by $$-4$$ and the first factor goes down by one each time:

3(-4) = -12 \qquad 2(-4) = -8 \qquad 1(-4) = -4 \qquad 0(-4) = 0

Each answer is $$4$$ more than the one before. If the pattern continues, $$(-1)(-4) = 4$$ and $$(-2)(-4) = 8$$ . The rule isn't arbitrary: it's the only one that keeps the pattern going.

More than two factors

Zero still behaves as before: any product with a factor of $$0$$ is $$0$$ , and dividing by $$0$$ is undefined.

Powers of negative numbers

Raising a negative number to a power is repeated multiplication, so the sign depends on how many negative factors there are.

(-3)^2 = 9 \qquad (-3)^3 = -27 \qquad (-3)^4 = 81 \qquad (-1)^{100} = 1 \qquad (-1)^{101} = -1

Order of operations with integers

PEMDAS works exactly the same way. The only extra job is keeping track of the signs.

Practice

Quick check

What is the sign of $$(-2)(-2)(-2)(-2)(-2)$$ ?

Quick check

Which expression equals $$-36$$ ?