Math Core

Multiplying and dividing signed numbers is actually easier than adding them. You work with the absolute values exactly as you always have, then decide the sign with one simple rule.

Multiplying as repeated jumps

Multiplication is repeated addition, and that still works with a negative number. $3 \times (-2)$ means three jumps of $-2$ , starting at $0$ :

3 × (−2) = −6

So a positive times a negative is negative. Because order does not matter in multiplication, $(-2) \times 3 = -6$ as well.

Why a negative times a negative is positive

Watch the pattern as the first factor goes down by $1$ each time:

product	value
$3 \times (-4)$	$-12$
$2 \times (-4)$	$-8$
$1 \times (-4)$	$-4$
$0 \times (-4)$	$0$
$-1 \times (-4)$	$4$
$-2 \times (-4)$	$8$

Each time the first factor drops by $1$ , the product goes up by $4$ . To keep the pattern going past zero, $(-1)(-4)$ must be $4$ and $(-2)(-4)$ must be $8$ . Multiplying by a negative flips the sign, so flipping twice brings you back to positive.

Signs when multiplying or dividing

Multiply or divide the absolute values, then choose the sign:

Same signs (both positive or both negative): the answer is positive.
Different signs: the answer is negative.

This rule is the same for multiplication and division, because division is multiplication by a reciprocal.

Examples: $(-5)(-6) = 30$ , $\;(-5)(6) = -30$ , $\;-42 \div (-7) = 6$ , $\;42 \div (-7) = -6$ .

More than two factors

When you multiply several numbers, count the negative factors.

An even number of negative factors gives a positive product (the negatives pair up).
An odd number of negative factors gives a negative product.

For example, $(-1)(-2)(-3)(4)$ has three negative factors, so it is negative: $-24$ .

Common mistake

Don't mix up the rules for adding and multiplying. $-3 + (-5) = -8$ (adding two negatives stays negative), but $(-3)(-5) = 15$ (multiplying two negatives is positive). Ask yourself which operation you are doing before you pick a sign.

Zero

Anything times $0$ is $0$ , and $0$ divided by any nonzero number is $0$ : $0 \div (-9) = 0$ . But division by $0$ is undefined. There is no number that times $0$ gives $-9$ , so $-9 \div 0$ has no answer.

Worked example: Two factors

Find each value.

$(-8)(7)$
$(-9)(-4)$
$-63 \div 9$
$\dfrac{-48}{-6}$

Solutions.

Different signs, so negative: $-56$ .
Same signs, so positive: $36$ .
Different signs, so negative: $-7$ .
Same signs, so positive: $8$ . (A fraction bar means divide.)

Worked example: Several factors

Find $(-2)(5)(-3)(-1)$ .

There are three negative factors, an odd number, so the product is negative. The absolute values multiply to $2 \cdot 5 \cdot 3 \cdot 1 = 30$ . The product is $-30$ .

Worked example: Working backward

What number times $-6$ equals $54$ ?

Rewrite as division: $54 \div (-6)$ . Different signs, so the answer is negative: $-9$ . Check: $(-9)(-6) = 54$ . ✓

Tip

A negative number to an even power is positive, and to an odd power is negative: $(-2)^4 = 16$ but $(-2)^3 = -8$ . It is just the counting rule, since $(-2)^4$ has four negative factors.