Math Core

If you lose $4 every day for 5 days, you've changed your money by $5 \times (-4)$ dollars. Multiplying and dividing with negative numbers comes down to one question: is the answer positive or negative? In this lesson you'll find the sign rules and see why they have to work the way they do.

A positive times a negative

Multiplication is repeated addition, so

3 \times (-4) = (-4) + (-4) + (-4) = -12

Three groups of $-4$ make $-12$ . Because you can multiply in either order, $(-4) \times 3 = -12$ too. A positive times a negative is negative.

A negative times a negative

What about $(-3) \times (-4)$ ? You can't make "negative 3 groups," so look at a pattern instead. Keep multiplying by $-4$ and make the first number smaller by $1$ each time:

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

Each time the first factor goes down by $1$ , the product goes up by $4$ . To keep the pattern going, $(-1) \times (-4)$ must be $4$ . A negative times a negative is positive.

Here's another way to see it. The number $-1$ times anything gives its opposite: $(-1) \times 4 = -4$ . So $(-1) \times (-4)$ is the opposite of $-4$ , which is $4$ .

Signs of products and quotients

Same signs give a positive answer: $\;(+)(+) = +\;$ and $\;(-)(-) = +$
Different signs give a negative answer: $\;(+)(-) = -\;$ and $\;(-)(+) = -$

Division follows the same rules, because division undoes multiplication.

Dividing integers

Every division fact comes from a multiplication fact. Since $(-3) \times (-4) = 12$ :

12 \div (-4) = -3 \qquad \text{and} \qquad 12 \div (-3) = -4

Since $3 \times (-4) = -12$ :

-12 \div (-4) = 3 \qquad \text{and} \qquad -12 \div 3 = -4

Same signs give a positive quotient, different signs give a negative one, just like multiplication.

One thing never changes: you can't divide by zero. An expression like $-8 \div 0$ is undefined, because no number times $0$ equals $-8$ .

Where the negative sign goes in a fraction

A fraction bar means division. Since $-12 \div 4$ , $12 \div (-4)$ and $-(12 \div 4)$ all equal $-3$ ,

\frac{-12}{4} = \frac{12}{-4} = -\frac{12}{4}

This works for any integers $p$ and $q$ with $q \ne 0$ : $-\dfrac{p}{q} = \dfrac{-p}{q} = \dfrac{p}{-q}$ . But $\dfrac{-12}{-4}$ is positive $3$ .

More than two factors

Pair the factors up. Each pair of negatives makes a positive, so you only need to count the negative signs.

Counting negatives

When you multiply several nonzero numbers, the product is positive if there is an even number of negative factors and negative if there is an odd number.

For example, $(-2)(-5)(-1) = 10 \times (-1) = -10$ . There are three negatives, an odd number, so the answer is negative.

Worked example: Products and quotients

$-7 \times 6$
$(-9)(-8)$
$-56 \div (-7)$
$\dfrac{45}{-9}$

Solutions.

Different signs, so negative: $-42$ .
Same signs, so positive: $72$ .
Same signs, so positive: $8$ .
Different signs, so negative: $-5$ .

Worked example: Several factors

Find $(-3)(4)(-2)(-5)$ .

There are three negative factors, an odd number, so the product is negative. The absolute values multiply to $3 \times 4 \times 2 \times 5 = 120$ . The answer is $-120$ .

Worked example: A real situation

A water tank loses $6$ gallons per hour. What is the change in the amount of water after $8$ hours? How many hours ago did the tank have $30$ more gallons than now?

A loss of $6$ gallons per hour is a rate of $-6$ . After $8$ hours the change is $8 \times (-6) = -48$ gallons.

Going back in time is the negative direction. The number of hours $h$ satisfies $h \times (-6) = 30$ , so $h = 30 \div (-6) = -5$ . That means $5$ hours ago the tank had $30$ more gallons.

Common mistake

The sign rules for multiplying are not the same as for adding. $-6 + (-3) = -9$ , but $(-6)(-3) = 18$ . Before you apply a rule, check which operation you're doing.

Tip

Decide the sign first, then multiply or divide the absolute values. It keeps the sign from getting lost.