Multiplying and Dividing Positive and Negative Numbers

This lesson covers how to multiply and divide positive and negative numbers. The key idea is a simple rule about signs: when the signs match the answer is positive, and when they differ the answer is negative.

The four sign rules for multiplication

There are four cases to remember:

$(+) \times (+) = (+)$ $(-) \times (-) = (+)$ $(+) \times (-) = (-)$ $(-) \times (+) = (-)$

The pattern: if both signs are the same, the product is positive. If the signs are different, the product is negative. Work out the sign first, then multiply the numbers.

Worked multiplication examples

Same signs give positive.

$(+8) \times (+1) = +8$ $(-12) \times (-8) = +96$

Different signs give negative.

$(-3) \times (+4) = -12$ $(+5) \times (-8) = -40$

Multiplying by zero. Any number times zero is zero:

$0 \times 5 = 0$

Chains of three numbers

For a string of numbers, settle the overall sign first, then multiply the sizes.

$(+8) \times (+1) \times (-3) = -24$

Here two positives and one negative give a negative result, and $8 \times 1 \times 3 = 24$.

$(-5) \times (-8) \times (-3) = -120$

Three negatives give a negative, and $5 \times 8 \times 3 = 120$.

$(-10) \times (-2) \times (+5) = +100$

Two negatives and a positive give a positive, and $10 \times 2 \times 5 = 100$.

$(-5) \times (-2) \times 0 = 0$

No matter the other signs, multiplying by zero gives zero.

$1 \times 5 \times (-8) = -40$

A number with no sign is treated as positive, so this is positive times positive times negative, which is negative, and $1 \times 5 \times 8 = 40$.

Sign rules for division

Division follows exactly the same sign rules:

$(+) \div (+) = (+)$ $(-) \div (-) = (+)$ $(+) \div (-) = (-)$ $(-) \div (+) = (-)$

Same signs give a positive quotient, different signs give a negative one.

Worked division examples

$(+8) \div (+2) = +4$ $(-16) \div (-8) = +2$ $(+10) \div (+1) = +10$ $(-15) \div (+3) = -5$ $(+25) \div (-5) = -5$

Zero and division by zero

Zero divided by any number is zero:

$0 \div (-6) = 0$

But dividing by zero is not defined:

$(+6) \div 0 = \text{undefined}$ $0 \div 0 = \text{undefined}$

More practice questions

$(+15) \div (+3) = +5$ $(-100) \div (-10) = +10$ $(+6) \div (-3) = -2$ $(-5) \div (+5) = -1$ $(+6) \div (+6) = +1$ $0 \div (-8) = 0$ $100 \div 0 = \text{undefined}$

Key takeaways

• For both multiplication and division, same signs give a positive result and different signs give a negative result.
• For a chain of numbers, decide the overall sign first, then multiply or divide the sizes.
• Zero multiplied or divided by any number is zero, but dividing by zero is undefined.