Multiplying and Dividing Integers

Once negative numbers enter the picture, every calculation has two parts: the digits and the sign. The digits work exactly as they always did. The sign follows one short rule that covers both multiplication and division.

Concept The rule of signs

Same signs give a positive answer. Different signs give a negative answer.

Four combinations, two outcomes. Matching signs cancel out to a positive; mismatched signs leave a negative. The same table serves for division.

Concept A number with no sign

When no sign is written, the number is positive. Writing $6$ is the same as writing $+6$ .

This matters when you apply the rule: in $-6 \times 3$ the second number counts as positive, so the signs differ and the answer is negative.

Example Multiplication, all four cases

6 \times 3 = 18

— both positive, so positive.

(-6) \times (-3) = 18

— both negative, so positive.

6 \times (-3) = -18

— signs differ, so negative.

(-6) \times 3 = -18

— signs differ, so negative.

In every line the digits give 18. Only the sign changes, and it changes according to the rule.

Example Division, all four cases

12 \div 4 = 3

— both positive, so positive.

(-12) \div (-4) = 3

— both negative, so positive.

(-12) \div 4 = -3

— signs differ, so negative.

12 \div (-4) = -3

— signs differ, so negative.

Division needs no separate rule. The pattern is identical to multiplication.

Example Working one out step by step

Work out $(-5) \times (-4)$ .

Handle the digits first:

5 \times 4 = 20

Now the signs: both are negative, so they match.

Matching signs give a positive answer.

⟹ 20

Separating the two decisions — digits, then sign — keeps things clear and stops one from interfering with the other.

Example A mixed set

(-20) \div 5 = -4

— different signs.

24 \div (-6) = -4

— different signs.

(-7) \times 3 = -21

— different signs.

Whenever exactly one negative is present, the answer is negative. Two negatives, or none at all, give a positive.

Note Mistakes to avoid

Thinking two negatives give a negative — they give a positive.

Forgetting that an unsigned number counts as positive.

Applying these sign rules to addition and subtraction, which behave differently.

Losing the minus sign partway through the working.

Deciding the sign before the digits and then muddling the two.

Summary

Same signs give a positive result; different signs give a negative one.
The rule is identical for multiplication and division.
A negative times a negative is positive.
A number written without a sign is positive.
Work out the digits first, then decide the sign.