Multiplying and Dividing Negatives

Multiplying and dividing with negative numbers comes down to a single rule about signs. Once you know what sign the answer should have, you work out the size exactly as you always have.

Same signs give a positive answer. Different signs give a negative answer. That holds for multiplication and division alike:

(+)(+) = +\qquad (-)(-) = +

(+)(-) = -\qquad (-)(+) = -

So (-3) \times 4 = -12 (different signs), while (-3) \times (-4) = 12 (same signs). Division behaves identically — (-12) \div (-4) = 3 and (-12) \div 4 = -3.

When you multiply or divide:

same signs give a positive answer;
different signs give a negative answer;
two negatives cancel — they make a positive;
counting the negative factors: an even number of them gives a positive result, an odd number gives a negative result.

The sign grid

Read off the sign of either operand along an edge; the cell where they meet is the sign of the answer. The diagonal of matching signs is positive; the off-diagonal of mixed signs is negative.