The sign rules

Set the four times-3 tables side by side: multiplier and multiplicand both positive; multiplier negative; multiplicand negative; and both negative. Read down them and four things stand out.

First, the MAGNITUDE of the product does not change with the signs. 4 × 3, (−4) × 3, 4 × (−3) and (−4) × (−3) all give 12 in size; only the sign differs.

Second, when BOTH are positive, the product is positive. Third, when BOTH are negative, the product is positive. Fourth, when one is positive and the other negative, the product is negative.

The tokens and patterns were there to help you understand; in the end, these are the rules to remember:

• (+) × (+) = (+)
• (−) × (−) = (+)
• (+) × (−) = (−)
• (−) × (+) = (−)

In one sentence: like signs give a positive, unlike signs give a negative, and the size is the same either way.

Two particular cases deserve checking. Consider 1 × a. For positive a we know it is a. Is it true for NEGATIVE a too? Use tokens: put a negatives into the bag just ONCE, and the bag contains a negatives. So 1 × a = a for every integer a, positive or negative.

Now consider −1 × a. When a is positive, the product has the same magnitude as a but is negative. When a is negative, the product has the same magnitude as a but is positive.

In each case the product is the additive inverse of a. So −1 × a = −a for all integers a. Multiplying by −1 is exactly the operation of flipping the sign — which is a useful thing to know by itself.