Multiplying integers

The most important rules in this whole chapter:

• Positive × Positive = Positive.
• Positive × Negative = Negative.
• Negative × Positive = Negative.
• Negative × Negative = Positive.

Idea

The "same signs $\to$ positive, different signs $\to$ negative" rule looks magic. Here's why it holds.

Pattern argument. Watch the sequence $3 \times 4 = 12, 3 \times 3 = 9, 3 \times 2 = 6, 3 \times 1 = 3, 3 \times 0 = 0$. Each step (decreasing the second factor by $1$) decreases the product by $3$. Continue: $3 \times (-1) = -3, 3 \times (-2) = -6$. Pattern confirms positive × negative = negative.

Now: $(-3) \times 4 = -12$ (by commutativity). And the sequence $(-3) \times 4 = -12, (-3) \times 3 = -9, \dots, (-3) \times 0 = 0$. Each step increases the product by $3$. Continue: $(-3) \times (-1) = 3, (-3) \times (-2) = 6, (-3) \times (-3) = 9$. So negative × negative = positive. ✓

Computational steps.

1. Multiply the magnitudes (ignore signs).
2. Decide the sign of the result by the rule above.
3. Combine.

For a product of many integers, count the negatives: if the count is even, result is positive; if odd, negative.

Worked examples

Example 1. Compute $(-6) \times 5$.

Different signs $\Rightarrow$ negative. $6 \times 5 = 30$. Result: $-30$.

Example 2. Compute $(-7) \times (-8)$.

Same signs (both negative) $\Rightarrow$ positive. $7 \times 8 = 56$. Result: $+56$.

Example 3. Compute $(-2) \times 3 \times (-4)$.

Three integers: two negatives (even count) $\Rightarrow$ positive sign. Magnitudes: $2 \times 3 \times 4 = 24$. Result: $+24$.

Example 4. Compute $(-1) \times (-1) \times (-1)$.

Three negatives (odd count) $\Rightarrow$ negative. $1 \times 1 \times 1 = 1$. Result: $-1$. (Equivalent to $(-1)^3 = -1$.)

Try it yourself

1. Compute $(-4) \times 7$.
2. Compute $(-6) \times (-9)$.
3. Compute $8 \times (-12)$.
4. Compute $(-1) \times 100$.
5. Compute $(-3) \times (-3) \times (-3)$.
6. Compute $5 \times (-4) \times (-2)$.
7. Find the sign of $(-2)^{10}$ without computing.
8. A diver descends $3$ m per second for $7$ seconds. Express the descent using a signed integer.

Activity

Make a "sign multiplication" wheel. Inside circle: two signs ($+, -$). Outside: result sign ($+, -$). Spin and pair: same gives plus, different gives minus.