Properties of integer multiplication

Integer multiplication obeys the same five "laws" you have already met for whole numbers , plus a small surprise about $-1$.

Idea

Closure. The product of any two integers is an integer. (You cannot leave the set $\mathbb{Z}$ by multiplying.)

Commutativity. $a \times b = b \times a$ for all integers $a, b$. Example: $(-3) \times 5 = 5 \times (-3) = -15$.

Associativity. $(a \times b) \times c = a \times (b \times c)$. Example: $((-2) \times 3) \times 4 = (-6) \times 4 = -24$, same as $(-2) \times (3 \times 4) = (-2) \times 12 = -24$.

Identity. $a \times 1 = 1 \times a = a$. So $1$ is the multiplicative identity.

Zero. $a \times 0 = 0 \times a = 0$. Multiplying by $0$ wipes everything out.

Multiplication by $-1$. $a \times (-1) = -a$. So multiplying by $-1$ flips the sign of any number.

Distributive over addition. $a \times (b + c) = a \times b + a \times c$. This works with signed numbers too. Example: $(-3) \times (4 + 5) = -27 = (-3)(4) + (-3)(5) = -12 + (-15) = -27$. ✓

These properties let you rearrange computations into easier forms , group commutatively, factor distributively, etc.

Worked examples

Example 1. Compute $(-25) \times 7 \times (-4)$.

Re-order: $(-25) \times (-4) \times 7 = 100 \times 7 = 700$.

Example 2. Use distribution: $(-8) \times 99$.

$(-8) \times 99 = (-8) \times (100 - 1) = -800 - (-8) = -800 + 8 = -792$.

Example 3. Verify associative law for $(-2), 3, (-4)$.

$((-2) \times 3) \times (-4) = (-6) \times (-4) = 24$. $(-2) \times (3 \times (-4)) = (-2) \times (-12) = 24$. ✓

Example 4. Use $a \times (-1) = -a$ to simplify $(-7) \times (-1)$.

$(-7) \times (-1) = -(-7) = 7$.

Try it yourself

1. State the closure property of integer multiplication.
2. Verify commutativity for $(-5) \times 8$.
3. Compute $(-12) \times (-25) \times 4$ , use grouping.
4. Use distribution: $(-7) \times 102$.
5. Verify the distributive law for $a=3, b=-2, c=4$.
6. State what $a \times 0$ equals.
7. Compute $(-9) \times 1$ and $(-9) \times (-1)$.
8. Simplify $(-5)(3) + (-5)(7)$.

Activity

Pick any three integers (with at least one negative). Compute the product in three different orders and groupings. Confirm all give the same result.