Lesson 4: Multiplying Integers

Property

Just as $3 \times 5$ can be understood as $(5) + (5) + (5) = 15$, so $3 \times (-5)$ can be understood as $(-5) + (-5) + (-5) = -15$.
For a product like $(-3) \times 5$, one way is to recognize that it is the same as $5 \times (-3)$ (e.g. the commutative property).
Another is to understand $-3$ as the 'opposite' of 3, so $(-3) \times 5$ is the opposite of $3 \times 5$, which is $-15$.

Examples

• Calculate $6 \times (-4)$. This is equivalent to adding $-4$ six times: $(-4) + (-4) + (-4) + (-4) + (-4) + (-4) = -24$.
• To find $(-8) \times 5$, you can find the opposite of $8 \times 5$. Since $8 \times 5 = 40$, the opposite is $-40$. Therefore, $(-8) \times 5 = -40$.
• A submarine's depth increases by 40 feet per minute (represented as $-40$). After 4 minutes, its total change in depth is $4 \times (-40) = -160$ feet.

Explanation

When multiplying a positive and a negative integer, the result is always negative. You can think of it as repeated addition of a negative number, or as finding the opposite of what the positive product would be. The signs are different, so the answer is negative.

Property

When multiplying integers, the product of two integers with the same sign is positive, and the product of two integers with opposite signs is negative. This follows from the distributive property and leads to results such as $(-1) \cdot (-1) = 1$.

Examples

Property

When multiplying three or more integers, multiply from left to right and apply sign rules at each step. The final sign depends on the number of negative factors: an even number of negative factors gives a positive product, an odd number of negative factors gives a negative product.

Examples