Lesson 5-4: Multiply Integers and Rational Numbers

Property

Multiplication is a shorthand for repeated addition. For example, $2(-3)$ means $(-3) + (-3)$.

Examples

To find $3(-2)$, we can show it on a number line as three jumps of -2, landing on -6.
This gives us the multiplication fact: $3(-2) = -6$.
This also helps us understand division: $\frac{-6}{3} = -2$.

Explanation

Imagine you're on a number line at zero. To solve $2(-3)$, you simply take two big steps in the negative direction, with each step being 3 units long. Where do you land? At -6! This shows why multiplying a positive number by a negative number always sends you into negative territory.

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

Multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as $(-1) \cdot (-1) = 1$ and the rules for multiplying signed numbers. The product of two numbers with the same sign is positive, and if the numbers have opposite signs, it is negative.

Examples

• For same signs, like $(-6) \times (-7)$, both numbers are negative, so the result is positive. $6 \times 7 = 42$, so $(-6) \times (-7) = 42$.
• For different signs, like $9 \times (-5)$, one is positive and one is negative, so the result is negative. $9 \times 5 = 45$, so $9 \times (-5) = -45$.
• For a series of multiplications like $(-3) \times (2) \times (-5)$, work step-by-step. First, $(-3) \times 2 = -6$. Then, $(-6) \times (-5) = 30$.

Explanation

A simple rule for signs: if the signs of the two numbers are the same, the product is positive. If the signs are different, the product is negative. Think of a 'double negative' becoming a positive.