Lesson 73: Multiplying and Dividing Positive and Negative Numbers

New Concept

Welcome to Saxon Math! This course builds your mathematical foundation by exploring the complete number system, including the rules for operating with negative numbers.

What’s next

This card introduces the big picture. Soon, we’ll dive into the specific rules and worked examples for multiplying and dividing positive and negative numbers.

Property

If the two numbers in a multiplication or division problem have the same sign, the answer is positive. If the two numbers have different signs, the answer is negative.

Examples

Same signs are positive: $(-6)(-2) = 12$ and $\frac{-12}{-4} = 3$.
Different signs are negative: $(+6)(-3) = -18$ and $\frac{25}{-5} = -5$.
Even with fractions the rule applies: $(-\frac{1}{2})(-\frac{1}{2}) = \frac{1}{4}$.

Explanation

Think of signs as friends (+) or enemies (-). An enemy of an enemy is a friend, so two negatives make a positive! It’s all about vibes: same signs are positive, and different signs are negative. This simple rule keeps your calculations on the right track, whether you're multiplying integers or dividing decimals.

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

An expression like $(-2)(-3)$ can be read as "the opposite of 2 times -3."

Examples

First, find the product of the positive and negative: $2(-3) = -6$.
Then, take the opposite of that result: $-(-6) = +6$. So, $(-2)(-3)=+6$.
This relationship also confirms our division rules: $\frac{+6}{-2} = -3$.

Explanation

Let's do a mental flip! We already know that $2 \times (-3)$ equals $-6$. So, when we see $(-2)(-3)$, it's asking for "the opposite" of that result. What's the opposite of -6? It's a positive 6! Multiplying by two negatives is like saying "I am not unhappy," which just means you're happy.