Multiplying Two Negative Numbers
Multiplying two negative numbers produces a positive result, which can be understood by reading the expression (-2)(-3) as 'the opposite of 2 times (-3).' Since 2 times (-3) = -6, the opposite is +6. This Grade 7 math skill from Saxon Math, Course 2 provides the conceptual justification for why negative times negative is positive — a rule that confuses many students when stated without explanation — and builds the integer fluency needed for algebra, coordinate graphing, and all higher mathematics.
Key Concepts
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.
Common Questions
Why does negative times negative equal positive?
One way to understand it: (-2)(-3) means 'the opposite of 2 times (-3).' Since 2 times (-3) = -6, the opposite is +6. The double negative becomes a positive.
What is (-5) times (-4)?
Negative times negative equals positive: (-5) times (-4) = +20.
What is the sign rule for multiplication?
Same signs (positive times positive OR negative times negative) give a positive product. Different signs (positive times negative OR negative times positive) give a negative product.
Does this rule apply to more than two negative numbers?
Yes. Count the negatives: an even number of negative factors gives a positive result, an odd number gives a negative result. (-2)(-3)(-1) = 6 times (-1) = -6.
When do students learn about multiplying two negative numbers?
This is a Grade 7 topic. Saxon Math, Course 2 covers it in Chapter 7 building on the earlier work with adding and subtracting signed numbers.
How does multiplying two negatives connect to the number line?
Repeated addition of a negative on a number line moves left. Multiplying a negative by a negative reverses direction (the 'opposite' meaning) which means moving right — back into positive territory.
Why is it important to understand why the rule works, not just memorize it?
Understanding the reason means you will apply the rule correctly even in complex situations involving variables, expressions, and algebraic proofs.