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# Multiplying two negative numbers

If 3(-8) can be 3 equal groups of -8, what does -3(-8) mean? What does it mean to multiply any two negative numbers? Let's use the distributive property and other properties of multiplication to find out.
When we multiply a positive number times a negative number, the product is the opposite of the product of the absolute values of the numbers. This means the result is always negative.
But what about when we multiply a negative number times a negative number? Let’s explore this idea using three different methods, starting with the distributive property.

## Multiplication with the distributive property: negative times negative

The distributive property works the same with negative numbers as with positive numbers and $0$. Let's use it to see what happens when we multiply two negative numbers, starting with the example $-7\left(-3\right)$.
Before we do, make a prediction.
What do you predict will be the value of $-7\left(-3\right)$?
This is an ungraded prediction, because we learn more when we make a guess before we get feedback.
Now let's use the zero-product property and the distributive property to reason about the product.
Fill each blank with a number to keep both sides of the equation equivalent.
$-7\left($
$\right)$
$=$$0$
$-7\left(-3+$
$\right)$
$=$$0$
$-7\left(-3\right)+\left(-7\right)\left(3\right)$$=$$0$
$-7\left(-3\right)+\left(-21\right)$$=$$0$
$+\left(-21\right)$
$=$$0$

## Multiplication by a negative as repeated subtraction from $0$‍

### Number lines

As a general trend, the symbol "$-$" changes the direction we move on a number line, whether we interpret it as a negative sign or a subtraction symbol.
Match the number lines to the expressions they represent.
Duplicate graphs will match to either equivalent expression.

Evaluate each expression.
$2\left(4\right)$$=$
$-2\left(4\right)$$=$
$-2\left(-4\right)$$=$

## Equal groups of objects

We represent multiplying by a positive number by adding equal groups of objects. We represent multiplying by a negative number by subtracting equal groups of objects.
So $-2\left(-5\right)$ is the value we have left after we take away $2$ groups of $-5$ objects. But how do we subtract groups of objects when we don't have any?
We can start with zero-pairs. The following diagram represents $0$ because there are $10$ positive integer chips and $10$ negative integer chips.
Now we can take away $2$ groups of $-5$.
Evaluate. $-2\left(-5\right)=$

## Conclusion

Now that we have explored multiplying a negative number times a negative number using three different methods, what conclusions can we draw?
Describe a general pattern for when we multiply two negative numbers.

## Want to join the conversation?

• Yes I indeed have a question. I get confused on diving the pairs and finding the answer. I need a breakdown on how to do it. Please and thankyou.
(1 vote)
• When you divide a negative number by another negative number the answer is positive. -54/-6 = -9

When you divide a positive number by another positive number the answer is positive. 54/6 = 9

When you divide a positive number by a negative number the answer is negative. 54/-6 = -9

When you divide a negative number by a positive number the answer is negative. -54/6 = -9

Hope this helps!
(pls vote up)
• In short: If two signs are the same, the result will be always positive. In contrast, if the two signs are different, the result will be always negative. This way you only have to multiply the numbers as you'd normally do (without taking in account the sign) and then prefix the sign to the result as same as i exposed before.
(1 vote)
• I understand this much more than his videos, this is only grade 7 math but kinda thanks for teacher Sal( did I get his name right)
• yeah his name is Sal Khan.
• Hi, My Name is Moses kaufman
(1 vote)
• yes it is Moses Kaufman
(1 vote)
• Normally, whenever two negative numbers decide to meet, the two negatives (- times -) decide to mash together to make a (+)
One negative stays same while the other turns side ways .
• - x - = +
• For the integer chips problem, you can easily add integer chips to nothing by simply adding a positive integer chips to 0. However, how can you remove negative integer chips from nothing?
Well, it says you have to take away 2 groups of negative 5 integer chips. Of course, you can't just have 5 negative integer chips so you have to derive those 5 negative integer chips from: NOTHING! What is a number that we can use to represent that you have nothing: 0! :). So you can just use 10 positive integer chips and 10 negative integer chips to represent zero and take out the 2 groups of negative 5 integer chips. This will leave you with 10 positive integer chips remaining. So positive 10 is your answer.
-2(-5)=10.