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# Why a negative times a negative is a positive

Use the distributive property to understand the products of negative numbers. Created by Sal Khan.

## Want to join the conversation?

• When you multiply a negative by a negative you get a positive, because the two negative signs are cancelled out. Why don't the positive signs cancel out?
(1 vote)
• Imagine it on the number line. Negative sing always changes direction, and if we change direction 2 times the outcome will be positive.
• I understand why negative multiplied by a negative is a positive, but why is a negative divided by a negative a positive?
• also because the rules are the same, but don't ask me why about that, it's just math I guess
• i dont understand how - x - = P and p x p = P its still confusing....

P= positive
- = negative
• A positive times a positive is positive because the positive value is being increased a positive number of times. For example, 5×3=15, since 5+5+5=15.

Let's visualize this on number lines. The 0 is where the 0 is, the left is negatives (decrease), and the right is positive (increase). Each individual arrow represents a step, the next whole number. Arrows pointing toward the right are positive, and arrows pointing toward the left are negative. (I meant to have the zeros lined up so they would be on top of each other, but for some reason, what I typed on the editor and the later displayed format are somewhat different; sorry about that.)

0. 0

1. 0>>>>>

2. 0>>>>>>>>>>

3. 0>>>>>>>>>>>>>>>

A negative times a negative will equal a positive because what was originally negative has been reversed in direction. For example, -2×-4=8, where we take away 4 negative 2s.
There, so we have <<<<<<<< (negative 8). But since the 4 is negative, we are having them removed (or erased). If we start at -8, take away two more <s (though we still remain at a negative amount until the very end, the "strength of reduction" is getting "weaker".
0.<<<<<<<<0
1.<<<<<<<<0<< = <<<<<<0
2.<<<<<<<<0<<<< = <<<<0
3.<<<<<<<<0<<<<<< = <<0
4.<<<<<<<<0<<<<<<<< = 0
From the point that would be -8, we removed 4 negative 2s, which means we went in the right/positive direction by 8. Remember that a subtraction is basically the reduction/removal of something.

In a similar idea, if we had 9 minus 2, it will equal 7.
• <<0>>>>>>>>>
•    0>>>>>>>
The nine arrows toward the right say the line moved "forward" 9 steps. But the 2 arrows toward the left are saying that the line also moved "backwards" 2 steps. We finish with 7 if we only want the total (instead of subtraction).

By the same reasoning, this is also why - × P and P × - equal negatives.

For example, if we have -3×4, it will be
0.        0
1.            0<<<
2.            0<<<<<<
3.            0<<<<<<<<<
4.            0<<<<<<<<<<<<
5.<<<<<<<<<<<<0
At each step we were adding 3 more negative ones (which is why they were placed on the positive side first).

And if we have 5×-2
0.          0
1.      >>>>>0
2. >>>>>>>>>>0
3. <<<<<<<<<<0
We were to have two 5s, but because the 2 is negative, we are actually going toward the left. The negative 2 says we need to erase 5 >s twice, but we don't have any available, so end up with -10.

As a final example, let's start at another value not 0, but is positive.
7+(-2×3).
We already have a 7. At each step, the 7 arrows going toward the right are "pushed back" by two more arrows going toward the left.
0.        0>>>>>>> [we start at positive 7]
1.       0>>>>>>> << = 0>>>>>
2.     0>>>>>>> <<<< = 0>>>
3.   0>>>>>>> <<<<<< = 0>
[And end up with a total of positive 1]

Hope this explains most of the things, and helps you understand your problems
• Why a negative times a negative is a positive. I can't able to understand this video clearly
• Think of it like this,

If I tell you, "do not NOT do this" then you WOULD do it. If that's hard to understand then look at this,
(Do not) NOT do this.
So like Sal said in one of his videos, neg and neg cancel each other out. In our case, the neg is not.
Therefore,
Do not NOT do this
= Do do this because we cancelled out the nots
When someone tells you, "do do this", it basically means "do this"
Finally
Do not NOT do this
= do this

Sorry for the English lesson
• Is this in fact how multiplication with negative numbers is proven? Using the distributive property of multiplication and what we know about addition and subtraction of negative numbers? Is it not possible to use an area model to prove multiplication with negative numbers?

That would be great, if possible, because many people understand math better through visual representation. Because after all, we are visual beings, we use sight more than any other of our senses. Not all of us have a brain wired like a computer. Can anyone imagine what it would be like to study functions without graphing them on a coordinate plane? Geometry and algebra combined takes things to a whole new level of understanding.
• At , where did Sal get the +3? I though the problem was 5 x -3, could someone help me understand where the positive 3 came from?
• Sal took the three like that so that the problem in the parentheses would be equal to zero. I hope that helps!
• Why do we even have negative numbers? Is there any history behind them?
(1 vote)
• Hi, I think that people came up with negative numbers when discounts were made or when someone owed money.

I hope this helps and please correct me if I’m wrong!
• What is the real life purpose of multiplying negative numbers?
• That is a really good question.

There are lots of examples for a negative times a positive, such as withdrawals from a bank account:

5 x -\$20 = -\$100
5 withdrawals of 20 dollars would leave your balance at \$100 lower than it was before.

It is much harder to imagine a situation that involves a negative multiplied by another negative. The reason may be because it is much easier to simplify those problems into both numbers being positive, since it makes more sense to us. For the money problem, if you were to DEPOSIT rather than withdraw the \$20 5 times, you COULD write it as:

If the first number represents number of withdrawals, and the second represents how much your balance changes:
-5 x -\$20 = \$100

But, usually we would just make it simpler by changing the definition to:
number of deposits x amount of deposit

5 x \$20 = \$100

Because, as you say, this makes more sense to us. And it does. A lot of times, when working with equations, you have to multiply 2 negatives together (or divide). So, 2 negatives is really more theoretical than practical.