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### Course: 7th grade>Unit 5

Lesson 1: Multiply with negatives

# 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?
• Good question. The trouble might be that the word "cancel" is misleading. Perhaps a better way to think about it is that multiplying by a negative number switches the sign of the number being multiplied. While multiplying by a positive does not.
• Why do we "distribute" the 5?
• That is one of the principle properties of multiplication: it is distributive.
• What will 5(-3-(-3)) be?
• What I always do is count the amount of lines. Even means plus, uneven means minus.
- = 1 line so minus
+ = 2 lines so plus
+- = 3 lines so minus
++ = 4 lines so plus

So 5 - 3 -- 3 =
5 - 3 + 3 =
The -3 and +3 cancel eachother out. So your answer is 5.

Do note that I assume you left out the + after the 5, otherwise it's 5 * 0 = 0
• I understand why negative multiplied by a negative is a positive, but why is a negative divided by a negative a positive?
• If a positive thing happens to a positive person, then that is positive(good), like a good scientist winning the Nobel Prize.

If a negative thing happens to a positive person, like someone getting robbed that's negative (bad).

If a positive thing happens to a negative person, that's negative (bad) like a robber being decreed innocent.

If a negative thing happens to a negative person, that's positive(good) like a robber being caught.

Apply this into division and multiplication,

Positive/Positive= Positive

Negative/Positive=Negative

Positive/Negative=Negative

Negative/Negative=Positive

Same for multiplication.

Hope this helps!
• Two wrongs do make a right.
• very true
(1 vote)
• At , Sal uses a dot to represent multiplying in the problem of "5 times 0 = 0," when right before that, he used an actual multiplication sign in the problem "5 times -3 = ?"

Why is that? Shouldn't he use the same sign always for multiplication? It is kind of confusing to me.

Any answer would be greatly appreciated!

Thank you!
Gleam of Dawn
• There are several ways to represent multiplication. The dot and the x mean the same thing, but in problems that involve a variable, especially if the the variable is x it can get confusing to use the x multiplication symbol, so the dot or parenthesis are typically used to avoid confusion with the variable x.
• What is the Distributive Property?
• The distributive property is the property of mathematics that states that a number multiplied by something in parenthesis can be distributed to everything between those parenthesis.

EXAMPLE:
3(x + 4) is the same as 3x + 12 when you distribute the 3 to the x and the 4.
• so basically their isn't a logical reason? its just because, it is just because
• Was it only me who heard the bug?
• no not only you
• 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