If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

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?

  • marcimus pink style avatar for user tessa.stowell336
    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)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Sam
    I understand why negative multiplied by a negative is a positive, but why is a negative divided by a negative a positive?
    (0 votes)
    Default Khan Academy avatar avatar for user
  • blobby blue style avatar for user Pranesh Prakashbabu
    i dont understand how - x - = P and p x p = P its still confusing....

    P= positive
    - = negative
    (4 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user Philip
      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
      (7 votes)
  • piceratops seed style avatar for user preetirathor01
    Why a negative times a negative is a positive. I can't able to understand this video clearly
    (4 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user MuntaqimC
      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

      Hopefully that made sense
      Sorry for the English lesson
      (6 votes)
  • spunky sam blue style avatar for user Samir Gunic
    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.
    (4 votes)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user Haylee Fowler
    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?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • leaf green style avatar for user jewel.hurtgen
    Why do we even have negative numbers? Is there any history behind them?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • piceratops sapling style avatar for user CynderStar
    What is the real life purpose of multiplying negative numbers?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Mr. James
      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.
      (2 votes)
  • female robot grace style avatar for user Hoo noze
    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?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • piceratops seed style avatar for user Michelle Campbell
    what is the product of a negative number by a positive number
    (2 votes)
    Default Khan Academy avatar avatar for user

Video transcript

Lets say you are an Ancient Philosopher who was building up mathematics who was building mathematics from the ground up And you already have a reasonable of what a negative number could or should represent and you know how to add and subtract negative numbers But now you are faced with a conundrum What happens when you multiply negative numbers? Either when you multiply a positive number times a negative number Or when you multiply two negative numbers So, for example You aren't quite sure what should happen if you were to multiply (and im just picking two numbers where one is positive and one is negative) What would happen if you were to multiply 5 times negative 3 You're not quite sure about this just yet You're also not quite sure what would happen if you multiply two negative numbers. So lets say negative two times negative 6 This is also unclear to you What you do know, because you are a mathematician, is however you define this or whatever this should be It should hopefully be consistant with all of the other properties of mathematics that you already know And preferably all of the other properties of multiplication That would make you feel comfortable that you are getting this right. and later we can think about other ways to get the intuition for what these might be allowed you to actually make sense but to make this make consistent with the rest of mathematics that you know, you go into a little bit of a thought experiment you say, well, what should five times three plus negative three equal well you already have a philosophy of adding negative numbers or adding positive numbers to negative numbers, you know negative three is the opposite of three, but you add three to negative three you're going to get zero, so this is going to be equal to five times zero based on how you already thought about adding a negative number to a positive, and anything times zero is going to be zero, so this expression right over here should be zero but you see, I want to multiply positive and negative numbers to be consistent with this distributive property so I should should be able to distribute this five and for math to be consistent, and math should be consistent, I should get the exact same answer, so let's distribute this five so we get five times three is going to write out as five times three let me write this multiplication sign, not this dot five times three, so i distributed there plus five times negative three i'll do that in yellow, five times negative three and this whole thing we just said should be equal to zero it should be equal to zero, well five times three those are two positive numbers, we should know what should should be, that is going to be fifteen now we get this thing, fifteen plus times whatever five times negative three is needs to be equal to zero in order to be consistent with all the other mathematics that we know, well what plus fifteen is going to be equal to zero, well the opposite of fifteen in order for this to be true, in order for this to be consistent with all the other mathematics we know this right over here needs to be equal to negative fifteen until you say five times negative three in order to be consistent with all other mathematics we know, needs to be equal to negative fifteen. That's also consistent with the intuition of adding negative three repeatedly five times, now look above above us slightly higher so you can see ideas of multiplying two negatives, but we can do the exact same product experiment. We want whatever this answer to be consistent with the rest of mathematics that we know so we can do the same product experiment. What would negative two times six plus negative six to be equal to. Well, six plus negative six is going to be zero. Negative two times zero, anything times zero, needs to be equal to zero, but then once again, we can distribute negative two times six so we get negative two times six, then plus negative two times negative six plus negative two times negative six, then once again all of this is going to be equal to zero, now based on the five experiment we just did, we said "well this needs to be equal to negative twelve" or we can view this as going to the six twice left direction on the number line which gets us to negative twelve or you could say repeatedly adding negative twos times six would also get you to negative twelve and now we also saw over here we want to multiply a positive and a negative we got the negative so this could be, you know, going to be equal to negative twelve so we have negative twelve plus whatever this business is going to have to be equal to zero (repeated) in order to be consistent with all the other mathematics that we know and so what plus negative twelve is going to equal to be zero Well, positive twelve plus negative twelve is going to equal to zero so this needs to be equal to positive twelve in order to be consistent with all the other mathematics we know so there we get the idea that this is going positive twelve. I'll leave you there and I'll see if I can make a few other videos that can also give you a conceptual understanding of why these are true