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Multiplying positive & negative numbers

Learn some rules of thumb for multiplying positive and negative numbers. Created by Sal Khan.

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  • old spice man green style avatar for user Pj Harry W Tice
    what would -5 x 1 be? 1 x any other number is that number but i do not know if that works for negatives.
    (253 votes)
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  • purple pi teal style avatar for user Ilina
    what happens if you have a negative times a positive times a positive or a negative times a negative times a positive?
    (21 votes)
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    • area 52 green style avatar for user Gilbert
      Good question. A negative times a negative times a positive will be a positive⏤the first two negatives cancel each other out to make a positive, so when you multiply them by a positive you will be multiplying a positive times a positive. Likewise a negative times a positive times a positive⏤the first two (neg. times pos.) will make a negative so when you multiply that by another positive you will end up with a negative. Does this make sense?
      (70 votes)
  • starky ultimate style avatar for user Valen
    Would two negatives being multiplied the same thing as thinking there both positive? Example: -a*-b = a*b true or false?
    (20 votes)
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    • duskpin ultimate style avatar for user Lawler
      stan is absolutely right, if you multiply or divide any two negative numbers, the answer will be positive. so yes your example is true. if you still cant figure it out, try to find a lesson about multiplying and dividing positive and negative integers
      (11 votes)
  • hopper jumping style avatar for user 🍔ⒷⓇⒾⒶⓃ🍔
    Why is negative times a negative a positive while positive times a positive is not negative?
    (13 votes)
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  • starky sapling style avatar for user Dood1421
    What is going on in the comments?
    (11 votes)
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  • starky sapling style avatar for user Ima Person
    If a=2

    And the problem I'm solving for is -2^a

    Then is this problem supposed to look like -(2 x 2) or (-2) (-2)?
    I'm so confused.
    (11 votes)
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  • leaf blue style avatar for user crookave000
    how to solve binomeals
    (5 votes)
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    • piceratops tree style avatar for user ericjin090
      Binomials, you ask?
      Here's how to solve them:

      For example, lets have the straight-up maths question (x + 7) (x + 12).
      First, you multiply the two unknowns into x^2, then you multiply the first 'x' with the '+12', turning it into '12x'. After that, you multiply the '+7' with the unknown in the second set of brackets, making it '7x', and finally, you multiply the '+7' with the '+12', making 84.

      But we're not done yet. You now need to chuck them all into an equation. So, the x^2' comes first, then the '+12x', then the '7x', then the '84', so the equation is as follows:
      (x + 7) (x + 12) = x^2 + 12x + 7x + 84.
      Of course, you can simplify it (slightly). Combine the only set of like terms, and voilà! Your equation turns into x^2 +19x +84. Pretty simple, innit?

      Sadly, life has its ups and downs, or should I say positives and negatives? It's the same with maths, or binomials, in our case. Here's another question:

      (x + 4) (2x - 6)
      Oh! what's that? It's a wild minus sign! (I really need to get friends...). You may have noticed that in the previous question, I put the respective signs in front of their respective numbers, regardless of if they're positive or negative. That helps me (and possibly you) distinguish the sign of them and not get jumbled up between the mind-boggling positives and negatives. Now, onto the question. It's the same procedure as before, but now with an extra challenge - the negative sign, the step-by-step is as follows:
      2x x x = +2x^2
      -6 x x = -6x
      4 x 2x = +8x
      -6 x 4 = -24
      Now to get them into the equation: (x + 4) (2x - 6) = 2x^2 - 6x + 8x -24, which simplifies to 2x^2 + 2x - 24.

      You also might come across some specials ones like (x + 3)^2, which if they are all represented in unknows, equals to a^2 + ab (a x b) + ab + b^2, which simplifies to a^2 + 2ab + b^2. In the opposite case, however, like (x - 3)^2, it's literally the same thing but instead of the second addition symbol in the simplified version, it's a subtraction symbol instead. You might be wondering, 'why isn't the last addition symbol negative as well?' Well, allow me to explain. With binomials, at the end of the equation with the two numbers, you multiply them together. But if the binomial is a duplicate of each other, you just square the last number, and put it in the equation, and their sign will always be positive (unlike me) because two positives make a positive, while two negatives also make a positive. Keep in mind that this only works for 'squared' binomials.

      Here's another special example of a binomial - (x + 6) (x - 6). This marvellous binomial is nicknamed 'the difference of two squares', and once you memorise the formula, it's really a cake walk to do these types of questions. (a + b) (a - b) = a^2 - ab + ab - b^2. The two 'ab's cancel each other out, so you're left with the first number's square minus the second number's square.

      That's the end of my hopefully educational and fun lecture, and I hope anyone who reads this has a wonderful week. Please consider upvoting as this took me half an hour to make, and thank you for staying this long to read what I have to say about binomials! I live in Australia by the way, so some of my spellings may be different from the rest of you.
      (15 votes)
  • blobby green style avatar for user Aundrea Estep
    If you multiply six positive numbers,the products sign will be ?
    (5 votes)
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  • purple pi teal style avatar for user Luciana Lazarte
    negative x negative = positive
    positive x positive = positive
    positive x negative = negative
    negative x positive = negative
    (9 votes)
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  • aqualine seed style avatar for user tyler.cable14
    how would we multiply fractions
    (6 votes)
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    • starky tree style avatar for user Oushnik Sarkar
      To multiply fractions, you need to multiply the numerators with the numerators and the denominators with the denominators, for example, 3/5 × -15/6 = (3 × -15)/(5 × 6) = -45/30, then you need to divide the numerator by the denominator using the short division method, therefore, -45/30 = -3/2 [since it is an improper fraction, you convert it into a mixed fraction].
      Alternatively, you can directly divide the numerators with the denominators and then multiply the result, like, 3/5 × -15/6 = (3 × -15)/(5 × 6) = (1 × -3) × (1 × 2) = -3/2.
      (4 votes)

Video transcript

We know that if we were to multiply 2 times 3, that would give us positive 6. And since we're going to start thinking about negative numbers in this video, one way to think about it I had a positive number times another positive number, and that gave me a positive number. So if I have a positive times a positive, that will give me a positive number. Now, let's mix it up a little bit, introduce some negative numbers. So what happens if I had negative 2 times 3? Well, one way to think about, and we'll talk more about the intuition in this video and in future videos, is, well, you could view this as negative 2 repeatedly added three times. So this could be negative 2 plus negative 2 plus negative 2-- not negative 6-- plus negative 2, which would be equal to-- well, negative 2 plus negative 2 is negative 4 plus another negative 2 is negative 6. So this would be equal to negative 6. Or another way to think about it is if I had 2 times 3, I would get 6. But because one of these two numbers is negative, then my product is going to be negative. So if I multiply a negative times a positive, I'm going to get a negative. Now, what if we swap the order in which we multiply? So if we were to multiply 3 times negative 2. Well, it shouldn't matter. The order in which we multiply things shouldn't change the product. Whether we multiply 2 times 3, we'll get 6, or if we multiply 3 times 2, we'll get 6. And so we should have the same property here. 3 times negative 2 should give us the same result. It's going to be equal to negative 6. And once again, we say 3 times 2 would be 6. One of these two numbers is negative. And so our product is going to be negative. So we could write a positive times a negative is also going to be a negative. And both of these are just the same thing with the order in which we're multiplying switched around. But this is one of the two numbers are negative, exactly one. So one negative, one positive number is being multiplied. Then you will get a negative product. Now let's think about the third circumstance when both of the numbers are negative. I'll just switch colors for fun here. If I were to multiply negative 2 times negative 3-- and this might be the least intuitive for you of all. And here I'm just going to introduce you to the rule. And in future videos, we'll explore why this is and why this makes mathematics more all fit together. But this is going to be, you say, well, 2 times 3 would be 6, and I have a negative times a negative. And one way you can think about it is that the negatives cancel out. And so you will actually end up with a positive 6. I actually don't have to write a positive here, but I'll write it here just to reemphasize. This right over here is a positive 6. So we have another rule of thumb here. If I have a negative times a negative, the negatives are going to cancel out. And that's going to give me a positive number. Now, with these out of the way, let's just do a bunch of examples. I encourage you to try them out before I do them. Pause the video, try them out, and see if you get the same answer. So let's try negative 1 times negative 1. Well, 1 times 1 would be 1, and we have a negative times a negative. They cancel out. Negative times a negative give me a positive, so this is going to be positive 1. I could just write 1, or I could literally write a plus sign there to emphasize that this is a positive 1. What happens if I did negative 1 times 0? Now, this might say, wait, this doesn't really fit into any of these circumstances. 0 is neither positive nor negative. And here you just have to remember anything times 0 is going to be 0. So negative 1 times 0 is going to be 0. Or I could have said 0 times negative 783, that is also going to be 0. Let me do some interesting ones. What about-- I'll pick a new color-- 12 times negative 4? Well, once again, 12 times positive 4 would be 48. And we're in the circumstance where one of these two numbers right over here is negative, this one right over here. If exactly one of the two numbers is negative, then the product is going to be negative. We are in this circumstance right over here. We have one negative, so the product is negative. You could imagine this as repeatedly adding negative 4 twelve times. And so you would get to negative 48. Let's do another one. What is 7 times 3? Well, this is a bit of a trick. There are no negative numbers here. This is just going to be 7 times 3, positive 7 times positive 3, the first circumstance, which you already knew how to do before this video. This would just be equal to 21. Let's do one more. So if I were to say negative 5 times negative 10, well, once again, negative times a negative, the negatives cancel out. Then you're just left with a positive product. So it's going to be 5 times 10. It's going to be 50. The negative and the negatives cancel out. Your product is going to be positive. That's this situation right over there.