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How can you multiply two negative numbers?

  1. May 24, 2008 #1
    Ok, I know this seems very obvious to most people, but I have a problem understanding it.
    First of all: I have not been taught in any way how something objective can have a negative amount, so I do not even understand results that ends up being negative (-1 apple, for example) - you can only consider it as potential negative; that it will in the future disappear: debt.
    I can, however, accept it to be debt, and I understand the math/theory of it (who doesn't, you're taught this when you're 7 years old).

    Ok, so now to the final: I get that you can multiply a negative with a positive.
    You have 3$ in debt, and it occurs 3 times: -3$*3$=-9$
    Now to what I don't understand, when you multiply TWO negative numbers: -3*-5=15
    I know the math-part, which is that they clear each other out.
    But in "practice", how the hell would it work out?
    Not necessarily objective; our reality - universe, but some sort of a practical way to explain how two negative numbers can be multiplied, it doesn't make sense to me.
    You have -3, -3 times, wtf!?
    You can't have something -3 times! xD
    The lowest value I can think of that you can multiply with is 0.

    My question sort of is: how did mathematicians come with the conclusion of creating the rule that two negative numbers multiplied with each others is a positive number?
    How? Why?
    Was it just to create a way for us to keep on using this theoretical mathematics?

    I'm sorry if this seem to be a foolish question, though I truly don't consider this foolish, regardless of how simple it might seem for the close-minded..
    I find this of importance, to reveal and understand every inch of something, how can you otherwise be able to understand it, when you don't understand the source?

    Thanks for potential replies and answers.
  2. jcsd
  3. May 24, 2008 #2
    Like you said, you can justify the rule for negative times positive easily enough with debt.

    For negative times negative I like a geometrical comparision, where a negative one multiplication is a rotation of a vector in space by 180 degrees maintaining the length of the original vector (if you have just a number line, this becomes an inversion). With this comparision a positive times -1 becomes negative as with debt, and two -1 multiplications get you back to your original position in space, justifying the rule -1x-1 = 1.

    ps. I don't think it's a foolish question. I had the same question back in grade school and didn't like the uninspired answer "it's just the rule" by my teacher either.
  4. May 24, 2008 #3


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    What if not? I mean - what (-1)*(-1) should be equal to if it will be not equal 1?
  5. May 24, 2008 #4
    Didn't get it.
    I google'd vector and found en.wikipedia.org/wiki/Vector_graphics, where I read the first lines, and read the description beneath/above/beside the picture, and I didn't get it.
    It just takes forever looking up English words from wikipedia, because the description of the word is again used with foreign words that I have to lookup..
    Thanks anyways, for being polite, and what seem to be a good explanation.

    As I wrote: "Was it just to create a way for us to keep on using this theoretical mathematics?" - as in, they had no greater answer, so instead of abandoning the idea of mathematics, they instead used their greatest answer.
    I was sort of just kidding with it though; I'm quite sure there's a logical clearcut reason behind it.
    I find this to be sort of alike to Imaginary numbers..
    You can't squareroot a negative number, but in my opinion, can you neither square a negative number, even if squaring is defined as ^2 and based on general mathematics it would work out, but this is so theoretical for me, that I got problems accepting it and working with it, because how can you know, with advanced mathematics, what's right and not, when it's never clearcut, and seem slightly illogical, or even very illogical at times?

    Btw, is it wrong to write -3*(-3)?
  6. May 24, 2008 #5
    You could think of it using division. If you have a debt of -10, and you divide it evenly between 5 people, then each of those people will have a debt of -2.

    In symbols:

    -10/5 = -2

    Therefore you could ask the question: how many people does it take to split a -10 debt into a -2 debt for each person:

    -10/-2 = 5
  7. May 24, 2008 #6
    Division works (= I get division), the issue is multiplication.
    I guess multiplication isn't very clearcut after all..
    I guess it is, as I suspected, very theoretical.

    I just feel that I'm like.. Playing with fire, like working with what I don't understand - and I just don't like that..
    Knowing what I'm working with, has always been my triumph card :P
  8. May 24, 2008 #7
    How about this (fictional) example:

    Every time I go to the nightclub, I spend a hundred dollars. For me this is represented as c = -$100 (let c stand for "change in my money").

    This means that if I go to the nightclub 5 times, I will spend 500 dollars, which is represented as c = 5*(-$100) = -$500.

    What do we mean if I say I went to the nightclub -5 times? It means that I decided not to go out 5 of the times that I usually would have i.e. my friends call, but I tell them I won't be going out. Then the change in my money is c = -5*(-$100) = $500 which means that at the end of the month I will have 500 more dollars than usual.
  9. May 24, 2008 #8


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    Division and multiplication are in a way the same operation. If division works, calculate 1/(-2) first - it gives -0.5. Now you multiply -0.5*(-10) and result must be the same 5, so 5 = -10*(-0.5) = -10/-2
    Last edited by a moderator: Aug 13, 2013
  10. May 24, 2008 #9


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    The problem is that you didn't ask what you wanted to know; "Why can you multiply two negative numbers?" is a much different question than "How might I use the arithmetic of negative numbers in real world situations?"
  11. May 24, 2008 #10
    I disagree with that way of putting it or something..
    I would rather put it like this: -100$*0(5)
    Do you lose your money? No, so it doesn't happen, which means multiplied 0 times.
    Now how many times does it (not) happen? 5.
    The conclusion is that you don't earn nor spend anything, because we're focusing at what really is happening, and not taking what you could have done with your money in consideration.
    I would set up how much you have earned/spent as: +-100$*5=+-500$, which means you earn/spend 100$ depending on how you put it/what counts.

    I get your example, but unfortunately it doesn't fit with my logic, however I was thinking like this when I was younger (which is why I accepted it earlier).
    Last edited: May 24, 2008
  12. May 24, 2008 #11


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    Consider North the positive direction, and an increase in temperature to be positive as well.

    If every mile north I drive, it gets 1 degree colder and I drive 10 miles north then the temperature has changed
    10 miles * -1 degree/mile = -10 degrees
    10 * -1 = -10, so the temperature has dropped 10 degrees

    If I drive south instead, then this would be represented by -10 miles since I consider north the positive direction.
    -10 * -1 = 10, so the temperature is 10 degrees warmer
  13. May 24, 2008 #12
    Dark Fire: Many times mathematics is developed for one thing (such as counting things or keeping track of accounting), and you find that you logically develop some thing that isn't used in that setting (for instance, multiplying two negative numbers together). In this case, the idea that two negatives multiplied together gives a positive is the simplest way of defining multiplication for two negative numbers.

    Often times though, there are other applications other than the ones that you originally thought of in which these developments are used. For instance, vectors have already been said as an application, so I'll try to explain what they are and why multiplying two negative numbers together works here.

    A vector can be intuitively thought of as a direction along with a magnitude. For instance, think of a velocity (a speed and direction): You can be driving north at 30 miles per hour.

    Let's say that v is your velocity (for instance, north at 30 miles per hour). You can think of multiplying by a number x as going x times faster. For instance, 1v = v since going 1 times faster is the same as not changing your speed or direction at all. 2v then would be going twice as fast, or north at 60 miles per hour.

    You can also multiply this by a negative number. So you can go -1*v, which is north at -30 miles per hour, but notice that this just means that every hour you're going -30 miles in the north direction, which is to say, that you went 30 miles south. Therefore, multiplying by a negative number turns you around.

    Then notice that multiplying by a negative number turns you around twice, so negative times negative is positive.


    One thing to note though is that this is a very different example than accounting. In accounting, there is no reason (at least as far as I can think of) why you would multiply two negative numbers together. However, in this new example, you do have situations in which you'd multiply two negative numbers together.

    However, in both of these situations, the numbers that you are used to are applicable.

    Note that I am saying that the numbers are "applicable". You should try to convince yourself that numbers are not the same as money or directions or anything like that; numbers themselves are not real and they are never the situation that you are modeling; they are only applicable to situations. And sometimes, some numbers are more or less applicable. For instance, you said that you can't take the square root of a negative number, but this really isn't accurate; it's just that in many of the real world situations you've thought of where you apply numbers, there is no interpretation of complex numbers. However, there was a post earlier where someone asked for physical applications of complex numbers, and people were able to come up with several.
  14. May 24, 2008 #13


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    I think that's a great example.

    A geometrical method would be to graph y = x * -1; if the straight line continues as usual to -1, it will be at y = 1.

    As pattern completion:

    4 * 2 = 8
    4 * 1 = 4
    4 * 0 = 0
    4 * -1 = -4
    4 * -2 = -8
    . . .

    3 * -2 = -6
    2 * -2 = -4
    1 * -2 = -2
    0 * -2 = 0
    -1 * -2 = 2
  15. May 24, 2008 #14
    Ah, I missed this or I would have said that it was a fairly good example of the use of multiplying negative numbers together in accounting.

    This only works if you set your 0 at your usual amount of spending.

    Though, this often is the way you'd think of your spendings if you've set a budget.
  16. May 24, 2008 #15
    How about this:

    A particle is moving left at a speed of 5 on a number line (so its velocity is -5). It starts at the origin at t = 0 s.

    Where is the particle at t = -2 s?

    x = v * t = -5 * -2 = 10.

    So, 2 seconds before it's at the origin at t = 0, it is 10 to the right at t = -2.
  17. May 24, 2008 #16
  18. May 25, 2008 #17


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  19. May 25, 2008 #18
    I sort of think of it this way:

    [tex]\ (-3)(-3)[/tex]

    (The opposite of) 3 copies of (the opposite of) 3.

    If you think of it on a number line, (the opposite of) means go in the other direction away from zero.

    Let's also look at it this way. [tex] (3 * -3)(-1) = 9[/tex]

    3 copies of -3 is negative nine, the opposite of -9 is 9.

    I don't mean to imply that these number are somehow opposed or anything; they're just on opposite sides of the number line.

    I also want to mention that sometimes negative numbers do have a physical representation. Take the charges of the electron and proton. Both are non-zero, but when you add them together you get zero, no net electric charge.
    Last edited: May 25, 2008
  20. May 25, 2008 #19
    Consider this example:
    you own 3$, then you have +3$.
    If you ow someone 3$, then you have -3$, or, lets say you have a bill on 3$, then you must sooner or later pay these 3$, so that "variable" is worth -3$.
    But if you ow someone a bill of 3$, ie you must give someone else a bill on 3$. Then you have -(-bill)=-(-3$).But when you give this bill to the other guy, he needs to pay you 3$, so "owing a bill to someone else" will give you 3$. Thus, -(-3$)=3$
    :smile: :smile: :smile:
    Last edited: May 25, 2008
  21. May 25, 2008 #20
    Oh it works. But as stated earlier you have to say I usually make a net profit of 0 dollars per week and in my budget I make room for my nightclub visits. If one week, you become to busy to go night clubbing and you review your budget, you are going to notice a 500 dollar increase. Thus not spending 500 dollars for not going out five days a week will produce 500 dollars more than you would have had otherwise.
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