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Why're two negs mult pos?

  1. Sep 19, 2005 #1
    Ok, so in discussion today we were talking about combinatonics and counting. We were going over binomial coeffients, and i didn't like the explanation, and i related it to "its like, when you're learning to multiply, and they tell you two negatives equal a positive because when you put one minus sign on top of the other it makes a plus sign." But moving the lines around doesn't actually explain why two negatives equal a positive, its just a conveinient way of learning it so you can do the problems right.

    ...but then someone asked me after class, "why DO two negatives equal a positive?" and i was like "err... cause multiplication is the addition of groups and when you have a negative and you put it into negative groups... Hmmph! i don't actually know dammit!"

    So, i've been chewing it over, and i think i've made some sense thinking about it in terms of negative meaning direction, and multiplying meaning you increase the magnitude... but i dunno.... i want a better explanation. Please.
  2. jcsd
  3. Sep 19, 2005 #2
    If a and b are two negative integers they can be written as two positive integers c and d such that

    [itex] ab = (-1)c(-1)d = (-1)(-1)cd = 1cd = cd [/tex] where a and c are equal in magnitude, as are b and d.

    edit: itex is prettier.
  4. Sep 19, 2005 #3


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    Perhaps this may help :smile:
  5. Sep 19, 2005 #4
    but why is neg one times neg one positive? just an identity? is there a better conceptual way to think of it?

    how about some number theory or set theory or something that describes what happens to negative numbers?
  6. Sep 19, 2005 #5
    read the page he sent you.
  7. Sep 19, 2005 #6


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    It contains multiple examples to make it 'understandable' as well as some mathematical details I believe.
  8. Sep 19, 2005 #7
    ya, ok, i guess that works... i have a better question to ask anyway...
  9. Sep 19, 2005 #8
    -1x=opposite of x. -1(-x)=opposite of a negative, a positive.
  10. Sep 19, 2005 #9


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    The integers are designed so that they form what is known as a ring. A ring must have (among other things) an additive identity and an additive inverse. The additive identity is the number we call 0, which has the property that 0 + x = x + 0 = x. Next we define the additive inverse of 1 as that element which when added to 1, gives the sum 0, or if x + 1 = 0, then x is the additive inverse of 1. We use the symbol "-1" to represent this number. Next we make use of the distributive property and the definition 0 = 1 + (-1) to write
    0 = 0*(-1) = {1 + (-1)}*(-1) = 1*(-1) + (-1)*(-1)
    Since 1 is the multiplicative identity, we know that 1*x = x*1 = x, for all x, and hence, 1*(-1) = -1. So we have 0 = -1 + (-1)*(-1). But we know that 0 = -1 + 1, therefore, from the uniqueness of addition (-1)*(-1) = 1.

    Note : The operation of subtraction is merely a shorthand for adding a negative number.
  11. Sep 19, 2005 #10
    Or the other way around from the page is to accept the field axioms, then prove yourself that (-1)(-1)=1 using them. You just need to use more precise terms of what a negative refers to and what rules multiplication follows.
    Last edited: Sep 19, 2005
  12. Sep 19, 2005 #11
    We debated this a while back on this thread https://www.physicsforums.com/showthread.php?t=82997.

    My approch was to use English grammer exmaples to explain why -- = +. Basicly, any negative statment such as "not" or "didnt" counts as a -1.

    [Quote = eNathan]Well that kinda gets into the very logic of what a negative is. Think of it as logic "not" operations. Think about using these in sentences and you will get the idea.

    -1*-1=1 ; No + No = Yes
    1*1=1 ; Yes + Yes = Yes
    -1*1=-1 ; No + Yes = No
    1*-1=-1 ; Yes + No = No

    For instance, if I said...
    I did not not go to the store. That really means, you DID go to the store. -1 * -1 = 1 ; No + No = Yes ... and so forth.

    You kinda see how there is a relationship between mathematics and grammer? lol Im not sure if this is an official way to present it, but I just though it up a while back.[/quote]
  13. Sep 19, 2005 #12


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    You've almost got the geometric picture, methinks. It's a nice one, and is important to know when you deal with the complex numbers.

    In the real case, multiplication by a positive means you leave the direction unchanged, and multiplication by a negative means that you flip the direction. So, if you have a negative, and multiply by another negative, the result is a positive.

    In the complex case, numbers can be seen as having a magnitude and a direction in the complex plane. Multiplying by a complex number that lies at an angle θ means rotation by θ. (Angles are measured counterclockwise from the positive x axis)
  14. Sep 20, 2005 #13

    ... multiplication is nothing more than a shorthand for repeated additions and in that light, there's no way to justify -1 * -1 = 1. Despite the convoluted logic the conventional explanation for multiplying two negative numbers amounts to this

    for negatives
    -1^2 = -1 + 2
    -2^2 = -2 + 6
    -3^2 = -3 + 12
    -4^2 = -4 + 20

    whereas for positives
    1^2 = 1 + 0
    2^2 = 2 + 2
    3^2 = 3 + 6
    4^2 = 4 + 12

    So we're supposed to buy the explanation that, through some magical process, the interval between a negative and it's square is always larger then the interval between the positive of the same number and it's square?

    Doesn't smell right.
  15. Sep 20, 2005 #14
    In the sentence examples you stated, the logic follows, but your logic table could also equal this:

    -1*-1=1 ; No + No = Emphatic No
    1*1=1 ; Yes + Yes = Emphatic Yes
    -1*1=-1 ; No + Yes = Terminating Yes
    1*-1=-1 ; Yes + No = Terminating No
  16. Sep 20, 2005 #15


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    Well, what is the better question?? We're waiting! :smile:
  17. Sep 20, 2005 #16
    That loses meaning when one looks beyond integers. What does it mean to repeat addition 1/2 times, or Pi times ? i times ?
    If you just want to stay within the integers, you first have to define what a negative integer is. In that case, it is the number that when added to the corresponding positive integer, returns 0 (called the additive inverse). Multiplication defined your way makes sense for positive multiples. A negative multiple would have to be translated to a positive multiple multiplied by -1, in which you would apply your repeated addition definition, then have the negative sign applied. A negative multiplied by a negative gives us the following problem then: (-a)*(-b) = (-1)*(-1)*a*b, so we now have the problem of figuring out (-1)*(-1). Now, we know from our definition that 1 + (-1) = 0, so we have (-1) + (-1)*(-1) = 0. It is easy to prove that each integer has only one additive inverse (uniqueness). Thus, (-1)*(-1) must be the additive inverse of -1, which is 1.
    Last edited: Sep 20, 2005
  18. Sep 21, 2005 #17
    That's because you are not following the rules!

    It would be like me saying,

    1 + 1 = 2 + 0


    2 + 2 = 5 - 1

    And then declaring that I have shown something smells fishy.

    In reality, however, you have established a false opposition. It is not as if
    -1^2 = -1 + 2
    in contrast to
    1^2 = 1 + 0

    To prove it, I could just as easily say:
    -1^2 = 1 + 0
    1^2 = -1 + 2

    The fact is, -1^2 = 1^2

    Why? Because multiplication by -1 is like 180 degree rotation (flipping) -- note that 90 degree rotation would be multiplication by i. Someone has already pointed us in this direction.
  19. Sep 21, 2005 #18


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    For k>0
    neg interval = (-k)2 - (-k) = k2 + k
    pos interval = k2 - k

    So, neg interval = pos interval + 2k, and since k>0, neg interval > pos interval.

    See ? No magic ! No bad smell !
  20. Sep 21, 2005 #19


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    I feel the need to comment that -1^2 = -1. Exponents before multiplication.
    :tongue2: (Of course, I know both of you meant (-1)^2)
  21. Sep 21, 2005 #20


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    Dearly Missed

    a) False
    b) Correct, your assertion in a) stinks.
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