why does a negative times a negative equal a positive?

I was reading a book on abstract algebra and came across the phrase "Minus times minus is plus. The reason for this we need not discuss"

is it that complicated?

Comments

  • Since you are reading up on abstract algebra, I'm assuming you know what the ring axioms are. Let a be an element of any ring. Then, by definition, -a is the additive inverse of a (the unique element with the property that a + (-a) = -a + a = 0). Using this and the ring axioms, we can prove why a product of negatives is a positive, but first, we need to prove two lemmas:

    Lemma 1: For any element a in a ring R, 0a = a0 = 0

    Proof: Let 0 be the additive identity of R and a be some element in R. Then,

    0 + 0a = 0a = (0 + 0)a = 0a + 0a

    Canceling a 0a from both sides (which we can do because any ring forms a group under it's addition operation and all groups have the cancellation property), we have 0 = 0a. A similar proof shows that 0 = a0.

    Lemma 2: For any elements a and b of a ring R, a(-b) = -ab = -(ab).

    Proof: Let a and b be elements of a ring R. Then,

    a(-b) + ab = a(-b + b) = a0 = 0 (by the above lemma)

    Canceling the ab on both sides yields

    a(-b) = -(ab)

    The proof that -ab = -(ab) is similar.

    With this, we can prove that (-a)(-b) = ab, for any elements a and b of a ring R.

    Let a and b be elements of a ring R. Then

    a + -a = 0, so

    (a + (-a))(-b) = 0, so

    a(-b) + (-a)(-b) = 0, so by lemma 2,

    -(ab) + (-a)(-b) = 0

    Canceling the -(ab) on both sides yields

    (-a)(-b) = ab

    And we're done.

  • A Mathematical Explanation

    If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way:

    (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab

    For example,

    -2 * -3 = (-1)(2)(-1)(3)

    = (-1)(-1)(2)(3)

    = (-1)(-1) * 6

    So the real question is,

    (-1)(-1) = ?

    and the answer is that the following convention has been adopted:

    (-1)(-1) = +1

    This convention has been adopted for the simple reason that any other convention would cause something to break.

    For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn't work for negative numbers:

    (-1)(1 + -1) = (-1)(1) + (-1)(-1)

    (-1)(0) = -1 + -1

    0 = -2

    As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth."

    Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1.

  • remember in lower grades they used to teach that multiplication is how many times you 'see' something. for example: 3 x 2 is: if you see three two times, what do you get? 6... so for negatives its the same way, only worded a little differently. for example: -3 x -2 is: if you see the negative three negative two times, the negatives cancel (just like in english you cant have any double negatives), and you get 6

    Another way to look at it is like this (I'll use english negatives): I did not not walk today. That means that the person has neglected to not walk, meaning, the person did walk.

    It is a very hard point to understand, you either get it or you dont and just go with it, u know?

  • It's not complicated. The reason is minus is found as undoing something. So if you undo, an undo, it doesn't change from the positive.

  • yes -2 x -2 would equal 4 for example

  • when i was taught we were told that there was no mathematical reason and that it was right so it may just be a given rule with no explination

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