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?
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
this should help:
http://www.graspr.com/videos/Math-Proof-Negative-T...