Lets have fun with integers shall we?
-3 + -3 = -6 and -3 * 2 = -6
Yet -3 * -2 = 6.
I know they’re not the same but I can’t put my finger on the concept behind why two negatives when multiplied together equals a positive number.
Is there a proof for this?
Re-direct me to the logic, it’s been 20 years since I’ve had any substantial math and I don’t remember asking why back then.
I’m sure Mathochist will be along any second with the technical background, I can only offer a layman’s opinion.
(Though, I feel like adding, I am good at math, I just didn’t pursue it especially far.)
It’s hard to wrap the brain around this because you can’t have negative-one objects.
-1 multiplied by A gives you A’s additive inverse. Also known as -A. That quantity, which, when added to A, makes 0. Why? I’m not sure as to the origins of the idea, but I think it’s just the way we’ve established our mathematical system.
So, if A is -2, and we multiply by -1, we get the additive inverse of -2. Which is 2.
Like this : A + (-1)A = 0; do the substitutions, and bang. From this property of -1, since it is a factor of all negative numbers, you can extrapolate the property to all negative numbers.
There’s probably more to it.
Ooh, wait. It’s coming back to me now.
Reflexive Property : A=A.
Subtract A from both sides : A - A = 0.
Factor out the -1 to make it an addition : A + (-1)A = 0.
There’s our proof of the Additive inverse. Doesn’t matter if A’s positive or negative.
I can’t provide a formal proof, but here’s some examples that might show you why a negative number times a negative number is a positive number. First of all, a given negative number is the additive inverse of its respective positive number. I.e. a positive number plus its respective negative number is zero. When multiplying a negative number times a positive number, we can show that the result is a negative number by solving a formula such as:
1 * 1 = 1
(2 - 1) * 1 = 1
2 * 1 + (-1) * 1 = 1
2 + (-1) * 1 = 1
-1 * 1 = 1 - 2 = -1
So, we can show that a negative number times another negative number is a positive by solving a formula like this:
1 * 1 = 1
(2 - 1) * (2 - 1) = 1
2 * 2 + (-1) * 2 + (-1) * 2 + (-1) * (-1) = 1
4 - 2 - 2 + (-1) * (-1) = 1
0 + (-1) * (-1) = 1
It’s a simple proof.
Suppose that a + b = 0, and that a + c = 0. Then a + b = a + c, and b = c.
By definition, a + (-a) = 0. Also by that definition, (-a) + (-(-a)) = 0. So a + (-a) = (-a) + (-(-a)), and a = (-(-a)).
It’s easy to show that -a = -1 * a, so (-(-a)) = -1 * -1 * a, and since that is equal to a, it follows that -1 * -1 = 1.
So if a, b > 0, what is -a * -b? It’s (-1 * a) * (-1 * b), which is (-1 * -1) * (a * b), and as we just showed above, that’s equal to a * b, which is greater than 0.
If I was writing that up for a math class, I’d have to include a little more detail, but that’s the underlying idea of the proof.
Or there’s the kindergarten version:
Think of “plus” and “minus” as directions on a number line. There being only two directions, one is left with the question “if ‘minus one’ times ‘minus one’ isn’t ‘plus one,’ what else is it?”
I’m sure this isn’t the formal logic behind it, but it helps to think of multiplication by -1 as “reflecting through zero on the number line”, e.g. if you are, say, three norches to the right (at +3), then you multiply by -1, you reflect that across onto three notches to the left of zero, i.e. -3. If multiplying by -2, then you are also doubling the length, as well as reflecting it, and so on.
Therefore, if you start off on the left of the zero, eg at -3, then you reflect onto the right hand side, so -3 x -1 = +3.
I should add that the proof I just gave depends on the existence of a multiplicative identity and an order relationship. In more general structures than the real numbers, you might not have that, so you’d need a different proof. That proof turns out to be shorter, so I thought I might present it.
0 = 0
0b = 0
(a + (-a))b = 0
ab + (-a)b = 0
(-a)(-b) + (-a)b = (-a)((-b) + b)
(-a)((-b) + b) = (-a)0
(-a)0 = 0.
ab + (-a)b = (-a)(-b) + (-a)b
ab = (-a)(-b)
Is that clear?
Yet another Board Favorite™
One way to answer “Why is a negative times a negative = a positive?” is simply to say that it has to be that way in order to be consistent with the rest of math.
Example: 3 + -3 = 0
so -2(3 + -3) = -2(0) = 0.
But, by the distributive property, -2(3 + -3) = -2(3) + -2(-3).
Now if you accept that -2(3) = -6, -2(-3) has to be positive 6 so that they’ll add up to 0.
If you want a nice, intuitive explanation, I’ve always thought this approach was the way to go, rather than an algebraic proof.
To extend this idea, consider the set of all complex numbers a+bi. The complex numbers are typically thought of as forming a plane; the number a+bi corresponds to the point in the plane given by the ordered pair (a,b). If you draw the line segment from the origin to (a,b), that segment will make some angle with the positive x-axis (or positive real axis, if you prefer). We’ll call this the “angle” of the complex number.
When we multiply two complex numbers, the angle of the product will be the sum of the two angles of the factors. For example, i has angle 90[sup]o[/sup]. i*i=-1, which has angle 90[sup]o[/sup]+90[sup]o[/sup]=180[sup]o[/sup].
When we multiply (-1)*(-1), the angle we get is 180[sup]o[/sup]+180[sup]o[/sup]=360[sup]o[/sup]=0[sup]o[/sup]; i.e., we get a positive real number.
So, when our view is restricted to the real line, multiplication by a negative has the property of reflecting the number line about zero. Extended to the complex plane, we see this reflection is in fact a 180[sup]o[/sup] rotation.
My brain works in very linear modes. I can sort of see what you guys are saying. I understand the math and all.
If you look at a number line and you notice that -3 is three spaces to the left of the zero. You then glance over at the number 9 which is 9 spaces to the right of zero. Common sense would tell you that the nine is twelve spaces from the -3. Yet you only need multiply the -3 by a -3 to get over the twelve spaces?
Explain this phenomenon please.
Multiplying by -3, as Colophon alludes to, is like adding a number to itself 3 times, but going in the opposite direction from 0. So, -3 times -1 is 3 away from 0, but in the positive direction instead of the negative direction that -3 is from 0, so it’s equal to 3. Similarly, -3 times -3 is 3 + 3 + 3 = 9.
I feel so honored. Yeah, the layman’s opinion is pretty much enough for the layman. For anyone who wants to see into the inner workings (though not as far as Principia Mathematica went or I’d be here until next Tuesday), read on.
The integers satisfy these rules, and my proof will follow from them. This means that anything else which satisfies these rules will also have “a negative times a negative equals a positive”.
[ol]
[li] There is an operation “+”, which satisfies these rules:[/li] [list=a]
[li] associative: (a+b)+c = a+(b+c) for all a, b, and c[/li] [li] identity: there is a “0” such that a+0 = 0+a = a for all a[/li] [li] inverse: for all a there is “-a” such that a+(-a) = (-a)+a = 0[/li] [li] commutative: a+b = b+a for all a and b[/li] [/ol]
[li] There is an operation "", which satisfies these rules:[/li] [list=a]
[li] associative: (ab)c = a(bc) for all a, b, and c[/li] [li] identity: there is a “1” such that a1 = 1a = a for all a[/li] [li] bilinear: a(b+c) = (ab)+(bc) and (a+b)c = (ac)+(bc) for all a, b, and c[/li] [li] commutative: ab = b*a for all a and b[/li] [/list]
[/list]
The identities are unique. Say 1 and 1’ were both multiplicative identities. Then
1 = 1*1’ = 1’
where the first equality is by the fact that 1’ is an identity and the second by the fact that 1 is an identity. The proof that 0 is unique is similar.
Inverses are unique. Say b and c are both additive inverses of a. Then
b = b+0 = b+(a+c) = (b+a)+c = 0+c = c
where justification of each equality is left to the reader.
Next, we show that 0*a = 0 for all a.
0a = (0a)+(a+(-a)) = ((0a)+a)+(-a) = ((0a)+(1*a))+(-a) = ((0+1)a)+(-a) = (1a)+(-a) = a+(-a) = 0
where again justification of each step is left to the reader.
Now, I assert that -a = (-1)*a. Indeed,
a+((-1)a) = (1a)+((-1)*a = (1+(-1))a = 0a = 0
and since additive inverses are unique, (-1)*a = a. Further, a similar argument shows that (-(-a)) = a.
Finally we may prove something near what the question asked: (-a)(-b) = ab. I’ll drop the parentheses where associativity lets me from here on
(-a)(-b) = (-1)a(-1)b = (-1)(-1)ab = (-(-1))ab = 1ab = ab
leaving us only to decide what we mean by “negative” and “positive”. I’ve left that part of the structure out, in case this is what the OP was really getting at. If you want me to go into it, though…
Say Bob gives you 7 payments of $4 for some reason. The amount of money you gain is 7 * $4 = $28.
It turns out Bob was only meant to give you 5 payments. Because you’re such a kind person, you decide to refund him two payments without a struggle. Refunding 2 payments is the same as gaining -2 payments. So the amount of money you gain in the refund is -2 * 4 = -8 (i.e. you lost $8).
Your wheelings and dealings with Bob remind you that you have to make 6 payments of $3 to Fred. So you do so. Making a payment of 3 is the same as receiving a payment of -3. So the amount of money you gain in your transaction with Fred is 6 * -3 = -18 (i.e. you lost $18).
Oops! It turns out you made the same mistake as Bob! You were only meant to make 4 payments to Fred. Luckily, Fred is as nice as you are, and gladly refunds 2 payments. As before, each payment is worth -3 to you, and refunding 2 payments is the same as making -2 payments. So the money you gain in this transaction is -2 * -3 = $6 (i.e. you actually gained $6).
If you want the distance between two numbers on the number line, you subtract. This applies whether the numbers are positive, negative, or one of each.
10 - 4 = 6, and 10 is 6 spaces over from 4 (i.e. the distance between 4 and 10 is 6 spaces).
9 - (-3) = 12, and 9 is (as you said) 12 spaces over from -3. (The distance between -3 and 9 is 12 spaces.)
So what does multiplication have to do with distance on the number line? Well, if you multiply a number by n, you end up with a number that’s n times as far from 0 as the original number was—in the same direction if n was positive, in the opposite direction if n was negative. So when you multiply -3 by -3, you get 9, which is 3 times as far from 0 as -3 was, but in the opposite direction.
This confounds all sorts of structures on Z. To put it most simply: why do the numbers you multiply by have anything to do with the distances being multiplied?
As I said, the key word (for me, anyway) is reflection. When something is reflected in a mirror, its (virtual) image is the same distance from the mirror as the object itself. If you stand facing a mirror three feet away, and Bob is three feet behind you, then your image appears to be six feet away from you, and Bob’s appears to be nine feet away. To Bob, your image appears to be nine feet away, and his image twelve feet away.
So, think of multiplication by minus one as like reflection in a mirror which cuts the number line at zero. Multiply -3 by -1 and it reflects across to +3, i.e. the same distance from zero but on the other side of the “mirror”. If you’re multiplying -3 by -3, you can think of that as multiplying -3 by -1 and then multiplying by 3 (because -3 = -1 x 3). So, you mutliply by -1 to get +3, then multiply by 3 to get +9.
I don’t think I understand your question, or what this has to do with Z. Should I have chosen a different letter than n, one that didn’t suggest I was specifically talking about integers?
The (signed) distances are integers and the scalars are integers, but why is the action of scalars on distances the same as the ring multiplication in Z? The question was asked symmetrically, with no difference between the kind of things a and b represent, so what justifies your switch to a being a signed distance and b being a scalar? It requires some extra argument to show that what you said proves anything about arithmetic.