(-3)(-3)=9. Any negative times a negative is a positive.
How come? It’s easy to see that if I gather up three apples thricely, I will have 9 apples. But if I give away 3 apples negative three times, I will end up with nine? Huh?
There must be a mathematical proof for this. What is it?
Does this work?
(-1)(-1)(3)(3)
e[sup]iPI[/sup]*e[sup]iPI[/sup]33
e[sup]2iPI[/sup]9
19
9
using your example to give away 3 apples -3 times. Negative 3 times of giving away. The opposite of giving away 3 times is gathering 3 times. That is to gather 3 times. Gather 3 apples 3 times you have 9.
How about this:
Let -3 x -3 = -9
But 3 x -3 = -9
So -3 x -3 = 3 x -3
Divide both sides by -3: -3 = 3
An obvious contradiction. Thus -3 x -3 <> -9
Well, you have to have things work this way for the distributive properties of the addition & multiplication, i.e. (a+b)c = ac + bc, to work out. 0 = 03 = (3 + (-3))3 = 33 + (-3)3, so (-3)3 = -9. Then 0 = 0(-3) = (3 + (-3))(-3) = 3*(-3) + (-3)(-3), so (-3)(-3) = 9.
Not as intuitive as it might be, but it works, I think. Here’s a recent thread on the subject.
How about looking at it in accounting terms;
Starting with a ‘balance’ of ten apples, I pay you three - this can be done by applying a debit of three to the account (reducing the balance)
Every time I apply a debit (essentially subtraction), the balance decreases
The opposing action is to apply a credit - adding apples to the account.
But suppose I credited some apples in error? I could apply a debit to correct it, but it would be better to ‘reverse’ the erroneous credit so a reverse(minus) credit(plus) is a minus.
Likewise, if I debited some apples in error, I could correct it by reversing the debit, so a reverse(minus) debit(minus) adds apples to the account - it has to, because it is the opposite of the simple action of taking them away.
Well, you can think of giving away apples a negative number of times as collecting apples that number of times, just like collecting negative apples is like giving them away.
Collect (-3) apples (-3) times.
Give away 3 apples (-3) times.
Collect 3 apples 3 times.
Collect 9 apples.
As for mathematics, it can be proven in what’s known as “ring theory”, since the integers are an example of a ring: a set with an addition rule and a multiplication rule defined to satisfy certain properties. 0 is the additive identity, 1 is the multiplicative identity, and -x is the additive inverse to x. Now, it can be shown that -x = (-1)*x.
x + (-1)x = 1x + (-1)*x = (1+(-1))x = 0x = 0
And since inverses are unique, -x = (-1)x. So (-x)(-y) = (-1)x(-1)y = (-1)(-1)xy and your question reduces to why (-1)*(-1)x=x. Well, this is (-1)(-x): the additive inverse of -x. But we know that the additive inverse of -x is just x! And there you go, as was to be shown.
[Bolding Mine]:
Your problem is flawed because -3 x -3 = 9, not negative nine, and the result of dividing both terms by -3 would be: 1x1 = -1x1, which is a contridiction. Unless you were trying to prove that -3 x -3 cannot equal -9 (I’m unfamiliar with the ‘<>’ notation).
“<>” = “!=” = [symbol]¹[/symbol]
He was.
Okay, so now RealityChuck has established that (-3)*(-3) != -9. Now he just has to do it for -10, -6, 8, 1/3, pi, e, and every other real number that isn’t 9.
Wouldn’t it just be easier to argue the positive from the basic properties?
That depends on whom you’re arguing to. If your audience is a junior or senior math major, then RealityChuck’s argument is neither necessary nor sufficient. Rigorously, proving that (-3)(-3) != -9 does not prove that (-3)(-3) = 9, and there are other methods which do rigorously prove it. However, a person interested in rigor probably would not have asked the question in the first place. A layman is probably willing to accept without rigorous proof that (-3)*(-3) is either 9 or -9, so once you show that it can’t be -9, the rest follows.
I think what the good poster in the OP would like someone to do is put the concept of multiplying two negative numbers into a descriptive concept he can understand.
Multiplying two positive numbers is easily shown by example: three boys have three apples apiece. How many apples do they have all together?
Multiplying a positive and a negative is slightly more difficult, but one can offer an example such as three people, all of whom owe the bank $3. How much do they owe the bank all together? The only difficult part is having the layperson understand the concept of negative numbers as they apply in the real world.
Multiplying a negative number and a negative number is quite a bit more difficult. With respect, the example offered by Mathochist would not satisfy most laypeople, because all you are offering is to suggest that the observer can equate two negatives to two positives (minus three people giving away minus three apples is the same as three people getting three apples). Meaningless gibberish, of no value to understanding what is going on mathematically when one engages in that operation.
I’m certainly not going to attempt to craft an analogy or word problem to properly describe the operation -3 x -3. But perhaps one of the math mites could craft a real-world situation which demonstrates relatively simply the concept of negative three sets of negative three items. 
Feh, mathematicians don’t care about the so-called “real world”.
To the best of my knowledge, there is absolutely no physical analogue of the multiplication of negative numbers. :shrug:
Actually, this is pretty good. I understand it, though now I need an aspirin.
Is this close to the proof that one might find in a college-level textbook?
I think of -1 as an operator that rotates the direction of a length on the number line by 180[sup]o[/sup]. By convention lengths measured to the right of 0 on the number line are positive. If I have lengths of line n and m the magnitude of their product is nm. If both are negative (-n-m) I have the 180[sup]o[/sup] operator, -1, applied twice which results in a 360[sup]o[/sup] rotation which is the same as 0[sup]o[/sup] so the result n*m is measured to the right on the number line which is the positive direction.
It wouldn’t sound so meaningless if you wouldn’t willfully misquote Mathochist and the others who offered identical examples. They satisfied this layperson just fine. The opposite of the the opposite of gathering three apples thrice is gathering three apples thrice, i.e., nine apples.
For those unfamiliar with complex numbers, I didn’t think up this “rotation” business. It is the standard geometric interpretation of the operators i, -i, i[sup]2[/sup] and so forth in complex number manipulation.
A standard geometric interpretation. I’d say the standard “geometric” interpretation is about a C[sup]*[/sup] structure on the algebra of continuous real-valued functions on a geometric object like a manifold.
Then again, if someone understands that, they already know C inside and out.