Gee, that’s sure helpful.
There’s also no good macroscopic analogue to a particle experiencing destructive interference of its waveform, but it happens all the time in the real world.
Which proves that common-sense ain’t worth shit in some fields.
Wow. That’s awesome. It almost makes negative numbers seem real somehow. I mean, I know they’re real, but almost something graspable to we denizens of the positively-signed universe.
Now just explain to me what zero is, and we’ll be all set. 
Wouldn’t releasing helium balloons qualify as I wrote in the thread MikeS referenced? Or have I misunderstood?
(-3)*(-3)
I used to teach math and was asked this question at least twice a year.
Like another poster said, they didn’t want a proof, just an example.
Initially I tried using money since it works so well in other examples and works well with 3*(3). However, (-3)*(-3) money doesn’t seem to ‘stick’ as well. I came up with this and it seemed to work well for many people:
If the tempature has been rising 3 degrees a day for many days and will continue to do so…How many degrees warmer will it be 3 days from now? 3*3=9
If the temperature has been falling 3 degrees a day for many days and will continue to do so…How many degree warmer will it be in 3 days? (-3)*3=-9
Now…If the temperature has been falling 3 degrees a day for many days how many degrees warmer was it 3 days ago? (-3)*(-3)=9.
For some reason, this example worked well.
Let ab > 0
(ab is positive)
ab + -ab = 0
(-ab is negative)
-ab + -(ab) = 0
(-(ab) is positive)
QED
I’ve read this five times and I can’t make it work. At the very least, it needs about 10 steps filled in, and even then, it’s unnecessarily complicated. Why not go with the simpler proof posted above?
It seems simple to me, but any proof will do. Why reject any?
You invoke an ordering relation, which doesn’t exist in arbitrary rings. You brought in an element b that’s not needed, and you haven’t actually proved that a * a = -a * -a. You also differentiate between -(ab) and -ab (unless you’ve made a typo), which are the same in any ring.
Consider this proof:
- a * a + -(-a * -a) = a * a + (–a) * -a (multiplicative associativity axiom)
- a * a + (–a) * -a = a * a + a * -a (uniqueness of additive inverses)
- a * a + a * -a = a * (a + -a) (distributive axiom)
- a * (a + -a) = a * 0 (definition of additive inverses)
- a * 0 = 0 (standard ring theorem)
- a * a + -(-a * -a) = 0 (transitive property of equality)
- a * a + -(-a * -a) + --(-a * -a) = 0 + --(-a * -a) (cancellation)
- a * a + -(-a * -a) + (-a * -a) = 0 + (-a * -a) (uniqueness of additive inverses)
- a * a = -a * -a (definition of additive inverses and the additive identity)
Short, no unneccessary elements, and every step follows directly from the ring axioms or an easily proved consequence thereof.
This isn’t proof, but maybe it will help you wrap your brain around this.
I have decided, in math in particular, and life in general, that you have to figure out 2 things: how far you’re going, and in which direction.
You understand that 3 * 3 = 9, and presumably that 3 * -3 = -9. Now why? Why does the -3 make the answer negative? Because anytime you multiply a number by a negative number you change (reverse actually) the direction. If you multiply that by another negative number, then the direction reverses again.
1 * -3 = -3
-3 * 2 = -6
-6 * -4 = 24
Have you ever played Uno? (I’m going to assume that you have, in lieu of posting the directions.) Say you’re playing with a large group of people, and the play is going clockwise. One player plays a reverse card, which makes the play go counter-clockwise. The player to his right now has to go, and plays another reverse card. The play now reverses direction and is going clockwise again. i.e. two reverses end up with the play going the way that it originally was, ergo 2 negatives make a positive*. A third reverse would make the play go counter-clockwise. A fourth - clockwise again, etc. If the number of Reverse cards played is even, then the play is clockwise. If it’s odd, then the play is counter-clockwise. Every Reverse card changes the direction, just like every multiplication by a negative changes the ‘direction’ (i.e. positive or negative).
Hope this helps. (Every proof posted by one of the math guys makes my brain explode.)
*If only I had known this when my mom told me that 2 wrongs don’t make a right.
slight hijack: You may or may not follow this, but I’m going to quote you in my mini lecture series on connections on principal bundles and associated vector bundles next week. Thanks.
Here’s how I demonstrate it to my students. (Actually, I get them to do it.)
Draw up a multiplication table going from -3 to 3 like so (see if I can get the formatting right)
x | -3 | -2 | -1 | 0 | 1 | 2 | 3
-------------------------------------
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
I then get the students to fill in the times tables that they know – that is, the positives.
x | -3 | -2 | -1 | 0 | 1 | 2 | 3
-------------------------------------
-3 |
-2 |
-1 |
0 |
1 | 1 2 3
2 | 2 4 6
3 | 3 6 9
From there it is simply a matter of following the patterns backwards to fill in the unknown regions. (-3)*(-3) = 9 because that’s the answer that makes sense without breaking the pattern.
This basically begs the question. To “fill in the pattern backwards” implies that changing the signs on the multiplicands can only change the sign of the product, not its magnitude. Stepping away from integers, even the concept of “magnitude” is hazy. Even so, the checkerboarding of negatives may be a natural pattern, but it’s hardly the only one. If your students are satisfied with this argument, so much the better, but really you haven’t justified the result any more than before.
The other quibble is that if the student won’t be satisfied with a rational argument (I can think offhand of one I wouldn’t be afraid to give to any student who’s learned the rules of arithmetic up to multiplying negative numbers) and needs a “real-world” example as motivation, this diagram won’t help at all.
I’m sure this is a day late, and -3 * -3 dollars short, but here’s somethin to chew on. Multiplication is nothing more than a description.
Think of a balance sheet. If I have 6 people that owe me 2 dollars, than I am owed 12 dollars. Easy, 12 describes the number of dollars I’m owed.
But if I have don’t have 3 people that DON’T owe me 3 dollars a piece, than I am NOT owed 9 dollars (albeit by the imaginary 3 people). 9 simply describes the amount I’m not owed.
I know this requires you to think of people that don’t exist not oweing you money, but the amount of money they don’t owe you is a positive amount.
On second thought maybe this isn’t easier. I’m gonna go collect from my imaginary friends.
Mathochist, I’m not really sure of your difficulty with this approach. If we understand multiplication as being repeated addition, then the pattern* “it goes up by threes”*, etc is an automatic result. Logic dictates that travelling in the other direction along the rows, one should be able to go down by threes. Applying this thinking to the columns and the result falls out. Students who are thinking about what they are doing generally do not have much difficulty with this in my experience. Those that do have difficulty are those that are hazy on the meaning of multiplication in the first place.
But this whole matter does raise another issue. What concrete meanings can be placed on negative numbers. This is by no means trivial. Common themes are temperatures, bank balances, metres below sea level.
In none of these does it make a whole lot of sense to multiply by another negative number. Therein lies the difficulty. It is possible to do create some contrived situations – walking backwards down the stairs, people who don’t owe money etc, but I have not found them particularly helpful for my students to grasp the concept using these.
If we step back and look more closely at subtraction, we find a few things to unlock the problem. We commonly think of subtraction as “take away”. With this mental construct it is necessary for negative numbers to be concrete
In real life however, subtraction is more often used to find “difference”. That is, comparing two items and deciding which is bigger and by how much. Golf scores are a whole lot more useful than temperatures on this one. With this construct, negative numbers are just as natural as positive numbers. Then when you begin to multiply, it is no major hurdle to follow a pattern backwards – the territory is already familiar.
It all comes together when expanding numerical and algebraic expressions. Consider the following:
50 - 2(6 - 4) = 50 - 26 + 24
It is not hard then to put this expression in some kind of context – be it money, area, length of cable or whatever. Example: I began with $50 and bought two widgets which normally cost $6, but have been discounted by $4 each.
Now, someone s going to pull me apart for that last sentence where I (deliberately) mix the concepts of negative numbers (noun) with the action of subtraction (verb). The two are isomorphic. From a pedagogical point of view it is quite good if students see that. I have never had any significant arguments over that one. Students who are sharp enough to ask the question are also perceptive enough to recevive the answer and mull it over for themselves. Excessive rigour is not helpful.
Oh, that pattern. Funny, you never mentioned it at all. I took the unspecified pattern to be something different entirely. Admittedly, my mistake, but still sloppy pedagogical technique.
Brilliant! In the unlikely event that I’ll ever need to explain this to anyone, I’ll use this example.
Daniel
I liked it too. Do we attribute it to Andy Murphy?
Awwww shucks 
When I taught, using example of money almost always worked well. People seem to intuitively grasp money flow very well 
However, money examples just didn’t seem to work with two negative numbers multiplied together giving a positive answer.
Well, they do in bookkeeping. Suppose I’m a business, and as part of the business I have 3 customers that I owe $3 each to. Now, as part of a deal with another business, I give those debts to that other business. (It’s not so odd – that other business may be a bank, and I’m writing 3 cheques for $3 on that bank, which the bank will honour by taking $9 out of my account with the bank. Now a debt is a negative asset, and transferring an asset to another person or business is a negative operation on that asset. But here I’m transferring 3 things (that’s the first -3) which are themselves negative (that’s the second -3), so I become $9 richer (against wich you have to offset the $9 that comes out of my bank account, so I also lose $9 at the same time in the total transaction). But I do have to become $9 better off on one side of the transaction for it to make business sense for me to lose the other $9 on the other side. So here (-3)x(-3)=+9.