The multiplication of negative numbers yields a positive number--what's the problem?

Factual Questions
1

I stumbled on [url=http://www.amazon.com/Negative-Math-Mathematical-Rules-Positively/dp/0691123098/sr=1-1/qid=1158171132/ref=pd_bbs_1/104-2140295-6612749?ie=UTF8&s=books]this book* at the library yesterday. I haven’t checked on the author’s credentials or anything, but anyway, it seemed like an interesting read so I checked it out.

Much of the book is premised on an idea that there is some conceptual difficulty in understanding what it could mean to multiply two negative numbers together, and why that should yield a positive result. In the book, in fact, the author constructs the fundamentals of an arithmetic (and an interpretation of its results, I guess) in which multiplying negative numbers yields a negative number. He seems to find this more intuitive, and seems to expect most people to find tha more intuitive as well.

The reviews at the amazon page seem to confirm (though with a very small sample size) that people find it more intuitive to expect negative numbers multiplied together to yield negative numbers.

This post is borderline IMHO, but I want to ask a genuine question here.

First, though, I’ll explain why I have always been perfectly intuitively comfortable with getting positive numbers out of a negative numbers multiplied together. I don’t remember now whether the explanation I’m going to give is something I figured out myself or if its something some elementary school teacher told me, but anyway:

If you think of a positive number as a “giving” and a negative number as a “taking away,” then if you ask “What happens when I take away a bunch of takings-away,” the answer is, “I end up with a givig.”

To clarify. Suppose I’m multiplying -3 x -3. The way I’ve interpreted this (when I’ve actively interpreted it at all) is as a way of asking the following question: “What happens when you take away three instances of a taking-away of three units?” Or in apples talk, “Say you’ve got some large number of apples, N, and someone takes away three of them. Say this happens three times in all. Now, you’ve got a new number of apples, M. The question is: What happens when you undo (i.e. ‘take away’) the three acts of taking-away which were just done to produce M? The answer is, you add 9 apples to M.” This means the answer to -3 x -3 is a positive 9.

Okay, so my question is: Does this work? Is this, as it seems to me, in keeping with the meaning and general use of negative numbers in arithmetic and in mathi in general? Or have I simply made up/been told a “just-so” story which, for some reason or other, doesn’t actually work once you think about it too deeply?

My further question, if this explanation of the multiplication of negative numbers is a good one, is this: Why did anyone ever think negative numbers should mutliply to give negative numbers in the first place? Apparently serious mathematicians had to debate this issue for at least a few decades before it was settled. Why? What’s the interpretation of negative numbers which makes them behave differently than the way we interpret them to behave today?

-FrL-

2

Sorry, that link was supposed to go like this:

I stumbled over this book at the library yesterday…

-FrL-

3

Are you going to sue?

4

weird

If multiplyign two negatives together is going to yield a negative, what the hell would they expect the result from multiplying a negative and a positive to be? It’s strange that someone would get hung up on this, because it follows grammar rules, too. Two negatives make a positive.

I always just learned negatives as “the opposite”. What happens if you take the opposite of three, three times? You get three times the opposite of three.

What happens if you take the opposite of the opposite of three, three times. You get the opposite of that.

I mean, that’s just a way to translate it into english and doesn’t really clarify the concept, but it’s never been a concept that I thought people found difficult, or that mathematicians found deep.

5

If you do, make sure to sue for a negative amount. Then when they judge against you (also a negative), you will end up with a positive product. That will really throw them for a loop.

6

I think you could redefine multiplication so that minus times minus equals minus, but you’re going to have to lose some other properties of multiplication. In particular, some numbers no longer have a multiplicative inverse, so the real numbers with that definition of multiplication are no longer a field. (I think they are still a ring, however: I can’t find any counterexamples to the ring axioms.)

7

I think you also lose distribution of multiplication over addition. I’m assuming that in this algebra, a negative times a positive is also a negative. Then,

-1*(1 + (-1)) = -1 * 0 = 0, but distributing, we get

-1*(1+(-1)) = -11 + (-1)(-1) = -1 + -1 = -2, a contradiction.

So multiplication doesn’t distribute over addition, and it’s not a ring either (I think).

Also, the multiplicative identity is screwy, since -1 * a = a, if a < 0. Are there any other rings where there is a second multiplicative identity over part of the ring?

-Rick (Long past formal math training. And I was a physicist, anyway.)

8

I read a history of algebra not long ago, and it seems that for centuries negative numbers were considered not quite kosher, and a great effort was made to avoid using them.

The way I see it (and I find it intuitive also) is that multiplication by -3 is 3* (-1), where -1 just flips the sign. So,
-3 * -3 = (10(3) * (-1)(3)
(3)(3)-1*-1 = 9 * -1 *-1. You flip the sign twice, and get back to 9. I find this far more intuitive than not taking away.

9

I agree, that’s a very intuitive way to express a very easy-to-follow rule. But I’m trying to ask about what recommends that rule in the first place. My “taking away/giving” was meant as a way to explain why the “sign flipping” rule is a good one in the first place.

In the book, upon looking ahead, I find that the arithmetic he comes up with is non-commutative when it comes to multiplication. A negative times a positive is a negative, and a positive times a negative is a positive. So in his system, the rule is, “The answer has the same sign as the first term in the multiplication.”

So there are two rules at hand:

“The sign of the answer is obtained by ‘flipping the sign’ whenever a negative multiplicand is encountered”

and

“The sign fo the answer is obtained by copying the sign of the first multiplicand.”

Both of these rules are very easy to follow, and so are “intuitive” in a sense. My questions have been aiming at finding out what recommends that we follow the first rule, and not the second. I gave the story about “giving and taking away” in order to motivate the first rule–your rule, of course–but I was wondering if anyone knows of any sensible stories that can be told to dismotivate that rule and to motivate, instead, some other rule. I was trying to figure out, in other words, why anyone (even serious mathematicians) ever thought the rule should be other than it is.

-FrL-

10

Maybe think of it terms of logic,which is similar to your giving/taking as well as the grammar rule (although it is not necesarily true that two negatives in grammar make a poisitve). You have A and you have B = not A. Well what is not B? not B = not (not A) it seems to only make sense that not not A is A. Well ,just trade a for a number and not for negative 1 and there you are.

11

The novice would ask: Then why don’t two negatives always make a negative in addition?

-FrL-

12

The equivilent in addition might be two evens or two odds make an even, but only an odd and an even make an odd.

13

Not really. Two negatives in a sentence is either still negative (e.g. Spanish), or it’s “ungrammatical” (e.g. English). We like to pretend that two negatives makes a positive because that makes logical sense, but people almost never actually speak that way.

14

Here’s a short proof:

First, a proof that 0a = 0: 0a = (0 + 0)a = 0a + 0a. Also, 0a = 0a + 0. So 0a + 0 = 0a + 0a, and it follows that 0a = 0.

Second, a proof that x + y = z + y implies x = z. x + y = y + z implies that x + y + (-y) = z + y + (-y), which implies that x = z.

So to prove the actual claim, we need that ab + (-a)b = 0. This is easy; ab + (-a)b = (a + (-a))b = 0b = 0. Then we need that (-a)b + (-a)(-b) = 0. As above, (-a)b + (-a)(-b) = (-a)(b + (-b)) = (-a)0 = 0.

Therefore, ab + (-a)b = (-a)b + (-a)(-b), and so ab = (-a)(-b). All we used is the distributive property of multiplication over addition, the associativity of addition, the existence of an additive identity, and the existence of additive inverses. If this result is false, then one of those properties fails to hold.

In a more formal scenario, we don’t assume that those properties hold for the number system in question. We start with the axioms of set theory, and then create sets of sets that have the properties we would like our number systems to have. So a failure of a high-level multiplication rule implies that some axiom of set theory is wrong, and that makes things very different.

You should also note that I have not assumed that addition or multiplication is commutative, and that I never used the words negative or positive. This theorem holds for all rings, not just ordered ones.

15

But can anyone comment on this part? Did it really take a few decades to settle on the (now) accepted interpretation? What was the reasoning behind the alternative interpretation that was good enough that it took so long to vanquish it?

16

There are worst concepts people can be hung up over. :wink:

Anyways, I once read that the existence of the universe can be ascribed to ones and zeroes. If you have a certain string combination you can get “something” from “nothing”.

17

Simplest proof:

Assume a negative time a negative is a negative.
Then:
-1 x -1 = -1
But -1 x 1 = -1 (this is one definition of a negative number).
So -1 x 1 = -1 x -1
Divide both sides by -1
1 = -1

Obviously not (again, by definition).

Thus -1 x -1 = 1

18

Is anyone else thinking that the multiplication of two negative numbers produces a positive number just because it’s how we defined multiplication? I feel like asking why is similiar to asking why Boardwalk is the most expensive piece of property on a standard Monopoly board. It is because that’s how somebody made up the game.

19

Well yeah, all of proven mathematics is a logical chain that stems from several independant definitions. You can always change the definitions if you want and still have valid mathematics, just most of the time it’s not going to be all that interesting.

As somebody noted above redefining arithmetic to have -1 * -1 = -1 would break the field and would leave us with a very useless number system. However, if you then redefine 1 * 1 to work out to -1, we’re back in a field (you might have to tweak addition) if I’m not mistaken (I’m tired, and haven’t had coffee).

Having a number system that’s not a field is crappy because any of the proven theorems about fields become unproven again and some (if not most) will probably wind up false.

20

When multiplying by a negative, negative literally means to travel in the opposite direction on a number line. So if you multiply -3 and -3, instead of traveling to the negative side of a number line 9 times, you do the exact opposite and travel to the positive side of the number line.

Also, if you multiply 3 and -3, instead of traveling to the positive side of a number line 9 times, you do the exact opposite and travel to the negative side of the number line.

Picture a number line like a set of stairs.