A logical (and simple) explanation for (-1*1=-1)?

Ntstanch

If you have one pool ball sitting on a table, and you're told to multiply the pool ball by negative one, how can it make sense that you're left with no ball on the table and also now lacking a pool ball? It seems more like it would result in zero.

Note: This is just an idea myself and a few others had some fun with last spring semester. Though the question was never really answered and it did bring up a lot of more interesting ideas.

gb7nash

Well, one way you could look at it is that instead of owning one pool ball, you now owe a pool ball to someone else. If you have -2 pool balls, you owe 2 pool balls to somebody else. This is the assets/liabilities way of looking at it.

Ntstanch

Yeah, the concept of ownership and debt was the first thing to come up when I introduced the question. And for those scenarios the tool (basic arithmetic) is accurate and useful.

However the fun part is looking at how mathematics and reality can conflict in logic. Like if someone says I owe them ten of my fingers plus interest, and that interest leads to me eventually owing them eleven of my fingers. Obviously I can't give someone something I don't have, or somehow manage to end up with -1 fingers. The whole thing settles around currency (and ownership, property, debt - etc), which is extremely flexible only because it has to be. Constantly demanding that imaginary 11th finger isn't going to produce it... and hopefully if this situation ever happened to me in real life the person I'm indebted to doesn't decide that two toes are close enough to one finger. Enter the invention of the barter system. :P

Anyway. If you remove the concept of debt and all that it becomes far more interesting.

zerkroz

Identity axoim of multiplication?
Any number multiplied by one will have a product of itself?


For any number a;
a * 1 = a

HallsofIvy

You are right. It makes no sense. In fact, it makes no sense to talk about multiplying a "pool ball", or any physical object, by -1.

Really? Perhaps if you would explain what you mean by "multiply a pool ball by -1"?

Did any one ever define their terms? It doesn't make much sense to argue about meaningless words.

awhalen

You have negitives in the physcal world. Space itself could be consitered a negitive. Its not the absence of an object its the "hole" where something can be. If you have a ball buried in the ground so that a hole is made then you take the ball out of that hole you would have a negitive ball; the hole where the ball was is the negitive. the was i look at a negitive is its not what was there its what could be there.

Ntstanch

I consider the "pool ball" as a physical representation of a number (so the pool ball is 1), and consider the -1 just as basic arithmetic does. I definitely do not have a logical representation of what a "negative one" pool ball is, which gets to the heart of the matter.

It's not so much an argument of "meaningless words" as it is critical thinking of what we're taught and how to look at these things with more understanding. The physical object considered as the number 1 when applied to the argument of (-1*1=-1) was used as a tool for that on account of a negative pool ball (or plane, or person, or whatever) doesn't make sense when trying to logically explain how it could equal -1.

The majority of playing with this idea was spent between myself and a student who plans on being a high school math teacher. Consideration of things like this, especially in his case, is exceptionally useful for answering a curious students questions and adding new perspectives and clarity on what is being taught. As saying "well it doesn't make sense" is less useful than illustrating why and how it makes sense for many things in life and doesn't in others.

A real life example would be from a physics professor that I speak with occasionally, much like how I hope to use this website. I was talking to him about this sort of thing and he mentioned that he had students who couldn't explain what a derivative was (in laymans) outside of the mathematical terms they had been taught. They had the general rote of it, but that was all.

Mark44

You're not discussing the operation you're doing - multiplication - and how it makes any sense to talk about multiplying a pool ball by -1.

It makes sense to talk about adding pool balls or subtracting them, or even multiplying them by positive numbers, when that makes sense.
Right, I get that, but if you are trying to make a somewhat abstract operation (e.g., -1 * 1) more understandable, you need to be able to represent all three things - the two numbers and the operation - in some way that makes sense. If you don't have that, then the example is not going to be very helpful.

Ntstanch

Yeah, the original post was admittedly weak. Was sort of testing the waters. One of the most practical and useful examples was a combination of how negatives operate on the number line and then comparing that with the debt concept, and then applying those points to what the original post weakly illustrates.

Also, I'm not a math major. Others were talking about the (-1*1) idea (this came after talking about sets... which came after the number line and debt applications) in terms of abstract algebra. Something about rings and grouping among other things which I have no background in at all. The main thing I contributed were thought experiments and logic inconsistencies of things not requiring four years of math knowledge/terminology at a technical college.

It's also been around six months since I last had a real discussion about this... so I'm a bit rusty. When I have time I will try and organize a more detailed post. Need to warm up my brain anyway, classes starts next week.

Ntstanch

For now I will just post the idea concerning multiplication of negative numbers that the physics prof sent me. It was one that had been brought up earlier, however he proposes the perspective well... and saves my slightly lazy self some time.


"Think of multiplication as a way to get to a specific point on the
number line, which stretches from negative infinity to positive
infinity. The integer is how far to go, the sign is direction.

So, 4*3 can be thought of as:
Take a group of positive 4, and go three of those groups in that same
direction. In other words, 4*3 = 12 (to the right of zero)
(This of course works the same for 3*4.)

4*(-1) would be take a group of positive 4, but go in the opposite
direction, that is to the left of zero. So 4*(-1) is -4.

-1*4 would be equivalent to 1 unit in the negative direction done four
times or -1*4 = -4.

OK, now two negative numbers...
-1*(-4) = 4 (as you have been taught).
Start with the -1: That's 1 unit to the left of zero on the number line.
The second number tells you how many sets of those to take. In this
case, the negative sign indicates that you should go in the direction
opposite to the sign of the first number, which is a net positive.

In essence, a negative sign tells you to go opposite the direction you
were going. If you are facing down the number line to the right, one
negative sign faces you down the number line to the left. Another one
simply reverses that, which turns you down the number line back to the
right.

I hope this helps a bit." - Dr.Atmospheric physics (not sure it's appropriate to use his name?)

I read over my response to him, which is fairly long, and now (around six months later) realize that a lot of my terms were off. So I will try and refine those. In the meantime his response to my response (I posted a bit about this in the set theory forum) was:

"You are, in fact, in pretty good company when it comes to not having an
intuitive grasp of negative numbers. And then there's the subjects of
irrational and imaginary numbers...

Apparently, Leopold Kronecker didn't believe in negative numbers.
http://en.wikipedia.org/wiki/Kronecker

Come by my office when you get a chance. I have a book that you can
borrow that I think you'll enjoy."

With that, does anyone here have fluent knowledge on Kronecker and some of his main ideas, or information that you think may be helpful in general? I read up on Kronecker awhile ago, but it was all still too deep in mathematics for me to be satisfied enough in my understanding to use it out loud.

JHamm

Since in this case multiplication is commutative it makes more sense to say you're multiplying your negative one pool ball by 1, since 1 is the identity for multiplication in this case you end with what you started with.