# Why do 2 negatives make a positive?

1. Jun 9, 2007

### stochastic

2x4 is the same as 4+4 right? they both equal 8 right? by what logic does -2 x -4 = 8. I know its a rule. but why?!?! shouldnt -4 + -4 = -8 or if it negates itself equal 0?

2. Jun 9, 2007

### Werg22

This has been discussed several times on these forums, you can find all the previous threads using the search function. There is no logic behind why a negative times a negative is a positive but the identity that all real numbers must answer

$$a - a = 0$$

Last edited: Jun 9, 2007
3. Jun 9, 2007

### stochastic

math without logic should not exist.

4. Jun 9, 2007

### Werg22

I don't think you're getting the point here... a negative times a negative dosen't mean anything intuitively nor physically like does a positive integer times another. It is defined and that is it.

Last edited: Jun 9, 2007
5. Jun 9, 2007

### ice109

math is tautological, some things are defined axiomatically and from those axioms other things are deduced.

6. Jun 9, 2007

### matt grime

You agree x-x=0? That x*0=0 for all x? Then you agree that (-x)*(-y)=xy if x and y are positive numbers. Try to work out why.

7. Jun 9, 2007

### fopc

I believe math presupposes logic.

Here's an "argument" you might take a look at.

Claim: (-x)*(-y) = x*y

1. (-y) + y = 0
2. (-x)*((-y) + y)) = (-x)*0
3. (-x)*(-y) + (-x)*y = 0
4. (-x)*(-y) + (-x)*y + x*y = 0 + x*y
5. (-x)*(-y) + (-(x*y)) + x*y = x*y
6. (-x)*(-y) + 0 = x*y
7. (-x)*(-y) = x*y

I'd say we have a theorem (with rough proof). Not a definiton. Not an axiom. Not a rule.

Can you justify the steps?

What's been presupposed here?
Equality axioms? Field axioms?
Some previously proved theorems from field? Like for example, (-x)*y = -(x*y)?

Anyway, don't take any of this stuff on blind faith.

Last edited: Jun 9, 2007
8. Jun 9, 2007

### stochastic

I'll just use math that makes sense.

No I don't agree at all. I did try working it out thats why im on your silly forum asking. Simply giving examples I've already seen doesn't answer my question. I guess I'm to logical for math. If you have 0 apples and then you receive 2 baskets of -4 apples you still don't have any god damn apples. :grumpy:

At least tell me what (-x)*(-y) is used for if nothing else.

Last edited: Jun 9, 2007
9. Jun 9, 2007

### apples

This becomes really childish, but since you seek logic like this, I'll go on.
Suppose you take 4 apples to school everyday, and there's a bully at your school who takes 4 apples away everyday, and sells them, makes apple pie, plays softball etc. You are then left with 0 apples. Suppose you don't take any apples with you one day, and thus the bully doesn't take any, but he tells you you owe him 4 apples. so now you have -4 apples, because the moment you have them he'll take them away (forget about the apples he takes each day for a while). So since you owe something you have a negative of it.

I used this example because you talked about it. But in Math, there are some things for specific purposes. For example there can't be negative time (unless you invent a time machine) but if a company takes a lot of loans but doesn't get enough profit, then it goes in loss, in other words the money it has in 'negative'.

10. Jun 9, 2007

### chroot

Staff Emeritus
apples,

Your example is justification for negative numbers, but contains nothing about multiplication of negative numbers!

Furthermore, there certainly can be negative time. If I declare that time t=0 is tomorrow at noon, then I can certainly say that it's about -25 hours right now.

stochastic,

Your question has been answered -- if the negative of a number is to be its own additive inverse, then -x * -y must equal x * y. If you do not understand the answer, ask for further clarification.

- Warren

11. Jun 9, 2007

### stochastic

apples: I'm not seeking logic I'm trying to have it logically explained which you couldn't do. As it seems you didn't even understand the 'childish' question. sigh

Someone just tell me what this can be used for. Maybe that will make me understand!

Last edited: Jun 9, 2007
12. Jun 9, 2007

### chroot

Staff Emeritus
Here's an example from the Math Forum FAQ

If a negative times a negative equaled a negative, then distribution of negative numbers wouldn't work:

(-1)(1 + -1) = (-1)(1) + (-1)(-1)
0 = -1 + -1
0 = -2

which is an absurdity. You can find many of these kinds of absurdities.

edit: The math forum FAQ also contains plenty of examples of "real world" negative number multiplications. Read them, and ask if you need any more clarification.

edit 2: please knock off the attitude.

- Warren

13. Jun 9, 2007

### matt grime

you tried to work what out? I posted 2 things that you surely agree with: x-x=0 and 0*y=0. From these it is very easy to deduce that 'minus times minus is plus.'

the 'examples' were there for you to deduce the result from.

if you're so logical do some deduction for yourself

what you just wrote has bugger all to do with logic.

it is required to make a logically consistent mathematical theory. Start with x-x=0 and 0*y=0 to show that (-1)*(-1)=1.

14. Jun 9, 2007

### matt grime

15. Jun 9, 2007

### stochastic

Actually it isn't contradictory. Logic, a noun, is something I already found and possess. My wanting someone to logically explain a topic is not the pursuit of logic but the request that they construct their explanation properly. Think before you act.

As for my deduction now that I have read more about this topic, there does not seem to be any logical explantaion or application for this other than to preserve some silly rule about the additive inverse.

(-2)*(2)= -4
vs
(-2)*(-2)= 4

and yet 2*2= 4 so easily

case closed.

Last edited: Jun 9, 2007
16. Jun 9, 2007

### arildno

Not at all.
What is silly, is your preconception as to what numbers should be, and what multiplication "means".

To see a "real-life" application of the "minus times minus" is plus, let us consider the relation between the physical quantities "velocity", "measured time" and "traversed distance".

Let a "positive" distance be how far to the right something is from a given point called the origin, whereas a "negative" distance is how far to the left of that origin something is placed.

Furthermore, let "positive" time denote the how much time AFTER some specified instant something occurs, and similarly for "negative" time.

Let a "positive" velocity be given to any object moving in the right direction, whereas a "negative" velocity is given to any object moving the left direction.

Now, suppose that at the given instant t=0, two moving objects are both present at the origin location x=0, the one object moving with positive velocity V>0, the other with negative velocity -V.

Now, after time T, the object with velocity V will be in location x=V*T.

But that must be the same location that the OTHER object had at time -T, that is: V*T=(-V)*(-T)

17. Jun 9, 2007

Does the phrase "don't feed the trolls" mean anything to anybody here?

I think it would be a good idea for this thread to be killed as the original poster is clearly not actually looking for answers to his question.

18. Jun 9, 2007

### stochastic

I see what your saying and yet I could still multiply all of that without the negative integers and get the same answer. This is vital information to be teaching children in the 7th grade. What a joke.

I agree with deadwolfe. trolls shouldnt feed trolls ;p Lets agree to disagree because i understand what your saying but i see absolutely no practical use for it.

Minus times minus results in a plus,
The reason for this, we needn't discuss.
- Ogden Nash

Last edited: Jun 9, 2007
19. Jun 9, 2007

### Dr. Proof

arildno has given an example that I have seen before; it is good to see that someone sees that Mathematics is not an autonomous subject in and of itself and that it must be connected to describing the physical world. Now, stochastic, you must understand that Mathematics has no meaning unless it is used to describe some quantity. For example, the symbol "3" has absolutely no meaning unless you say "3 nails", "3 cars", "3 inches", "3 newtons of force", "3 meters", etc. There is a reason why a negative number times a negative number is a postive number; I will explain it in a way similar to arildno's method:

Picture a lever sitting on a fulcrum. If I apply a downward force to the lever at a position to the right of the fulcrum , the lever will rotate clockwise about the fulcrum. Now let's call downward forces "negative forces", upward forces "positive forces", distance to the right of the fulcrum "positive distance", distance to the left of the fulcrum "negative distance", clockwise rotation "negative rotation", and counterclockwise rotation "positive rotation.

If I multiply an upward force (positive force) by a distance to the right of the fulcrum (positive distance) I will have the magnitude of a torque that causes a counterclockwise (positive) rotation about the fulcrum; therefore, the torque is positive. Now, if I multiply an upward force (positive force) by a distance to the left of the fulcrum (negative distance) I will have the magnitude of a torque that causes a clockwise (negative) rotation about the fulcrum; therefore, the torque is negative. Now, if I multiply a downward force (negative force) by a distance to the left of the fulcrum (negative distance) I will have the magnitude of a torque that causes a counterclockwise (positive) rotation about the fulcrum; therefore, the torque is negative; therefore, a negative number times a negative number is a positive number.

Multiply all the different combinations of forces and distances in this lever example and you will see that a positive times a positive is always a positive; that a positive times a negative is always a negative; that a negative times a positive is always a negative; and lastly, that a negative times a negative is always a positive.

Lastly, stochastic, let me say this. You probably didn't understand negative numbers because you didn't know in what areas of science they are actually used. Don't let anyone tell you that mathematicians just make up rules that we have to follow, and that negative numbers is just one of those rules; THAT'S A LOAD OF CRAP. Negative numbers exist for a reason, and they exist so that we can describe physical quantities correctly; just look back at the example I just gave: there is more to defining torque than just magnitude (such as 5Nm or 15 ftlb), there is also what we engineers call "sense", that is, the direction of the rotation. A negative torque tends to rotate a body clockwise, whereas a positive torque tends to rotate a body counterclockwise.

20. Jun 9, 2007

### stochastic

Thank you so much. I beseech thee, do the world a favor and write a pre-algebra textbook and in the chapter dedicated to this subject use this exact example but also build up to it with an explanation of the terms fulcrum, torque and so on as children at that age will have no clue what this means.

21. Jun 9, 2007

### Werg22

Dr. Proof, the definition of multiplication for negative numbers was put in place far before anyone ever spelled out the word vector, let alone torque. This definition has a far more important raison d'être; it is one of the foundation of algebra.

22. Jun 9, 2007

### arildno

Help!!
Why are all these rats emerging from the sewer??
Maths is first and formost the study of logically consistent systems, irrespective of their "practical" applications.

those who don't understand that has a long way to go.

23. Jun 9, 2007

### Werg22

The confusion mostly comes from the embedded idea that math has to be strictly relevant to the physical world that is taught at the elementary levels. While math did evolve out of the human mind for practical reasons at first, it was later refined into a domain in which practice had no relevance. That is not to say that definitions are chosen arbitrarily; like mentioned, one could say that the definition of a negative times a negative is so to make algebra what it is. However, this is turning in circle, for one could define a negative times a negative and then deduce the rules of algebra instead of the other way around.

Last edited: Jun 9, 2007
24. Jun 9, 2007

### stochastic

ahhhhh, and my last comment was the perfect end to this thread. since that has now been ruined let me ask a new question. it seems that this form of multiplication ( i mean (-x)*(-y) ) is more for conveying a mathematical idea from one person to another rather than to figure out something for yourself. is that true?

Last edited: Jun 9, 2007
25. Jun 9, 2007

### morphism

Well one can define what one wants. An axiom/definition is adopted depending on the richness of the mathematical theory it produces. Of course coming up with the 'right' or 'most useful' definition is a very challenging task. (Take for instance the foundations of topology: it took a very long time and quite some experimentation before the 'proper' definitions were discovered.)

It turns out if we assume some basic facts about how real numbers behave, then x*y=(-x)*(-y) immediately follows as a consequence. As matt and fopc pointed out to you previously in this thread, the two innocent-looking facts that x*0=0 and x-x=0 are sufficient to deduce that (-x)*(-y)=xy. (Although there is slightly more going on here, but this is roughly how the argument goes.)

I recommend you consult the first chapter of Apostol or Spivak (both books having the title Calculus). Then you yourself will be able to provide a logical proof that (-x)*(-y)=x*y. Plus, you'll learn how the real numbers are actually defined. For a more difficult read, you can try Landau's classic: Foundations of Analysis.

Last edited: Jun 9, 2007