Proving (-1)(-1)=1: A Curious Solution

[JasonRox]

It seems so simple, but I can't think of a formal way of doing it.

I never did proofs, so this would be good practice.

This was a question the prof asked for curious ones to answer for the next lecture (he may forget about it), and it is not important.

This is what I got, but there is some conflict:

If x^2=1 - x=\pm 1, than (-1)^2=(-1)(-1)=1.

It does make some assumptions, but I can't see a way around it.

I can also use the definition of multiply, which means that it is the sum of a number taken a stated number of times.

Because of this, the stated amount of times is -1.

Adding this would be like this -(-1)=0. If it were -2(-1)=2, than -(-2)=2. Something a little more complicated (-2)(-3)=6, than -(-2)-(-2)-(-2)=6.

Can someone help me here?

PS. I want to impress the class. j/k!

 

[HallsofIvy]

In any ring (the most general thing in which we have both addition and multiplication) -1 is defined as the additive inverse of 1: 1+ -1= 0.
(and therefore 1 is the additive inverse of -1)

-1(1+ -1)= -1*1+ -1*-1 (by the distributive law)

But -1(1+ -1)= -1*0= 0 (do you know how to prove this?)

So -1*1+ -1*-1= 0

Of course, -1*1= -1 (since 1 is the additive identity) so this is the same as

-1+ -1*-1= 0 which says that -1*-1 is the additive inverse of -1. Since we already know that the additive inverse of -1 is 1 (and that the additive inverse is unique), we have -1*-1= 1.

1+ -

 

[Alkatran]

How do determine what you can and can't use when verifying a proof? For example, I would use -1*-1 proving some other proof, but I obviously can't use it here.

If you can show that something you used to prove -1*-1 depends on a 'higher level' proof, is it unsound?

 

[robert Ihnot]

HallsofIvy has given a really good demonstration, which is what the student needs.

But I can not resist stating that when I read Courant's "What is Mathematics," that, if I can remember correctly, he tended to debunk these kind of proofs, which he said could be done by arguing that either (-1)(-1) = 1 or -1, and it can not equal -1. Courant believed that such things were embedded in the axiomatic system and were really not proofs.

His institue got a lot of funding from the government and devoted itself to applied mathematics, so it was somewhat skeptical of axiomatic approaches and too much abstract theory.

Someone said Auden, the poet was fond of this:
Minus times minus equals plus, The reason for this we need not discuss.

It is then said: I therefore think this verse should be taken as a parody designed to ridicule secondary-school pedagogy in mathematics.

Check it out: http://mathforum.org/epigone/math-h...01140943.D19676-0100000@math.math.ucdavis.edu

Please excuse this, as I know the student needs to know, and proofs are important. In my case, I never really understood Abstract Algebra very much anyway.

 

[ahrkron]

It depends on what you are doing. In homeworks you usually know what material you can and cannot use.

In other situations, you may be interested in the properties of one operation given some assumptions. You may need to work a little to isolate all the assumptions you are using, but the context usually helps you in defining what it is that you can keep as your lowest level.

 

[recon]

I'm not (-) going to not (-) help you = I am going to help you (+).

This is not a sound mathematical proof. But it is how I think of the problem anyway.

 

[matt grime]

Well, for the Courant reader, the "proof" must be embedded in the axioms otherwise it wouldn't be true. However when axiomatizing a system we tend to stick the minimal number of axioms, though sometimes there is still redundant information.
We can agree that it is a tedious proof, but it is one of those things everyone should do once and never again.

 

[robert Ihnot]

In any ring (the most general thing in which we have both addition and multiplication) -1 is defined as the additive inverse of 1: 1+ -1= 0.
(and therefore 1 is the additive inverse of -1)

The great thing about this proof is the assumption of number properties that are very basic, but this could be a very sticky question if we had to go over the actual axioms supplied by his teacher. However, JasonRoy stated that this was not an important problem and possibly there was no axioms even mentioned!

If we had to use axioms we could begin with the Peano postulates, but Henri Poincare tells us they are only useful if consistant.

 

[JasonRox]

Exactly, he did not mention axioms, and also did not mention the proof either. He did not forgot; he just doesn't really care.

I do understand what you were saying above.

I never did proofs before, but now that I know we need axioms to do proofs, I know a little bit more. :)

I didn't meet any students who tried yet, so I guess it's no big deal. I just liked the idea of practice.

Note: I don't have to impress the class because one guy is always trying, yet he does not know it makes him look like an ass.

 

[kronecker]

Here i think formal to me.
Let x=-1 (*), also means -1=x and x^x=(-1)(-1)
plus 2 for bothe sides of (*), we then have 2+x=1
multiply x for both sides 2x+x^2=x
this means (-1)(-1) +x=0;
So, (-1)(-1)-1=0 <=> (-1)(-1)=1

 

[BobG]

Here i think formal to me.
Let x=-1 (*), also means -1=x and x^x=(-1)(-1)
plus 2 for bothe sides of (*), we then have 2+x=1
multiply x for both sides 2x+x^2=x
this means (-1)(-1) +x=0;
So, (-1)(-1)-1=0 <=> (-1)(-1)=1

A person with the handle 'kronecker' shouldn't believe in negative numbers.

 

[Damned charming :)]

IMO -*- =+ is basically an axiom. You can show that it is a corollary of basic axioms like HallsOfIvy but I would also give examples that show that the definition works well in reality and it is reasonable to define -1*-1. In the most basic sense if if you owe 5 people 10 dollars each killing them does the same for for your paper assets as earning 50 dollars so -5 *-10 = 50 (Its mathematically the same but not ethically the same).

To show you why the definition works better for physics a better example consider that you see a magic meteor fly past your window 1000m up.
at a time you call time = 0. The meteor flies at 100m per second.
height of meteor = 1000m - 100* time
so h = 1000 - 100t
quite reasonable since when t = 0, h = 1000 and when t=10 h=0. now how high was the meteor 1 second before it passed your window. well t = -1.
so h = 1000 -100*(-1) = 1100.

basically all that humans have a great deal of "common sense" and world understanding you can see the answer with actually thinking -*- is positive.
for Algebra has to partially capture common sense it needs a few axioms (and corollaries of axioms) that people find a bit strange.

Another example is coulombs law

http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Kevin_Jones/coulombslaw.html


q1 and q2 are charges since opposites attract and likes repel for this formula to make sense -*- must have the same sign as +*+ and +*+ is clearly positive. For the formula to work -*+ and +*-

 

[jxiemail]

Let me try a proof

1. Definition of (-1)

The most popular definition is: 1 + (-1) = 0

Let's start from here.


2. Theorem: 0-1 = (-1)

From the definition 1 + (-1) = 0, we have:

1 + (-1) -1 = 0 - 1 [Subtraction Property, both sides subtract "1"]

(-1) + 1 -1 = 0 - 1 [Commutative Property]

(-1) +(1-1) = 0 -1 [Associative Property]

(-1) = 0 -1


3. Prove (-1) x (-1) = 1

Base on the above Theorem, we have

(-1) x (-1) = (0-1) x (0-1) [Substition Property]

= 0 x (0-1) - 1 x (0-1) [Distributive Property]
=0 - (0-1)
=0 - 0 + 1
=1

[Note: "-" here is "to subtract", and has nothing to
do with the concept of "negative"]

Thus proved.

IMPORTANT: There has been an ASSUMPTION here:

The new defined number "negative numbers" follows the
associative, distributive and commutative properties.

If you don't force the "new number" to follow these
laws, then (-1) x (-1) doesn't need to be equal to "1".

However, if you REQUIRE this new number follow these
laws just as the "positive numbers", then, it has to be "1".

 

[arildno]

"subtract" has no place in the axioms of arithmetic.
Nice try, though

 

[robert Ihnot]

Have you guys every heard of Parkerson's Law, "Work expands so as to fill the time allowed for its completion"? Here is another one, When the committee meets the time spent on an expenditure is inversely proportion to the expense.

This is illustrated by a committee which meets to decide to vote $2 billion for a nuclear reactor and $30 for coffee.

Obviously, the $2 billion is unanimously agreed upon without debate, and the day is spend arguing over a $30 expense for coffee.

 

[arildno]

Methinks you think the issue outthinked somehow..

 

[robert Ihnot]

I was just overcome by the humor of that parkenson's Law, and could not resist. Of course, it is an important and interesting question (-1)(-1) = 1. There is no reason not to continue with this subject. However if the student has to learn to total the grocery bill, I doubt his parents would be happy to hear how he spent this week learning about (-1)(-1) =1. That is why I made that comparison.

Gottieb Frege did a tremendous amount of work in the field of foundations, but encountered contradictions, one pointed out by Russell. He put a lot of effort into studying basics like the number one. He influenced mathematicans such as Peano. You might want to look at http://plato.stanford.edu/entries/frege/#proof