Is there proof for -1*-1 = 1

Syth

It's just something that's been bugging me, is there actually a proof for -1*-1=1. It's fundamental to many thing, and it's never really explained, except appealing to intiuition. Is ther actually a formal proof.

Cyan

When we say -1 we mean the additive inverse of 1 so we have:

1 + -1 = 0 (since they are inverses)

1 + (1)*-1 = 0 (1 multiplied by anything is that thing)

1 = -(1)*-1 (subtract (1)*-1 from both sides)

1 = -1*-1 (the parentheses don't mean anything)


So a negative times a negative means a positive.

A minute's googling turns up: http://mathforum.org/dr.math/faq/faq.negxneg.html

:-)

Cyan

Fair play for asking questions like this, though, and not leaving it down to intuition. Mathematics is only fully useful when it's rigorous. Your questioning of fundamental things and trying to find proofs is admirable. :-)

seamus

Afaik, it's just one of those mathematical fundamentals with no real proof, except through mathematical application.

Infinity would fit into the same category. For example, we can't prove or define that
100 / Infinity = 0
yet, when applied into real-life equations, it works out properly, so it must be true.

I'm wide open to be proven wrong here btw

smiles

Originally posted by seamus
Infinity would fit into the same category. For example, we can't prove or define that
100 / Infinity = 0
yet, when applied into real-life equations, it works out properly, so it must be true.

Well the way i see it it kinda makes sense....

what i do is think that infinity is a *huge* number and getting bigger all the time... so 100/1000000000000000000 is very very near 0, so as the big bit gets bigger then the rest gets smaller.

I think I'll always say 100/infinity "tends to" 0 rather than "equals"

<< Fio >>

Sev

Syth

Cyan: I think there might be a flaw in your argument. Going from
1 + (1)*-1 = 0 (1 multiplied by anything is that thing)
to 1 = -(1)*-1 (subtract (1)*-1 from both sides).
What your actully doing is adding on the inverse of (1)*-1. Since (1)*-1 = -1, you're thus adding on 1.
But you say that the inverse (which is 1) is -(1)*-1. So you're using the result you're trying to prove. Your logic is circular.

Again I'm not really doubting -1*-1=1, it's just that mathematicians are normally so concerned that every little step, every contingency is taken care of, and I'm sure there's a proof out there some where.

ballooba

Originally posted by Syth
Cyan: I think there might be a flaw in your argument. Going from
1 + (1)*-1 = 0 (1 multiplied by anything is that thing)
to 1 = -(1)*-1 (subtract (1)*-1 from both sides).

1+(1)*-1=0
therefore
1=-((1)*-1) ≠ -(1)*-1

Capt'n Midnight

By definition i * i = -1

hence -i * -i = -1
guess what i ^ 4 = 1

but that's another day's working out...

Assuming we agree that you change sides when crossing equations and tell normal a+b = b+a stuff

Take x = -y and agree this means x+y=0
now square both sides and solve

0*0 = x2 + 2xy +y2 =0

so put numbers in for x =1

0 = 1 +2y +y2

-2y = 1 + y2

put in -1 for y

-(-2) = 1 + ( -1 * -1)

1 = -1 * -1

If you just replace y for -x then there are more brackets but you end up with the much more satisfying

x * x = -x * -x

Syth

Capt'n Midnight: I think you may also have made the same mistakes going from
-(-2) = 1 + ( -1 * -1)
to
1 = -1 * -1
What your doing is
-(-2)=1 +(-1*-1)
2=1+(-1*-1)
2-1=(-1*-1)
1=-1*-1
no?
But the only way to show that -(-2)=2 is if -1*-1=1. Which is what is being tried to prove?

Again I know intuitivly that it's true, but there must be a proof, and now I'm determined to find it

Capt'n Midnight

Using only positives and saying X=1 , I got to
0 = 1 +2y +y2
But we already know Y = -1

BUT WE STILL DON'T KNOW what (-1 * -1) is until we the SOLVE THE EQUATION
0=1 + (-1*2) + (-1 * -1)



======================
(y is of course 0 - but I don't want to be accused of having nothing up my sleeve)

x + (-x) = y

now squre both sides

x ( x -x) + -x ( x -x) =y2

x2 + 2*(x *(-x)) + (-x)2 =y2

So let X=1

1 + (2 * -1 )+ (-1*-1) = y2

1 -2 + (-1*-1)=y2

And remembering y=0 and so y2 =0

-1 + (-1 * -1) = 0

ADD 1 to both sides
(since I'm not allowed to change sign when moving across the '=' )

1 + 1 + (-1 * -1) = 1

ie. -1 * -1 = 1

And I absolutely refuse to prove associativity...
or that -1 *1 = -1

Kwizatz Anorak

Nice one Midnight, I like your style. Shame about the slight arithmetic slip at the end but we can see where you got to. Arithmetic slips are quite common among genii anyway.

Good Work!

Kwizatz Anorak

-1 + (-1 * -1) = 0

ADD 1 to both sides
(since I'm not allowed to change sign when moving across the '=' )

1 + 1 + (-1 * -1) = 1

should have gotten

-1 = -(-1 *-1)
we know that -1 = -1 so -(-1 * -1) must also = -1, means that everything in the brackets is equal to 1 as the negative is outside hence -1 * -1 must equal 1. Your mistake was having too many 1's on the LHS

My genii comment, I think it's the plural of genius, I was trying to give you a compliment... :rolleyes:

Capt'n Midnight

From -1 + (-1 * -1) = 0

ADD 1 to both sides (remembering to type in the -Ve sign - D'oh)

-1 + 1 + (-1 * -1) = 1

so (-1 * -1) = 1

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The administrator has specified that you can only edit messages for 2880 minutes after you have posted. This limit has expired, so you must contact the administrator to make alterations on your message - But I'm lazy.

Syth

That seems to be OK. Thank's Cap'n Midnight. It's just something that was annoying me.

And the pural of genius is genii or geniuses.