What is the Flaw in the 'Proof' That -1 Equals 1?

[spec138]

Hi all. I found this "proof" and was just wondering if there is an error in it or not, because I couldn't find it. Any ideas?

-1/1=1/-1
sqrt(-1/1)=sqrt(1/-1)
sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)
i/1=1/i
i*i/1=i*1/i
i^2/1=i/i
-1=1 ?

 

[Coin]

sqrt(-1/1)=sqrt(1/-1)
sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)

The transition between these two steps is invalid. There is no reason you would be allowed to do this.

There is a basic property of the square root function that sqrt(a)*sqrt(b)=sqrt(a*b) where a and b are positive, but -1/1 isn't positive so we don't get to use that here.

I've actually seen this one before a page promoting "Time Cube" theory!

 

[belliott4488]

The problem comes from writing simple "sqrt" functions rather than "+/-sqrt". IOW, whenever you take the square root, you have to take into consideration that there are two square roots of any complex number, which differ by a factor of -1. Deciding which roots to take to maintain an equality often requires working through exactly the kind of computation you have shown.

I would have looked at the second line in your "proof" and decide which sign each sqrt should have. To do that I would go through pretty much the proof you have, and when I got "-1=1", I would say, "Oh - that's not it - I guess the two sqrts have to have opposite signs to maintain the equality."

I know that looks circular, but it's really how you decide which root to take. It might be clearer with pure real numbers:

(-2)^2 = (2)^2
sqrt((-2)^2) = sqrt((2)^2)
-2 = 2

oops ... should have had a minus sign in line 2!

 

[spec138]

Just to add this wasn't my proof and I knew it wasn't true. I just couldn't find the mistake. Thanks guys.