I think it's useful and interesting to point out exactly why fake proofs like the one in the OP don't work. Someone who isn't already familiar with basic logic could learn a lot about it that way. But in the spirit of trolling: Like you're probably aware, the square function defined over the reals isn't bijective and thus doesn't have an inverse. Actually in mathematics and most programming languages I know, the power operator binds more tightly than the negative sign, so \(-1^2 = -1\). Problem solved!
Another good one (for kids): Assume a = b Then it follows that 2=1 because... Multiplying both sides by a, we get: a^2 = ab Subtracting b^2 from both sides: a^2 - b^2 = ab - b^2 Factoring, that is: (a + b)(a - b) = b (a - b) Simplifying, we get: (a + b) = b Since a = b, by assumption: (a + b) = (b + b), and so we get: (b + b) = b That is: 2b = b Dividing both sides by "b": 2 = 1
This is a very nice example of what przyk was talking about, and it's a lot more subtle than the OP. I'll give you a clue as to where the logical misstep lies so as not to spoil it for anyone who wants to think it through. In this step: What operation is being done?
Because there's no logical reason to say it. If sqrtr(1)= sqrt(1) then all we know is that 1 = 1. You must be confusing this with (-1)[sup]2[/sup] = (+1)[sup]2[/sup] and then taking the sqrt of both sides. Once you do that you'd get |-1| = |+1| ===> 1 = 1
