# Canonical “-1 = 1” answer?

There are quite a few questions which involves "showing" $1=2$ or $0 = 1$ or $-1 = 1$ via incorrect algebraic manipulations of $i = \sqrt{-1}$.

Does anyone know if there is a "canonical" question/answer (and if not, would one of our great educators write an answer) that is easily generalisable to at least the usual arithmetic operations?

I'm wondering because of the question " Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$? ". The suggested "duplicate" target is " Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$? ", but I think that for someone who is having difficulties seeing why his/her algebraic manipulations are wrong, the connection between the two questions may not be immediately apparent. (In other words, the answer to the second question may not help.)

And ideally a question of this type should be added to our list here: http://meta.math.stackexchange.com/questions/1868/list-of-generalizations-of-common-questions

• I think that canonical answer should be $\mathbb Z/2\mathbb Z$ :-D – Asaf Karagila Jul 3 '11 at 18:28

I $\TeX$-ified my answer to -1 is not 1, so where is the mistake? and added explicit mention of the newer question that you're asking about and the multiplication-of-radicals property. Offhand, I think that trying to get much more general than what I've written there is, as you suggest, likely to be too general for the people asking such questions.