1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = i * i = -1

Thanks.

On the non-negative reals, for every x there is exactly one y with y^2 = x. So the function sqrt is a natural thing to work with. it has nice properties, for example sqrt(ab) = sqrt(a)sqrt(b).

When you move on to the reals, things are trickier. Any strictly negative real has no square roots in this domain, but any positive real has 2. So it is normal to do without a sqrt function on this domain, since it would have no value at -1. Instead the function sqrt from the non-negative reals to the non-negative reals suffices. For positive reals, this selects 1 of the two possible roots in a nice way: as I mentioned above, sqrt(ab) = sqrt(a)sqrt(b).

But when you get to the complex numbers you have a problem. For every non-zero complex number there are two numbers which have that number as their square. It would be nice to have a function sqrt from C to C which:
1. Satisfies (sqrt(x))^2 = x for all x.
2. Satisfies sqrt(ab) = sqrt(a)sqrt(b).
However, as you have shown in your post, there is no such function.

This gives a minor problem to those who say 'We define i to be sqrt(-1)'. So it is better (if you are a pedant) to stick with 'We let i be such that i^2 = -1'.

I'm sure there are good discussions of these problems and of how mathematicians have tried to cope with them in most textbooks on the subject, but I don't know any specific references myself.

Thankyou

sfwc
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