Implied Functions

a^2 + sqrt(b^4*i^4) = c^2

a^2 + sqrt(i^4)*sqrt(b^4) = c^2

For complex numbers, the formula



is not valid, because of the discontinuous nature of the square root in the complex plane.

In essence, my question is this: can the following ever exist, either as a real number or a complex number?

sqrt(a) = -1

What happens when something that doesn't exist arises? Imaginary numbers were made to handle sqrt(-1). Limits were made to handle indeterminate forms. So what about this?

No. No solutions exist to that equation. And no, imaginary numbers were not made to handle sqrt(-1), they were made to handle equations on the form x^2 = -y. The definition of i is not i = sqrt(-1), but rather, i^2 = -1. You cannot really use the former, though it sometimes work.

Noah