Implied Functions

EDIT: @ Noah, sqrt(i) = sqrt(2)/2 + sqrt(2)/2 * i, sqrt(-1) = i, sqrt(a) = -1 has no solution because -1 is outside the range of the square root function.

I meant the opposite, thanks - sqrt(-1) = i. Thanks for the notification.

I haven't really worked with complex numbers, but when I plotted it, I don't see how the discontinuities affect the function other than separate the real from the imaginary parts:

 Spoiler

The imaginary values are not continuous.

i = sqrt(-1)

How is it not used to handle sqrt(-1)?

Because sqrt(-1) = -i too. You could use the property sqrt(-1) = i, but then you also have to solve the equation for sqrt(-1) = -i as well. It is thus easier to just use the i^2 = -1 property. This also usually removes possible square root problems you might run into, such as complex square roots.

Noah