Implied Functions

What? i is defined to be sqrt(-1). The only way I can possibly see this being possible is:

 Spoiler

sqrt(-1) = i

sqrt(-i^4) = i

sqrt(i^4)*sqrt(-1) = i

i^2*sqrt(-1) = i

-sqrt(-1) = i

sqrt(-1) = -i

You already said that this is false, because the property:



doesn't hold in the complex domain.

The precise definition of i varies from source to source. However, all definitions leads to i^2 = -1, which is what we're after.

Now, (-i)(-i) = (-1)(-1)(i)(i) = (i)(i) = -1
and (i)(i) = -1. Therefore, both -i and i is "the" square root of -1. (That's why people don't say that i = sqrt(-1), because that implies that i = -i, which is untrue. It is however true that i is one possible value of sqrt(-1).)

So, if you have a sqrt(-1) somewhere, you need to either square it to get it to i^2, or you need to substitute it with ±i instead.

Noah