Question

I am trying to get the value of imaginary numbers. The expression is
`i^2`

and now if I evaluate it like this,

`i^2 = (√(-1) )^2\\= -1`
I get -1
Again, if I evaluate it like this,

`i^2 = i * i\\= √(-1) * √(-1)\\= √(-1 * -1)\\= √(1)\\= 1\\`
I get 1
So there are two values -1 and 1. Which one is correct? If either of these two is incorrect, why?

Answer 1

Obviously 1 ≠ -1, so both conclusions cannot be true.

The short answer is that √(ab) = √a √b is always true if a and b are both non-negative integers, but not always true otherwise. The case here is one in which the identity does not apply. "√a" literally means "the positive number such that its square is a". But you're using the convention that √(-1) = i, and i is not a positive number, so this identity does not apply.

The correct result is i ² = -1.

That's not to say that the identity above is always wrong whenever a or b are not non-negative. For example,

√(-25) = √((-1) × 5²) = √(-1) × √(5²) = 5i