I'm a high school student, who started learning about complex numbers literally today. Sorry if I say anything naive.
My question is this: sqrt(-1) x sqrt(-1) = sqrt(1) = 1 [simple algebraic manipulation]
however i = sqrt(-1), and is literally defined as i^2 = -1
yet the algebraic result says that i^2 = 1 ??
have i done a mistake somewhere? if not, can someome explain this?
You are making a mistake first made by Euler in 1770 (see https://www.jstor.org/stable/27642191). In fact, the rule $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ does not apply when both $a$ and $b$ are negative. Therefore, you cannot say $\sqrt{-1} \times \sqrt{-1} = \sqrt{1}$.