Doesn't $i=-i$ from the definition $i^2=-1$?

Question

There is 2 definition about imaginary numbers:-

1. Imaginary unit, $i=\sqrt{-1}$
2. The square of Imaginary unit, $i^2=-1$

But the later is used mostly, as known to me, because of many reasons. And my question is from the later(2nd) definition. Now consider this:- $$ i^2=-1$$ $$=> (1) i=\sqrt{-1} or (2) i=-\sqrt{-1}$$ Now, $$ i=-\sqrt{-1}$$ $$=> i=-i \text{ [from (1)]}$$

But actually $i$ cannot be equal to $-i$. So, where did I actually gone; where is my fault.

Answer 1

You can replace $i$ with $-i$ and get exactly the same algebraic structure and since we have no reason to prefer one over the other we choose the one with less symbols.