How is $i$ defined, really? And why is $i \neq -i$? $i$ is generally defined as $\sqrt{-1}$ which is ambiguous because $\sqrt{x}$ is defined as the positive number whose square is $x$; however $i$ can't be positive since it isn't real.
Ok, so what if we defined $i$ as the number whose square is $-1$? This wouldn't make sense either because it would imply either of the following statements:


*

*$i = -i$

*$(-i)^2 \neq -1$


since the number whose square is $-1$ implies unicity.
This seems pretty basic point but none of the textbooks I've seen give more rigorous definitions than $i = \sqrt{-1}$.

The most common way to see the complex numbers is as pairs $(x,y) $ of reals with the operations
$$
(a,b)+(c,d)=(a+c,b+d),\ \ \ \ (a,b)(c,d)=(ac-bd,ad+bc).
$$
Then you identify the reals with the numbers $(a,0) $ and define $i=(0,1) $. In this case, $-i=(0,-1) $.
