I want to know what is the best way to define the imaginary unit $i$. In many books we see $$i^2 = -1$$ But sometimes there is a $$i = \sqrt{-1}$$ there. Isn't the first definition a little problematic? Because if $i^2 = -1$ then $|i| = \sqrt{-1}$implying that $i$ could be both $-\sqrt{-1}$or $\sqrt{-1}$ which honestly i don't know if its true) So,
It is better to define $i$ directly as $\sqrt{-1}$?
why would $i$ be defined seperately from the other complex numbers? I think it makes more sense to define $\mathbb C$ as the set $\mathbb R\times \mathbb R$ with the operation $(a,b)+(c,d)=(a+c,b+d)$ and $(a,b)\cdot(c,d)=(ac-ad,ad+bc)$.
And then identify the pair $(a,b)$ with the expression $a+bi$. so $(0,1)$ is associated with $0+1i$ which we shorten to $i$.