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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 $\sqrt{-1}$?

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    Observe that $|i|=1$. The general formula for the absolute value for complex numbers is $\sqrt{z\bar z}$ where $\bar z$ is the conjugate of $z$. If $z$ is real the conjugate is itself, hence if $r$ is a real number then $|r|=\sqrt{r^2}$. – 2017-01-28

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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 $i$.

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    You could add for clarity that in this case $i = (0,1)$ – 2017-01-28
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    Defining $i$ is defining complex numbers, so this may be the same. A duplicate anyway. – 2017-01-28
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    My point is that we don't really have to define what $i$ "is", we can treat it the same way as any element of a field. We don't need to "define" arbitrary elements in fields, we just say what they do. – 2017-01-28