Refining my knowledge of the imaginary number So I am about halfway through complex analysis (using Churchill amd Brown's book) right now.  I began thinking some more about the nature and behavior of $i$ and ran into some confusion.  I have seen the definition of $i$ in two different forms; $i = \sqrt{-1} $ and $i^2 = -1$.  Now I know that these two statements are not equivalent, so I am confused as to which is the 'correct' definition.  I see quite frequently that the first form is a common mistake, but then again Wolfram Math World says otherwise.  So my questions are:


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*What is the 'correct' definition of $i$ and why?  Or are both definitions correct and you can view the first one as a principal branch?

*It seems that if we are treating $i$ as the number with the property $i^2 = -1$, it is implied that we are treating $i$ as a concept and not necessarily as a "quantity"?

*If we are indeed treating $i$ as a concept rather than a "quantity", how would things such as $i^i$ and other equations/expressions involving $i$ be viewed?  How would such an equation have value if we treat $i$ like a concept?
I've checked around on the various imaginary number posts on this site, so please don't mark this as a duplicate.  My questions are different than those that have already been asked.

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*As you probably know, there are two solutions to $x^2+1=0$. We arbitrarily call one of them $i$ and the other $-i$, but this choice could have been made the other way (which is why complex conjugation is an automorphism of $\mathbb C$). So, both definitions are essentially correct, but of course, the first one is slightly less correct.

*What is the difference between a "concept" and a "quantity"? This question seems more philosophical than mathematical...

*Well, $i^i$ is usually defined as $e^{i\log i}$, where $e^{a+bi}=e^a(\cos b+i\sin b)$ and $\log(re^{i\theta})=\ln r+i(\theta+2n\pi)$, where $n$ is any integer (the logarithm is a multivalued function since the exponential function is not injective)
