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?
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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"?
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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.
Best Answer
The definition of complex number is given on page 1 of Churchill and Brown's book:
The definition of $i$ is given on page 2:
So to answer your question, $i$ is not defined by the equation $i=\sqrt{-1}$, nor is it defined by the equation $i^2=-1$. Instead, it is defined as a particular ordered pair of real numbers, $i=(0,1)$. Then, given the definition of complex multiplication, one proves that $i^2=-1$.