Often when I see a formula containing $\mathbf {i}$, it will be accompanied by the definition $\mathbf {i^2 = -1}$. Why don't we just assume that most students of advanced math know what $\mathbf {i}$ is, like we do with other constants?
For instance, the wikipedia article on Gaussian Integers defines $\mathbf {i}$. But in the wikipedia article for Natural Logarithms, it's assumed that the reader knows what $\mathbf {e}$ is.
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Sometimes it is defined $i=\sqrt{-1}$ and sometimes it is defined $i^2=-1$. In some applications of math, the $i$ is used as a subscript for initial conditions so just to avoid any confusion, textbooks often make sure to define it.
Not only that, but some students don't study complex numbers until 2nd or 3rd year university when they have to take courses like differential equations or complex analysis. Some students who see the variable $i$ in a 2nd or 3rd year textbook may be seeing it for the first time so its good to define.
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If you are looking for a formal definition (as given in the wikipedia page) it is best to state the algebraic relation that $i$ satisfies.
Furthermore, if you look at your wikipedia article for the natural logarithms, $e$ is defined in the first sentence. This reflects a general principle where even the most advanced texts define things such as groups even though the reader has definitely seen groups before. This is to just create a general sense of completeness of the text.
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I just wanted to mention: The relation $i^2 = -1$ does not define $i$, since e.g. $(-i)^2 = -1$. Also, it doesn't make any sense in the reals, because there is no real number satisfying this property. Why do I mention that?
Well, somehow you have to construct the complex numbers, and during this construction you just "naturally stumble across" $i$ (one particular element), and that $i^2=-1$ is just an immediate fact. Usually you would define addition and multiplication on $\mathbb{R}^2$ by: $$(a,b)+(c,d)=(a+c,b+d)$$ $$(a,b)\cdot(c,d) = (ac-bd, ad+bc)$$ Then you just set $i=(0,1)$ and that's it, that's a proper definition. Now we identify $a:=(a,0)$ and we have our regular representation of complex numbers:
$$a+ib = (a,b)$$
As for why they mention $i^2 = -1$ so often: Don't know, maybe because $i$ is often used denoting an integer, but surely it can't hurt mentioning it.
There's no harm in defining it.
BTW if you are studying Alternating Current, $i$ is for current so $j^2=-1$ is used in analysis.