Is this rule/convention something mathematicians simply agreed upon, like always rationalizing the denominator when it contains a radical expression, or is there a logical/mathematical reason for this?
My current understanding of math is only up to pre-calculus, so any explanation beyond that will most likely go over my head.
Best Answer
There can be complex non-real numbers in the denominator. It just happens that if you have an expression of the type$$\frac{a+bi}{c+di},$$with $d\neq0$, if you convert it to$$\frac{(a+bi)(c-di)}{(c+di)(c-di)}=\frac{ac+bd+i(bc-ad)}{c^2+d^2},$$you get, in general, a more readable expression.
But you are not required to do this. Complex Analysis has lots of formulas that begin with $\frac1{2\pi i}$ and, as far as I know, nobody ever suggested that one should use $-\frac i{2\pi}$ instead.