There's this maths joke, where $i$ says to $π$, "get rational!" while $π$ says to $i$, "get real!" (I like to say that $e$ says to the both of them, "join me, and we will absolutely be one!" (don't forget that $abs(-1) = 1$, and if needed, look at Euler's identity.))
I thought, however, "is $i$ rational?" If not, they're being a hypocrite in the joke above!
Now: more mathematics; less... sociology.
The definition of a rational number, to the best of my knowledge, is a number which can be represented as $$\frac{p}{q}$$ where $p$ and $q$ are both integers (which, as may be the reason I'm asking the question, is a term I don't understand on the purely abstract level (after all, I could make a number system where "0.5" is equal to $1$.)
So then I was thinking, "well, if $i$'s rational, I can represent it like the $\frac{p}{q}$ above. I know that $i = \frac{i}{1}$, but of course, although every rational number can be represented like that, not every number that can be represented like that is rational." So this is where I got stuck. If $i$ and $1$ are integers ($1$ definitely is,) $i$ is rational.
But... is $i$ an integer‽
I'd like to think so, for the sake of the joke, and because my natural gut feelings think so, but I'm really not sure. There's a lot of floating cloudy stuff in my mind saying so, but I'm looking for some solid evidence that says so or otherwise.
Thanks.
As Morgan Rodgers says, $i$ is not an integer. (You are correct that this is a sociological question and not a mathematical one; there is a convention that we agree upon for how to use the word "integer" and the convention is that the integers are the numbers $0, 1, -1, 2, -2, 3, -3, \dots$.) Instead it is what is called a Gaussian integer.