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.


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    $\begingroup$ No, $\mathrm{i}$ is not a rational number in the traditional sense (though there is the idea of a Gaussian rational, which includes numbers of the form $a+b \mathrm{i}$ with $a,b \in \mathbb{Q}$, and the Gaussian integers). $\endgroup$ – Morgan Rodgers May 10 '17 at 18:35
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    $\begingroup$ But "get Gaussian" doesn't work in a joke. $\endgroup$ – Robert Israel May 10 '17 at 18:36
  • $\begingroup$ (FWIW, I've always thought this joke was bad for precisely this reason) $\endgroup$ – Morgan Rodgers May 10 '17 at 18:37
  • $\begingroup$ "Get rational" isn't an idiomatic expression; the joke usually goes "Be rational". $\endgroup$ – Rahul May 10 '17 at 19:08

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.

  • $\begingroup$ So... does that make $i$ a... Gaussian rational or something? $\endgroup$ – Tachytaenius May 11 '17 at 10:49
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    $\begingroup$ Yes, that's right. Gaussian rationals are numbers that can be written as a ratio of Gaussian integers. They form what is called a number field, and these sorts of things are studied in algebraic number theory. $\endgroup$ – Qiaochu Yuan May 11 '17 at 17:46

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