Edit:
To those voting to close as 'opinion based', the core question (which should satisfy the 'opinion based'-criteria) is this: Is there any specific reason why an imaginary number can't be classified irrational; additionally, are the properties of traditionally irrational numbers, when on the imaginary-plane, different from those numbers when on the real-plane?
Original Post Follows:
I was discussing irrational numbers and mentioned $πi$ when, to my surprise, I was confronted by a colleague who declared that $πi$ was not irrational since irrational numbers have to be real. I didn't agree with him, but after some googling, to my surprise he was right.
By current definitions of irrational, a number must be real to be irrational. Ironically, transcendental numbers are defined as being able to be real or complex.
Online Definitions:
In mathematics, the irrational numbers are all the real numbers which are not rational numbers
In mathematics, a transcendental number is a real or complex number that is not algebraic
Is there any specific reason why an imaginary number can't be irrational (other than the definition of irrational). As far as I know, $πi$ contains all of the properties of an irrational number (other than not being on the real-axis).
If you defined irrational numbers as $\mathbb C \setminus \mathbb Q$ rather than $\mathbb R \setminus \mathbb Q$, then you would be in the uncomfortable position of calling both $i+1$ and $\sqrt 2+\pi i$ irrational, even though the first looks almost like a rational, even an integer, whereas the second looks more like what we expect from an irrational.
Instead, it's cleaner to define Gaussian rationals as those complex numbers $a+bi$ where both $a$ and $b$ are rational. So the first example above is a Gaussian rational (in fact a Gaussian integer), whereas the second is not.