TITLE: Are imaginary numbers also classified into algebraic and non-algebraic numbers, just as real numbers?
QUESTION [6 upvotes]: I know that real numbers can be algebraic or non-algebraic (if they cannot be the solution to an algebraic equation).
Are imaginary numbers also divided into these two categories?

REPLY [4 votes]: Absolutely. The same goes for complex numbers with nonzero real part and nonzero imaginary part.
Consider for example $\pi$, a purely real transcendental number. Multiply it by $i$ to get the purely imaginary number $\pi i$, which is not algebraic either, since it's not a solution to any algebraic equation just as $\pi$ is not either.
If we round $\pi i$ "down" to $3i$, we get not just an algebraic number but an algebraic integer, a solution to the equation $x^2 + 9 = 0$. Let's try $\frac{3i}{2}$, that's also an algebraic number since it's a solution to the equation $4x^2 + 9$.
The main thing to keep in mind is that a lot (but not all) of the properties of real numbers are "rotated" from the real number line to the imaginary number line.
You can play around with Wolfram Alpha, asking it questions like is (1 + sqrt(-7))/2 an algebraic number?