A complex number that has transcendental real part is always transcendental? How about in the case of imaginary part?
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The answer to your first question, "A complex number that has transcendental real part is always transcendental?" is yes, since the conjugate of an algebraic number $r$ is a root of the same polynomial that witnesses that $r$ is algebraic. But then the sum of $r$ and its complex conjugate (twice the real part) is algebraic as well -- the algebraic numbers form a field, so they are closed under addition, multiplication, inverses...
Similarly, if the imaginary part of a number is transcendental, then the number is transcendental, by essentially the same argument (now you would consider the difference between $r$ and its conjugate).
On the other hand, if the real part of a number is transcendental, we cannot conclude that the imaginary part is transcendental as well. For example, look at $\pi$, or $\pi+i$. Similarly, if the imaginary part is transcendental, we cannot conclude anything about the real part.
Yes, because the set of algebraic numbers is closed under complex conjugation, addition, and multiplication.