TITLE: Are transcendental numbers computable?
QUESTION [27 upvotes]: Wikipedia states: "The computable numbers include many of the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers." 
I remember my professor saying incomputable numbers are transcendental. Is this so? Or am I to suggest that some transcendental numbers such as π and e are computable but others are not. If so are there any examples of transcendental numbers that are not computable? Or any numbers which are not computable?
Thanks in advance!

REPLY [23 votes]: Yes, every incomputable number is transcendental, or, differently said, every algebraic number is computable. (Because it is possible to compute an arbitrary close rational approximation to every algebraic number).
As you noted, not every transcendental is incomputable. Consider for instance $0.101001000100001\dots$ (or $\pi$ or $e$).
A counting argument shows that nearly all numbers are incomputable: there are only countable many algorithms, so only countable many computable numbers, but there are uncountable many real (or complex) numbers.