TITLE: Are some real numbers "uncomputable"?
QUESTION [28 upvotes]: Is there an algorithm to calculate any real number. I mean given $a \in \mathbb{R}$ is there an algorithm to calculate $a$ at any degree of accuracy ? 
I read somewhere (I cannot find the paper) that the answer is no, because $\mathbb{R}$ is not a recursive set since it's not a countable set.
But I'm missing something... can someone explain me in detail why so ?
Moreover it means that there exist real number that are uncomputable, and that you cannot neither write down nor express... it's like they don't exist.

REPLY [4 votes]: Precisely, a real number $x$ is computable if there is a total computable function $\varphi_{e}$ such that $|x-q_{\varphi_{e}(n)}|<2^{-n}$, where $\langle q_i \rangle_{i \in \omega}$ is an effective numbering of the rationals. So $\varphi_{e}$ picks out a computably-fast converging sequence of rationals whose limit is $x$. Note that one can then compute the decimal expansion of $x$ to any desired precision (and know when that precision has been reached).
Since each $\varphi_{e}$ corresponds to at most one computable real and there are only countably many such $\varphi_{e}$'s, there can be at most countably many computable reals. Of course, each rational is computable, so there are a countable infinity of computable reals.