Hartmanis-Stearns conjecture and the computable transcendental numbers In the 1965 article "On the computational complexity of algorithms" by Hartmanis and Stearns, the authors conjecture that if a real-time Turing Machine computes the real number $r$ in, for example, base 10, then $r$ is either a rational number or a transcendental number.

Is there a computable transcendental number that is not computable by a real-time Turing machine in, for example, base 10?
Let $L$ be an EXPTIME-complete language, and let $r \in (0,1)$ be the corresponding real. Clearly $r$ is computable. The number $r$ cannot be algebraic since the $n$th bit of an algebraic number can be computed in time $n^{O(1)}$ (Datta and Pratap). Since the $n$th bit of any number computable by a real-time Turing machine can be computed in time $O(n)$, $r$ cannot be computed by a real-time Turing machine.