  In this work we study the space complexity of computable real numbers
represented by fast convergent Cauchy sequences. We show the existence of
families of trascendental numbers which are logspace computable, as opposed to
algebraic irrational numbers which seem to required linear space. We
characterized the complexity of space-bounded real numbers by quantifying the
space complexities of tally sets. The latter result introduces a technique to
prove the space complexity of real numbers by studying its corresponding tally
sets, which is arguably a more natural approach. Results of this work present a
new approach to study real numbers whose transcendence is unknown.
