This question arises from a comment I recently read in another question.

My question is whether we can represent every real number using only finite memory. I will clarify what I mean by *represent using only finite memory* by use of examples:

- $5$ can be represented in finite memory simply by itself as a one-character string.
- Similarly for $1.234583$, which can also be represented by a string of finite length.
- $\pi$ can also be adequately represented in finite memory:
*it is the ratio of any circle's circumference to its diameter.* - $e$ we can represent as $\displaystyle\lim_{n \rightarrow \infty} \left(1+\frac1n\right)^n$
- $0.818181\ldots$ can be represented as $0.\overline{81}$ or $\frac{9}{11}$.
- $0.010011000111\ldots$ can be represented as the sum of some sequence $a_n$ as $n\rightarrow \infty$.

For all the examples above, an adequate representation of the given real is possible using only finite memory, because we can describe/define exactly the given real using a string of finite length.

So do any reals that cannot be described/represented in finite memory exist? For which their only closed-form expression requires a string of infinite length? (Infinitely many digits?)

Relevant Reading Material Includes:

Is it possible to represent every huge number in abbreviated form?