Are there methods to recursively calculate the decimal expansion of real numbers?

Question

Using the concept of self-similarity, it's possible to encode the decimal expansion of a number as a sort of 'fractal' object. For instance, consider the sequence,

$$(1) \quad C_0=0.1, \ C_1=0.101, \ C_2=0.101000101, 0.101000101000000000101000101,...,C_n$$

The astute reader will notice this is analogous to the construction of the Cantor Set. The number I'd assume is irrational. However there is a fairly simple way to construct the number, and thus find it's decimal expansion. In fact, the number $C_n$ satisfies,

$$(2) \quad C_{n+1}=C_n+C_n \cdot 10^{-2 \cdot 3^{n}}$$

Do similar methods exist for other reals such as $\sqrt{2}$ or $\pi$? If they do, how are these methods developed?

Answer 1

It depends on the real number, but most "normal" irrational numbers will not have such constructions. This is intuitive as any such method would finitely expressed and there would only be countably many of them. $\pi$ and $\sqrt 2$ have several series expansions that allow as to calculate the decimal expansions. But your "arbitrary" real number will not. We do not even have any method to describe an arbitrary real number.

Answer 2

It depends on the definition of the real in question. Some, like $\sqrt 2$ and $\pi$, are defined precisely enough that their decimal expansions can be computed to arbitrary precision. For the two reals just mentioned, the methods used aren't the same, and each takes advantage of aspects of the defined real peculiar to that real (or, a class of analogous reals).

However, there are clearly-defined reals whose decimal expansions are not computable: Chaitin's constants. The wiki article just cited summarizes these as defining a "halting probability" — "a real number that informally represents the probability that a randomly constructed program [in a particular formalism, e.g. Turing machines, systems of equations, etc.] will halt". Each formalism for computation has its own particular halting probability, i.e. Chaitin constant. It goes on to mention that "each halting probability is a normal and transcendental real number that is not computable, which means that there is no algorithm to compute its digits".

In general, there are (many, many) more reals than there are algorithms, and for the "typical" irrational, there is no procedure for computing its decimal expansion.