Closed form representation of an irrational number

Question

Can an arbitrary non-terminating and non-repeating decimal be represented in any other way? For example if I construct such a number like 0.1 01 001 0001 ... (which is irrational by definition), can it be represented in a closed form using algebraic operators? Can it have any other representation for that matter?

Answer 1

In general, no, since, for one thing, there's an uncountable infinity of such decimals, and only a countable infinity of closed forms (under any reasonable definition).

Answer 2

Since $0.1 = \frac{1}{10}$, $0.001 = \frac{1}{10^3}$, $0.0000001 = \frac{1}{10^6}$. Making a guess that $n$-th term is $10^{-n(n+1)/2}$ the sum, representing the irrational number becomes $$ 0.1010010001\ldots = \sum_{k=0}^\infty \frac{1}{10^{\frac{k(k+1)}{2}}} = \left.\frac{1}{2 q^{1/4}} \theta_2\left(0, q\right)-1\right|_{q=\frac{1}{\sqrt{10}}} $$ where $\theta_2(u,q) =2 q^{1/4} \sum_{n=0}^\infty q^{n(n+1)} \cos((2n+1)u)$ is the elliptic theta function.

Answer 3

Some irrational numbers can be expressed in a closed form using algebraic operations; $\sqrt7$ is a very simple example. Some can be expressed in other ways, like $\pi$ for which a multitude of formulas is known. Most real numbers however cannot be expressed (using a finite amount of information, but that is implicit in "expressing") at all, since there are just too many of them.