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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?

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Given that there are uncountably infinite real numbers, and only countably infinite closed form expressions, you can say that "most" real numbers do not have a closed form expression. –  Thomas Andrews Aug 31 '12 at 13:30
    
"Can it have any other representation for that matter?" - you could represent it as a simple continued fraction, or as an Engel/Pierce expansion, for instance... –  J. M. Aug 31 '12 at 14:38

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up vote 7 down vote accepted

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).

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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.

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Are there any references on the impossibility of representing all irrational numbers as products of algebraic or similar operations? –  Phonon Apr 6 '13 at 22:34

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.

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what is $\theta_2$? –  ajay Aug 31 '12 at 13:05
    
It's a theta function, q.v. –  Gerry Myerson Aug 31 '12 at 13:30

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