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There are many ways to express $\pi$ by infinite application of some simple operation (+,-,/,*,^) .

Is there a method that represents all real numbers uniquely? By method, I mean a restriction to certain operations applied in a certain way, such that all real numbers are expressible, but they cannot be expressed in two different ways.

Examples of representations of $\pi$:

$\pi= {3 + \cfrac{1^2}{6 + \cfrac{3^2}{6 + \cfrac{5^2}{6 + \cfrac{7^2}{6 + \ddots\,}}}}}$

$\pi = \cfrac{4}{1 + \cfrac{1^2}{3 + \cfrac{2^2}{5 + \cfrac{3^2}{7 + \cfrac{4^2}{9 + \ddots}}}}}$

What I would like the most is a natural way to restrict the class of infinite products to one that is unique for every real number. The generalized prime factorization representation $x = \prod_{i \in \mathbb N} p_i^{f(i)}$ where $p_i$ is the $i$th prime number, and $f : \mathbb N \rightarrow \mathbb Z$ is a function is an example of an infinite product representation. I.e.: What should $f$ satisfy, if it should be possible to represent all real numbers uniquely? Here, it would be nice if the representation coincided with the following property: The only $f_a$ describing the elements $a \in \mathbb Q$ corresponds to the exponents of the primefactors in the numerator and denominator in the representation of $x = p/q$.

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You can use the continued fraction representation. You can use the decimal expansion, if you avoid infinitely repeating $9$'s. You can use a harmonic sum: for $x \gt 0$ add up all the terms in $\sum_n \frac 1n$ that keep it less than $x$, for negative $x$ subtract them. None of these are products, but the last two are sums, so you can represent $\ln x$ and then you can take the exponential.

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    I was confident you were thinking that (and it is indeed what I describe also) but it wasn't explicit so I made it so.2012-10-19