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 $x = p/q$.