Notation for representing ANY number? i'm working on a mathematics/number-manipulation program, and i was wondering if you could practically have a representation that could holds the value of any number. This would need to include complex numbers, irrational numbers, and any other kind of number. For example, you can represent any rational number with a prime factorization(assuming you'd be willing to use negative exponents). I know that there are also some irrationals that can be represented in a prime factorization/polynomial equation, so i'd be great to have a representation for those. If such a thing is possible(and can be answered without an overly complicated explanation), then i'd wonder about that too. Can you add any number representation with another of the same notation? what about subtract, multiply, use exponents, etc.
Update:
i recently talked with someone who has a very clear understanding, and i now have a much better idea of(in addition to the answer to my question), my question itself. I had assumed that any number that can be referred to, could be calculated to any precision. That was what i meant when i asked my question. It turns out that people argue that this is not true, so my question was perceived as general. I also now know that the answer to my question would be REALLY complicated.
 A: It's impossible, because there are simply too many numbers. Even if your question is a bit imprecise, a "notation" has to be a finite sequence of symbols (you can imagine a text using the 128 characters of ASCII, or similar things).
The problem with that is you will then only be able to name a countable set of numbers, which means that there will be as many numbers adequately described by your notation as there are natural numbers. (It does not mean that you will only describe natural numbers: the set of rational numbers, for instance, is also countable, and that's certainly bigger than $\mathbb N$).
However, Cantor proved that there are uncountably many real numbers. It's not even that hard, the key breakthrough was to have the audacity to conceptualise infinite sets of numbers and compare their "sizes". 
So you will not be able to name most real numbers. To be honest, that also means that most real numbers will never be used by anyone... You can for example take a look at the definition of a computable number to make these things a bit more concrete.
Cantor's first application of this result was quite similar to the question you asked. (Roughly:) Since you can describe algebraic numbers with a finite number of symbols (the golden ratio, for example, is "the second root of x^2 - x - 1", and it's easy to extend this system of notation to all real algebraic numbers), there are countably many algebraic numbers. Because $\mathbb R$ is uncountable, there must be transcendental numbers! That was not the first proof of this fact, but it's a remarkable argument, and its impact on the history of mathematics is hard to overstate.
A: I guess you may take a look at continued fractions. 
It is a representation that lacks versatility and I know nothing about approximation by truncating those fractions (since I guess you were looking for some application), but it is a universal repr for real numbers both rational and irrational.
