Systematic way to represent any irrational number

I'm wondering if there's a way to symbolically (or is there a more lose constraint?) represent ANY irrational number in a systematic way.

You can represent any rational number as two integers and I know you can represent some irrational numbers as sums of some infinite series. Is there any representation that can surely be found for any irrational number?

Now, I'm pretty sure there's an interesting observation that since the set of all possible English sentences is countable, there has to be a great set of numbers that can't be expressed as an English sentence. What I'm looking for is perhaps a way to loosen that rule to allow an uncountable infinity of numbers to be expressed.

My previous research into similar topics has usually led to people saying a question like this is ambiguous because a "representation" is not a very well defined idea. So take your best guess at what the definition might be so that this question makes sense (if there are any).

• If you are trying to systematically enumerate the reals, then of course this can't be done since no enumeration exists at all. What then do you mean by "systematically representing the reals"? – Ittay Weiss Oct 18 '14 at 1:34
• You have described the concept of computable numbers—numbers for which there is some algorithm that describes how to compute the number. It is possible to describe (some) other numbers, and some can even be unambiguously and formally defined, but those irrational numbers can never to computed to arbitrary precision, making (in my opinion) such descriptions not qualify as representations. – Fengyang Wang Oct 18 '14 at 1:42

Every irrational number $x$ has a single unique decimal expansion, in a straightforward way. There is a non-negative integer $n$ and a sequence $d_n, d_{n-1}, d_{n-2}, \ldots, d_0, d_{-1}, d_{-2}\ldots$ of decimal digits for which $$x = \sum_{i\le n} d_i 10^i.$$
Every irrational number has a single unique continued fraction expansion: there is a sequence of positive integers $c_0, c_1, c_2,\ldots$ for which $$x = c_0 +\cfrac1{c_1 + \cfrac1{c_2+\cdots}}$$
But each of these representations gives every irrational number an infinite representation, in terms of the infinite sequence of $d_i$ in the first case, or the $c_i$ in the second case. There is no method for giving every irrational number a finite representation, because it can be shown that for any such system, the family of such representations is countably infinite, and so there must be uncountably many irrationals with no finite representation. (This has been discussed many times on this site; try a search for "finite strings countable".)