Peter: Thank you for your "plenty" answer, and for wishing me.

Let us not overlook the simplest answer of all: every (natural) number can be written in binary, that is, a finite string of 0s and 1s. That is exactly how computer memory looks theoretically. You can store anything in it: Shakespeare's Hamlet or a picture of your wife.

I wouldn't use Goedel numbering for that.

While in this era of computing, the more simpler way a computer understands is the binary language, though it may be tough enough(in one or other way) for us, but all and every number is converted to the very lower level of Binary 1s and 0s . And it's easy for us to implement in the electronics field while designing certain logic. 

So, if we are to implement some more complex numbering system that would require may be a library of definitions or a much larger computing time but whatever be it computer will always change it to the very basic of binary.

So I agree more with Mr. Marcel Van de Vel . 

I can't disagree with any of these approaches, because to every process, there is some logic associated.

I would suggest using Lambda Calculus to design a model that would understand the significance of numbers, and their properties: quantitative and qualitative.

Peter: you can't encode every real within a single real number: you can only encode countably many of them!

There are also other amazing encodings in the naturals. Consider the Cantor pairing function C(x,y) = (x+y)(x+y+1)/2 + y, which is a bijection N x N ---> N. Now every finite string of natural numbers, of arbitrary positive length,  can be uniquely encoded in a natural number in the following way: (x_1, ..., x_n) ---> C(n-1, C(x_1, C(x_2, C(....., C(x_(n-1), x_n)))))...). I think that the decoding is faster as by using the prime factor decomposition encoding, because now one has only to compare with triangular numbers, which is equivalent to repeatedly computing integer parts of square roots. 

Dear Mihai, thank you for your beautiful answer. The function you've discussed about is computationally not so expensive too. I checked that it was a bijection, which serves a great purpose, as for every natural number we can obtain a finite string of natural numbers, following the inverse mapping of:

(x1, ..., xn) -> C(n - 1, C(x1, C(x2, C(..., C(xn-1))))).

Hello Kumar! Yes, I programed this using Visual Studio 2013 (which is a freeware). It is important to use the reference System.Numerics and to use the type "BigInteger" from this reference. Even very normal short sequences, like 1, 2, 3, 4, 5 - get hudge codes, so we really need BigInteger. The only difficult task is to program the square root for BigIntegers. I take 10^((length of number in digits) / 2). By eventually dividing and multiplying with 10, I get numbers a and b such that a^2

You might also look at the bijection from the natural numbers to the positive rationals in my paper with Wilf, Recounting the Rationals.  If you define 

a_0=1

a_{2n+1} =a_n

a_{2n}=a_n + a_{n-1}

then a_n/a_{n+1}, n>=0,  enumerates all rationals.  This is computationally much more efficient than the Cantor approach if you are erasing the duplications (like 2/2 is a duplicate of 1/1).

Also, if you are not aware of Chaitin's constant, it's something I'd recommend to you: it's almost the reverse of what you asked for: a number about which we know almost nothing!  

http://en.wikipedia.org/wiki/Chaitin%27s_constant

Mentioning Cantor's function rang a bell with me. In a remarkable paper of Tarski (posthumous) and Givant, A formalization of set theory without variables, Colloquium Publications, AMS, 1987, the authors formulated an important concept of "conjugated quasiprojections" (Q-system) as a pair of functions F1, F2, such that for each pair of elements y, z there is an element x with F1(x)=y and F2(x)=z. Such function pairs obviously exist in set theory (projections from a Cartesian product), but also in Peano Arithmetic. Tarski and Givant give the following function pair (all numbers are natural):

F1(x,y) means (exists a) x = (y+a)2 + y

F2(x,y) means (exists b) x = (y+b)2 + b

(page 216). It can easily be shown that, given x, there is one y with F1(x,y), making F1 a function. The same holds for F2.

Given any natural numbers y, z, the unique x with F1(x)=y and F2(x)=z is (y+z)2 + y. I interprete this unique x as a record with two fields, holding the respective values y and z. From this fact the (lengthy) paper derives (among other things) that Peano Arithmetics can be interpreted into some kind of hereditarily finite set theory.

The question in this thread is unexpectedly rich in content. In attachment my 2002 paper (available on RG) in Studia Logica on enriching logic theories with record-like structures.

Article Interpreting First-order Theories into a Logic of Records

 @ Marcel

The left and right projections of the Cantor bijection (defined by C(L(n), R(n)) = n) have the same properties as your F1 and F2, and are also definable. I suppose that Tarski and Givant decided to use this easier form - based on squares rather than triangular numbers - because they didn't need every number n to be the code of a pair. Also, their formulas are not using any constants, (neither 1, nor 2) and also do not need division or subtraction - just addition and multiplication.