# Can you have a numeral system with infinite digits?

If you were working in a number system where there was a one-to-one and onto mapping from each natural to a symbol in the system, what would it mean to have a representation in the system that involved more than one digit?

For example, if we let $a_0$ represent $0$, and $a_n$ represent the number $n$ for any $n$ in $\mathbb{N}$, would 'a_1$$a_0' represent a number? Is such a system well defined or useful for anything? - add comment ## 4 Answers The set$${\mathbb N}[X]$$is exactly a system as you describe. The constant polynomials are the whole numbers {\mathbb N}, and they represent the infinite "symbols" in your system, while a_0a_1 is actually the polynomials a_0+a_1X. If you replace \mathbb N by \mathbb Z or \mathbb Q you get some rings which are actually often studied in mathematics. Added When studying the prime factorization of integers, the same type of system actually comes in play. Look at the primes p_1=2,p_2=3,... Then any n >2 can be written as 2^{a_1}3^{a_2}5^{a_3}....p_k^{a_k} where p_k is the last prime appearing in the prime factorization of n. Then the "symbols" would correspond to the powers of 2, the elements of teh form a_0a_1 correspond to the integers divisible by no other prime than (maybe) 2 and 3 and so on. Interesting, the example is similar to \mathbb N[X] and the addition of polynomials in \mathbb N[X] corresponds to multiplication in positive integers. - Very cool! I was expecting a negative answer. – enthdegree Dec 5 at 3:45 add comment In a sense, our usual system is like that. In \LaTeX if you put braces around something it gets treated as a single character. I know that isn't what you are thinking, but to do what you are thinking you would need a countably infinite set of characters, which is what we get with the decimal (or other base) system. If you had them, you could define concatenation to be some operation like multiplication if you wanted. - add comment In base 10, we represent a number n as a sequence of digits n_0, n_1, \ldots such that$$n = \sum_{i=0}^\infty n_i 10^i\qquad\text{where } 0\le n_i<10$$and we require that the sequence of n_i must be eventually zero. By changing the representation a little bit, we get the so-called factorial base:$$n = \sum_{i=1}^\infty n_i i!\qquad\text{where } 0\le n_i<i+1$$and again the sequence n_i must be eventually zero. There is no upper bound on the size of the digits n_i. In this representation, the number 718 is represented as \langle 0,2, 3,4,5\rangle since$$\begin{align}5\cdot 5! + 4\cdot 4! + 3\cdot 3! + 2\cdot 2! + 0\cdot 1! & = \\ 5\cdot120+4\cdot24 + 3\cdot6 + 2\cdot 2 + 0\cdot 1 & =\\ 600 + 96 + 18 + 4 + 0 & = 718.\end{align}\$

This has actual applications; for example it is a useful way to represent a permutation of a list.

-