The number system Simple question(or maybe not...): Its quite fascinating that with only 10 symbols (for base 10) we can represent any possible natural number. In particular, it seems that no matter how you arrange any two or more numbers, you always get a number. For example: "1233" + "56" + "0000122" = "1233560000122". The same does not apply for the Roman numeral system because of the 3-rule. You can't have "I" + "I" + "I" + "I" = "IIII" because that's and illegal statement. My question is, where can I find a formal proof that the number system we have is "unbreakable" or the opposite, which could be possible as well.

This comes from the division remainder theorem, which tells that for any two integers a, b we can write a = k*b + r where r < b. Hence, for any chosen base we can call every number between 0 and h an algarism, and concatenate them to produce any natural number. Any string of digits will then represent a multiple of the base plus some remainder. This is simply not true for roman numerals, which do not have a fixed base. 
