I ran across a book on p-adic numbers whose first few pages present ideas which I repeat here even less formally. First, p is some fixed positive integer that we will soon require to be prime. Consider integers expressed in radix p. Such an expression is a finite sequence of digits numbered from the right end. In school we learned to add, subtract and multiply such numerals using procedures that proceeded from right to left and did not really require that the sequence be finite on the left end. By this I mean we can produce the right n digits of the output be seeing only the right n digits of the inputs. p-adic numbers admit an infinite number of non-zero digits to the left. Addition and multiplication are as in normal arithmetic. Each digit of the answer can be computed finitely, and thus any and all digits can be computed. They might have been called infinite numerals, but so too might the infinite decimal representations of the reals have been called that. We can allow digits to the right of the radix point as well, but only a finite number of them. In normal numbers 1/3 = 0.33333... but an infinite number of digits is required so this limitation is real. We can still add, subtract, and multiply. Since there are an infinite number of digits to the left there is no room for a sign. In 10-adic numbers 5-7 = ...99998. It is like an odometer.

If p is prime and we allow a finite number of digits to the right, multiplication has a unique inverse. If p = 5 then 2.∙(...2223.) = 1 and thus ...2223. is the inverse of 2. (By “...2223” I mean an infinite string of “2”s to the left.) 10∙0.1 = 1. When p is prime we have a field. We henceforth require p to be prime. To see that each non-zero p-adic number z has an inverse consider its right most non-zero digit x. The rightmost non-zero digit of its inverse is y chosen so that xy=1 mod p. This is possible since p is prime. Position y so that the product will be ...qqq1.0000 . You can now tediously work out the digits to the left of q, one by one by a proces reminiscent of long division.

The book then proceeds to introduce a topology on these numbers. Whereas digits to the left are more significant in determining nearness in the familiar topology of the reals, digits to the right are more significant in the p-adic topology. Addition and multiplication, as functions, are continuous.

The paper An Introduction to p-adic Numbers and p-adic Analysis introduces the same ideas from the opposite direction. It reviews how to get to the reals via the rationals and generalizes to rings, and then introduces a norm on the rationals where 27 is small and 81 is smaller if p=3. It carries out the Cauchy completion using this norm in place of the absolute value which Cauchy used as a norm and gets, voilá, a structure isomorphic to that from the book. Matthew Watkins has an interesting approach to p-adic numbers along these lines.

P-adic numbers form a field but not an ordered field, since GF(p) is a subfield which is not ordered.

There are thick volumes on p-adic numbers. I started this search to understand why some think that p-adic numbers might be useful in physics. You can define the reals in terms of infinite numerals and binary reals and decimal reals are isomorphic. But 2-adic numbers are not isomorphic with 3-adic numbers. Unlike the reals, p-adic numbers require specifying p. Presumably we must discover the value of p if p-adic numbers are useful in explaining the universe. The Watkins link above addresses these issues. Lest one think that p-adic numbers aren’t that different: If a disc, or ball, is the set of points within some distance from a given point, then two balls are either nested one in the other, or are disjoint.

The Wikipedia article introduces p-adic numbers in different ways and has good info!