Detail #337: A System for Encoding Numbers

Consider a positional system of numbers based on some form of ordinal thinking. We assume for now a decimal system.

1 is really the first number ^{in the first decad in the first centad in the first millenniad ...} ... this means 1 is really ...1111, but we omit leading ones and thus obtain 1. The range ten to nineteen is the second decad. Thus ....11121, ...11122, ...., or as we'd rather have them: 21, 22.

I am not particularly interested in forcing a particular base onto this either, any integer would do... it's just that I want a system where you get the following kind of pattern, given that Z = the base (which also needs a symbol of its own)

1	2	3	4		...				Z
21	22	23	24		...				2Z
					...
Z1	Z2				...
211	212	213	214		...				21Z

We don't need a zero, since we're not interested in those at all: the second '1' may very well come directly after the first '7' for all I am concerned, as long as the pattern is kept intact.

This is fairly similar to bijective numeration in some way, but adds the twist of being slightly off.

Fun thing: there's always an infinite string of 1s to the left of any 'regular' number. One could, however, imagine exceptional numbers where, for instance, there's an infinite string of some other numbers to the left, or an infinite regular pattern (e.g. ...123123123), or even an infinite irregular pattern (reverse your favourite irrational number and drop the decimal mark).

Challenge: develop easy rules for arithmetic for this, without involving conversion back to and from regular numbers.