b is the base of the digit and is the symbol count

b symbols

1 0

2 0,1

3 0,1,2

...

The symbol count is necessary. The symbols can be any set of marks.

If you count and come to the end of the symbols you have, there are 3

choices:

1: give up

2: make another symbol

3: add another digit

If you add another digit you must have a carry mechanism and your digits make a polynomial in b.

It is easy to show that any multi digit number can be made into a single digit by choosing a new base that is old b to the digit power.

I make machines that do math. Anyone interested in this kind of stuff?