The Ring-a-Ding Numeration System by Jim Henle
science.smith.edu
science.smith.edu
Anyone know where to find a rigorous maths treatment for it?
(They correspond to things you can do with the expanded algebraic representation of a base-2 number, Sum a_k 2^k, but that's a necessary prerequisite for such a number system to work. After all, a way of writing numbers that did not ultimately correspond to the existing ways of writing numbers would be wrong.)
1×2 + 1×4 + 0×8 + 1×16, 1×8
and substitute the second for every 1 in the first,
8×2 + 8×4 + 0×8 + 8×16
then finally un-distribute,
(1×8)(1×2 + 1×4 + 0×8 + 1×16)
and that's why putting small copies of the second number in the place of every "ding" in the first number results in multiplication.
a ring is 0
a ring,ding is 1
The ring represents a doubling of its contents. The special case for the leading bit could be avoided by always having an innermost empty ring. That would also give you a better (i.e. visible) representation for 0. The addition rule ensures that you always have exactly 1 leading 0.
> The rings must be nested. The innermost ring must have a ding. No region can have more than one ding. Those are the only rules.
According to these rules the intermediate results are not valid numbers