I have no idea why, but I love alternate number systems (p-adics, surreals, base -2, split-complex, etc.). This one is playful.

Anyone know where to find a rigorous maths treatment for it?

Uh...isn't this just binary?

It's binary with the central dot/ding being the leading 1 and then toward the right

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.

It's not binary because all of the intermediate results shown during the videos are also valid numbers. It's actually an (upside down so as not to spoil it for anyone) ˙ǝuo ɟo uoıʇıppɐ ʇuǝsǝɹdǝɹ sʇop ǝɥʇ puɐ 'oʍʇ ʎq pǝıןdıʇןnɯ sʇuǝʇuoɔ ǝɥʇ ɥʇıʍ sısǝɥʇuǝɹɐd ʇuǝsǝɹdǝɹ sǝןɔɹıɔ ǝɹǝɥʍ uoıssǝɹdxǝ ɔıʇǝɯɥʇıɹ∀

From the article:

> 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

It's binary in "fully reduced" form, what's interesting are the graphical rules for addition, multiplication and reduction.

(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.)

For addition I can see the binary addition at work. The ding "carries" if it get put in a ring that already has a ding by getting its own ring etc... For multiplication I just WAT

Take a couple expanded algebraic representations of base-2 numbers:

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.

Oh yeah, thanks!

I don't think there's anything resembling a unifying theory of alternative number systems. But it's a fun topic for sure.