The system is simple. A ding (a dot) counts as 1. But anything inside a ring (a circle) is doubled. A ding inside two rings is doubled and doubled again. Here's the numeral for 13:

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.

View a few numbers of your choosing at

Ring-a-Ding Numeration

The fun comes when you add and multiply numerals. To add two numerals, you put them together and watch. They will put on show getting together. I won't spoil the show by describing it.

You can see numerals (of your choosing) doing addition in real time if you go to

Ring-a-Ding Addition

Multiplication is even more exciting. To multiply one numeral by another, you simply replace each ding in one numeral with a copy of the other numeral. Then as before the rings and dings will then sort themselves out. It's a quite a show.

You can watch numerals (your choice!) do their stuff at

Ring-a-Ding Multiplication.

Special thanks to Fred Henle for technical assistance, tasteful suggestions, and timely encouragement!