A few years ago I ran into Abhijit Bhattacharjee of India via the web. He had a site that can still be found at http://replay.web.archive.org/20090312094241/http://abhijit.info/tristate/tristate.html with much more information on this idea. In his bibliography still found at http://replay.web.archive.org/20090221122459/http://abhijit.info/tristate/biblio.htm Abhijit says of Donald Knuth:

In 1981, in his book “The Art of Computer Programming”, Vol 2: Seminumerical Algorithms. Second Edition. Reading Mass: Addison-Wesley, pp 190-193 calls it the “prettiest number system of all”…

I now realize this is where I first encountered the idea, as this book was text for a programming course I took in the 1970′s.

Here’s are diagrams representing the “standard” and balanced versions of base 3. Each step lower in a diagram represents adding one more digit of precision to a number.  In the upper diagram, all numbers that begin with “0.1…” are equal to or larger than 0.1. In the lower diagram, the value 0.1 is the precise middle of all numbers that begin with “0.1…”.

Diagram, two forms of Base 3

In the first form, the descending lines from a single point represent the digits 0, 1, or 2. In the second form, the descending lines represent the digits -1 (to the left), 0 (vertical), or 1 (to the right). To get a feel for how this works, here’s how one would count from -5 to +13 in balanced base 3, using “<” for a digit with value -1, and “>” for a digit with value +1: <>>, 0<<, 0<0, 0<>, 00<, 000, 00>, 0><, 0>0, 0>>, ><<, ><0, ><>, >0<, >00, >0>, >><, >>0, >>>. In these three-digit numbers the first is the 9′s place, the second is the 3′s place, and the third is the 1′s place. So the first number in this sequence, <>> represents (-9)+(+3)+(+1)=-5.

At www.solbakkn.com/math you can find more on this topic. It also includes information on N-Grams, and a pointer to information (in a PDF file) on Dimensionalities, two other math topic of interest to me. Eventually I plan to move and expand information about all three math topics on this site.

In looking for archived copies of Abhijit’s materials, I found a site on Finding old Web Pages that suggests a number of ways of finding archived web information. There’s more info on the tool I used at the Web Archiving at archive.org site.