Balanced Ternary
Introduction
The usual range of digits in a number expressed in base “b” (bases 2 and larger) is from zero to b-1. The range becomes: 0 to 9 in base ten; 0 to 7 in octal (base eight);0 to 1 in base two; 0 to F (15) in hexadecimal (base sixteen).
Some years ago I hit on the idea of a “balanced” base — one that is zero-centered. For base 3, digits would have values -1, 0 and +1 (rather than 0, 1, and2). The material world seems to me to “use” zero-centered arithmetic. Values are “on target” or to one side or the other. For numbers expressed in this system, there is no difference between rounding and truncating a number.
Only the odd-valued bases: 3, 5, 7, etc. can be balanced. In even-valued bases, there’d need to be two zeroes (-0 and +0) or the most negative and most positive digits would have different magnitudes (e.g., -4 to +5 in base ten). Balanced ternary is the smallest balanced base. It has digits with values -1, 0, and +1.
Here’s a diagram representing the “standard” and balanced versions of base 3. Each step lower in the top (standard) or bottom (centered) diagram represents adding one more digit of precision to a number.
- 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.
