Beautifully Complicated

This post is a very brief introduction to the “Ternary Tau” number system.

Before I introduce the new system, let’s go over the basics:
Binary is a number system with radix 2 and two possible symbols {0,1}. Ternery is a number system with radix 3 and three possible symbols {0,1,2}. BALANCED Ternary is a number system with radix 3 and three possible symbols {1,0,1}. Examples:

Binary: ${1011}_2 = 1*2^3 + 0*2^2 + 1*2^1 + 1*2^0 = {11}_{10}$
Ternary:${120}_3 = 1*3^2 + 2*3^1 + 0*3^0 = {15}_{10}$
Balanced Ternary: ${1\underline{1}0}_3 = 1*3^2 - 1*3^1 + 0*3^0 = 6_{10}$

Ok, got that? good. Let’s quickly touch on the intermediate:
Bergman’s number system (also known as phinary or the golden ratio base) has an irrational radix equal to (1+ √5)/2 or roughly 1.61803… Using the rules of beta expansion, phinary has two possible symbols {0,1}, example:

(Minimal representation only)
Phinary: ${100.01}_\varphi = (1*\varphi^2) + (0*\varphi^1) + (0*\varphi^0) + (0*\varphi^{-1}) + (1*\varphi^{-2}) = 3_{10}$
Phinary: ${1000.1001}_\varphi = (1*\varphi^3) + 0 + 0 + 0 + (1*\varphi^{-1}) + 0 + 0 + (1*\varphi^{-4}) = 5_{10}$

Now for the advanced stuff!:
In a paper in 2001 (link below), balanced ternary and phinary were molded into one very impressive number system, using properties already aparant in phinary the system was able to become balanced. With a radix of (1+√5)/2 and three posible symbols {-1,0,1} here is an example of Balanced Phinary:

${10\underline{1}01.0\underline{1}01}_\varphi = (1*\varphi^4) + 0 + (-1*\varphi^2) + 0 + (1*\varphi^0) + 0 + (-1*\varphi^{-2}) + 0 + (1*\varphi^-4) = 5_{10}$

An example of the Ternary Tau System:
If the radix is changed to τ = (3+√5)/2 then a more efficient output is given:

$1\underline{1}1.\underline{1}1_\tau = (1 * t^2) + (-1 * t^1) + (1 * t^0) + (-1 * t^{-1}) + (1 * t^{-2}) = 5_{10}$
Which correspondes to radix e being the most efficient base for a number system.

Along with code redundancy, zero-property, double-property and negative representation the Ternary Tau System has some extremely interesting factors and is by far my favourite number system, I’d even go as far to say it’s more interesting than Donald Knuth’s Quater-Imaginary!

Useful links relating to this post:
All this information was from this paper apart from e (2.71828…) being the most efficient base which I got from the book.
http://en.wikipedia.org/wiki/Numeral_system
http://en.wikipedia.org/wiki/Golden_ratio_base
http://en.wikipedia.org/wiki/Balanced_ternary

4 thoughts on “Beautifully Complicated”

1. Pingback: Complex Bases | Neural Outlet..

2. Correction: set {1,0,1} (which isn’t a set properly) has two symbols in it. I’m sure you meant -1.

• Yeah cheers, it was supposed to be an underlined 1 (to, like you said, indicate -1). I’ve changed it now.