# Caleb Stanford Blog

## 2017

math random-facts

The current year, translated to various systems for your convenience:

\begin{align*} 11111100001 &\quad \text{ Binary}\\ 1100000100001 &\quad \text{ Negabinary}\\ 1001000010010001 &\quad \text{ Zeckendorf (Fibonacci base)} \\ 1,0,-1,1,0,-1,0,1 &\quad \text{ Offset base 3} \\ 2,4,4,0,0,1 &\quad \text{ Factorial base} \end{align*}

Fun fact: All of these number systems have a unique representation theorem of some kind.$$^{1}$$ However, the only which offer a unique representation for all integers, and not just for all nonnegative integers, are negabinary and offset base 3. For traditional decimal or binary numbers, you either only have representations for nonnegative integers, or you allow a possible minus sign in front at which point you get two representations of zero.

$$^{1}$$For uniqueness of Zeckendorf representation, one requires that there are no two adjacent 1s.