A numeral system with 3 as the base. The usual decimal system uses 10 and binary uses 2. If we specify that the representation be non-terminating, each real number can be represented uniquely.
A number system with 3 as the standard base. The digits are known as trits, short for trinary bits. For instance, $16_{10}=1\cdot 3^2+2\cdot3+1\cdot 1= 121_3$; likewise, $$\left(\frac{1}{5}\right)_{10}=\frac{1}{9}+\frac{2}{27}+\frac{1}{81}+\cdots= 0.\overline{0121}_3,$$as can also be seen from the geometric series. If we specify a that the representation be non-terminating, each real number can be represented uniquely. In computer programming, instead of taking the trits to be $\{0,1,2\}$, $\{-1,0,1\}$ are also occasionally used; here we would write $16_{10}=1 -1-1 1_{\text{bal} 3}$.
Perhaps the most well-known application of the ternary system is in the construction of the ternary Cantor set $\mathcal{C}$. Indeed, one characterization of $\mathcal{C}$ is those numbers in $[0,1]$ whose non-terminating ternary expansion contains no $1$s.
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