# Representation of Ternary Expansions

## Theorem

Let $x \in \R$ be a real number.

Let $x$ be represented in base $3$ notation.

While it may be possible for $x$ to have two different such representations, for example:

- $\dfrac 1 3 = 0.100000\ldots_3 = 0.022222\ldots_3$

it is not possible for $x$ be written in more than one way without using the digit $1$.

## Proof

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 29$