# Representation of Ternary Expansions

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## 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

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $29$. The Cantor Set