# Definition:Zero Digit

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## Definition

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

Let $b \in \Z$ such that $b > 1$ be a number base in which $x$ is represented.

By the Basis Representation Theorem, $x$ can be expressed uniquely in the form:

- $\displaystyle x = \sum_{j \mathop \in \Z}^m r_j b^j$

Any instance of $r_j$ being equal to $0$ is known as a **zero (digit)** of $n$.

## Also known as

The somewhat dated term **cipher** or **cypher** can on occasion be seen for the **zero digit**.

The word **nought** can commonly be seen.

## Linguistic Note

The word **cipher** can also be found in its less common spelling: **cypher**.

The word ultimately derives from the Arabic **صِفْر** (**ṣifr**), meaning **zero** or **empty**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**zero**:**1a.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**cipher (cypher)**:**1.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**cipher (cypher)**