# Definition:Zero Digit

## 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:

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

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

## Also known as

The somewhat outdated term **cipher** or **cypher** can on occasion be seen for the number **zero**, especially when used in the context of a **zero digit** in a basis representation.

The words **nought**, or its somewhat old-fashioned form **naught**, can also be seen.

Younger children often use the word **nothing**.

## Historical Note

The earliest use of a special symbol to be used as a placeholder for a missing denomination in a number expressed in a positional numeral system was in the Babylonian number system.

However, the technique was not transmitted to other cultures.

The concept was reinvented by the mathematicians of the Hindu culture in the first few centuries CE.

The *Bakhshali Manuscript* (dated from between $200$ and $1100$) uses it, as a heavy dot.

It is also used in the *Lokavibhaga* of $458$, but not as a symbol as such.

Aryabhata the Elder used a positional numeral system around $500$, but it did not have a zero digit.

The earliest certain use of a zero digit in a positional numeral system appears on a stone tablet dated to $876$.

## 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):**cipher**or**cypher**:**1.** - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**zero**:**1a.** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**cipher (cypher)**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**cipher (cypher)**:**1.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**cipher (cypher)** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**cipher (cypher)**