# 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:

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

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.

## 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.