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 $n$.

The somewhat outdated term cipher or cypher can on occasion be seen for the number zero, especially when used in the context of a zero digit in a basis representation.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

The earliest use of a special symbol to be used as a placeholder for a missing denomination in a number expressed in a positional number system was in the Babylonian number system, at around $\text {300}$$\text { BCE}$.

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 $\text {200}$$\text { CE}$ and $\text {1100}$$\text { CE}$) uses it, as a heavy dot.

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

The earliest certain use of a zero digit in a positional number system appears on a stone tablet dated to $\text {876}$$\text { CE}$.

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.