# Definition:Zero (Number)

## Contents

## Definition

The number **zero** is defined as being the cardinal of the empty set.

### Naturally Ordered Semigroup

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.

Then from axiom $(NO 1)$, $\left({S, \circ, \preceq}\right)$ has a smallest element.

This smallest element of $\left({S, \circ, \preceq}\right)$ is called **zero** and has the symbol $0$.

That is:

- $\forall n \in S: 0 \preceq n$

### Natural Numbers

### Integers

### Rational Numbers

### Real Numbers

### Complex Numbers

Let $\C$ denote the set of complex numbers.

The **zero** of $\C$ is the complex number:

- $0 + 0 i$

## Historical Note

The Babylonians from the $2$nd century BCE used a number base system of arithmetic, with a placeholder to indicate that a particular place within a number was empty, but its use was inconsistent. However, they had no actual recognition of zero as a mathematical concept in its own right.

The Ancient Greeks had no conception of zero as a number.

The concept of zero was invented by the mathematicians of India. The Bahkshali manuscript from the $3$rd century CE contains the first reference to it.

However, even then there were reservations about its existence, and misunderstanding about how it behaved.

In *Ganita Sara Samgraha* of Mahāvīrāchārya, c. $850$ CE appears:

*A number multiplied by zero is zero and that number remains unchanged which is divided by, added to or diminished by zero.*

It was not until the propagation of Arabic numbers, where its use as a placeholder made it important, that it became commonplace.

## Linguistic Note

The Sanskrit word used by the early Indian mathematicians for zero was **sunya**, which means **empty**, or **blank**.

In Arabic this was translated as **sifr**.

This was translated via the Latin **zephirum** into various European languages as **zero**, **cifre**, **cifra**, and into English as **zero** and **cipher**.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems - 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $2$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*: $\S 8$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $2$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $-1$ and $i$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $-1$ and $i$