# Definition:Number

## Definition

There are five main classes of number:

$(1): \quad$ The natural numbers: $\N = \set {0, 1, 2, 3, \ldots}$
$(2): \quad$ The integers: $\Z = \set {\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}$
$(3): \quad$ The rational numbers: $\Q = \set {p / q: p, q \in \Z, q \ne 0}$
$(4): \quad$ The real numbers: $\R = \set {x: x = \sequence {s_n} }$ where $\sequence {s_n}$ is a Cauchy sequence in $\Q$
$(5): \quad$ The complex numbers: $\C = \set {a + i b: a, b \in \R, i^2 = -1}$

It is possible to categorize numbers further, for example:

The set of algebraic numbers $\mathbb A$ is the subset of the complex numbers which are roots of polynomials with rational coefficients. The algebraic numbers include the rational numbers, $\sqrt 2$, and the golden section $\varphi$.
The set of transcendental numbers is the set of all the real numbers which are not algebraic. The transcendental numbers include $\pi, e$ and $\sqrt 2^{\sqrt 2}$.
The set of prime numbers (sometimes referred to as $\mathbb P$) is the subset of the integers which have exactly two positive divisors, $1$ and the number itself. The first several positive primes are $2, 3, 5, 7, 11, 13, \ldots$

## Number Sets as Algebraic Structures

Note that:

$\struct {\N, +, \le}$ can be defined as a naturally ordered semigroup.
$\struct {\Z, +, \times, \le}$ is a totally ordered integral domain.
$\struct {\Q, +, \times, \le}$ is a totally ordered field, and also a metric space.
$\struct {\R, +, \times, \le}$ is a totally ordered field, and also a complete metric space.
$\struct {\C, +, \times}$ is a field, but cannot be ordered. However, it can be treated as a metric space.

## Also see

It is possible to continue from the concept of complex numbers and define:

The quaternions $\H$ (labelled $\H$ for William Rowan Hamilton who discovered / invented them, as $\Q$ was already taken)
The octonions $\mathbb O$
The sedenions $\mathbb S$

and so forth.

Thence follows an entire branch of mathematics: see Cayley-Dickson construction.

In a different direction, the concept of natural numbers can be extended to the ordinals or the cardinals.

## Comment

Note that (up to isomorphism):

$\N \subseteq \Z \subseteq \Q \subseteq \R \subseteq \C$

and of course $\mathbb P \subseteq \Z$.

## Linguistic Note

The word number comes from an Indo-European word meaning share or portion.

It appears to have been originally associated with the division of land.

Hence we have the derived terms:

nimble: descriptive of one who is quick to take his share
numb: originally meaning seized or taken
nomad: a person who wanders in search of some pasture he can take a share of
Supernumerary, which means redundant, but originally had the sense of meaning over and above the numbers stated by the rules

The root nom can be found in the following examples of technical terms:

Binomial: a mathematical object with two numbers
Astronomy: the science of numbering the stars
Economy and economics
Autonomy

The German word nehmen means to take, which has the imperative form nimm.

This word is found slightly modified in archaic English as nim, which by the time of Shakespeare had evolved to mean to steal or to pilfer.

The word nim still lives on as the name of a game whose mechanics consist of taking objects from a heap.

In Latin and Greek, the word nomisma meant coin.

The word lives on in the English word numismatist, a collector of coins.

We also have nummulite, which is a coin-shaped fossil.