# 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 an ordered field, and also a metric space.
- $\struct {\R, +, \times, \le}$ is an 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**

**nemesis**: your**share**of fate

**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**

**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.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (next): $\S 1.1$. Sets - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (next): Introduction - 1971: Wilfred Kaplan and Donald J. Lewis:
*Calculus and Linear Algebra*... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers - 1973: G. Stephenson:
*Mathematical Methods for Science Students*(2nd ed.) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): Notation and Terminology - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.2$: The set of real numbers - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 2$: Introductory remarks on sets: $\text{(b)}$ - 1994: H.E. Rose:
*A Course in Number Theory*(2nd ed.) ... (previous) ... (next): Preface to first edition: Prerequisites - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets - 2001: John H. Conway:
*On Numbers And Games*(2nd ed.) ... (next): Chapter $0$: All Numbers Great and Small