Definition:Number

From ProofWiki
Jump to: navigation, search

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 $2, 3, 5, 7, 11, 13, \ldots$


Extension of concepts

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

The quaternions $\mathbb H$ (labelled $\mathbb H$ for William Rowan Hamilton who discovered / invented them)
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.


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.


Comment

Note that (disregarding isomorphisms):

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

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


Sources