Definition:Number

Definition

There are five main classes of number:

$(1): \quad$ The natural numbers: $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$

$(2): \quad$ The integers: $\Z = \left\{{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}\right\}$

$(3): \quad$ The rational numbers: $\Q = \left\{{p / q: p, q \in \Z, q \ne 0}\right\}$

$(4): \quad$ The real numbers: $\R = \{{x: x = \left \langle {s_n} \right \rangle}\}$ where $\left \langle {s_n} \right \rangle$ is a Cauchy sequence in $\Q$

$(5): \quad$ The complex numbers: $\C = \left\{{a + i b: a, b \in \R, i^2 = -1}\right\}$

It is possible to categorize numbers further, for example:

• The set of algebraic numbers $\mathbb A$ is the subset of the real 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 primes are $2, 3, 5, 7, 11, 13, \ldots$

Extension to the concept

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

• The quaternions $\mathbb H$ (labelled $\mathbb H$ for William Hamilton who discovered / invented them)
• The octonions $\mathbb O$
• The sedenions $\mathbb S$

and so forth.

Thence follows an entire branch of mathematics: see Clifford algebras.

Number Sets as Algebraic Structures

Note that:

• $\left({\N, +, \le}\right)$ is a naturally ordered semigroup.
• $\left({\Z, +, \times, \le}\right)$ is a totally ordered integral domain.
• $\left({\Q, +, \times, \le}\right)$ is a totally ordered field, and also a metric space.
• $\left({\R, +, \times, \le}\right)$ is a totally ordered field, and also a complete metric space.
• $\left({\C, +, \times}\right)$ is a field, but can not be ordered. However, it can be treated as a metric space.

Comment

Note that (disregarding isomorphisms):

$\N \subset \Z \subset \Q \subset \mathbb A \subset \R \subset \C$

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

Sources

• Seth Warner: Modern Algebra (1965): $\S 1$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Introduction
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977): $\S 1.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 2 \ \text{(b)}$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.1$