Definition:Number

There are five main classes of number:


 * 1) The natural numbers: $$\N = \left\{{0, 1, 2, 3, \ldots}\right\}$$;
 * 2) The integers: $$\Z = \left\{{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}\right\}$$;
 * 3) The rational numbers: $$\Q = \left\{{p / q: p, q \in \mathbb{Z}, q \ne 0}\right\}$$;
 * 4) 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) The complex numbers: $$\C = \left\{{a + i b: a, b \in \R, i^2 = -1}\right\}$$.

It is possible to categorize numbers further:


 * 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 $$\mathbb{H}$$ (so labelled as they are named from William Hamilton who discovered 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 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 $$\mathbb{P} \subset \Z$$.