\begin{definition}[Definition:Number]
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 $2, 3, 5, 7, 11, 13, \ldots$
\end{definition}