Definition:Number

There are five main classes of number:


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

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 $$2,3,5,7,11,13...$$

Comment
Note that (disregarding isomorphisms):

$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb {A} \subset \mathbb {R} \subset \mathbb{C}$$,

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