Definition:Standard Number Field

Definition
The standard number fields are the following sets of numbers:


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

These sets are indeed fields:


 * $\left({\Q, +, \times, \le}\right)$ 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 cannot be (totally) ordered. However, it can be treated as a metric space.

Also see
Neither the set $\N$ of natural numbers nor the set $\Z$ of integers are fields.

However:


 * $\left({\N, +, \le}\right)$ is a naturally ordered semigroup.
 * $\left({\Z, +, \times, \le}\right)$ is an ordered integral domain.