Definition:Standard Number Field

From ProofWiki
Jump to navigation Jump to search


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.


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