Definition:Standard Number Field

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Definition

The standard number fields are the following sets of numbers:

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


These sets are indeed fields:

$\struct {\Q, +, \times, \le}$ is an ordered field, and also a metric space.
$\struct {\R, +, \times, \le}$ is an ordered field, and also a complete metric space.
$\struct {\C, +, \times}$ is a field, but cannot be ordered compatibly with $+$ and $\times$. 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:

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