# 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: $\C = \set {a + i b: a, b \in \R, i^2 = -1}$

- $\struct {\Q, +, \times, \le}$ 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.

- Results about
**the standard number fields**can be found**here**.

## Linguistic Note

The term **standard number field** was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ so as to be able to refer to elements of $\set {\Q, \R, \C}$ conveniently and unambiguously.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.