Definition:Gaussian Rational

Definition

Definition 1

A Gaussian rational is a complex number whose real and imaginary parts are both rational numbers.

That is, a Gaussian rational is a number in the form:

$a + b i: a, b \in \Q$

Definition 2

The field $\struct {\Q \sqbrk i, +, \times}$ of Gaussian rationals is the field of quotients of the integral domain $\struct {\Z \sqbrk i, +, \times}$ of Gaussian integers.

This is shown to exist in Existence of Field of Quotients.

In view of Field of Quotients is Unique, we construct the field of quotients of $\Z \sqbrk i$, give it a label $\Q \sqbrk i$ and call its elements Gaussian rationals.

Notation

The set of all Gaussian rationals can be denoted $\Q \sqbrk i$, and hence we have:

$\Q \sqbrk i = \set {z \in \C: z = a + b i: a, b \in \Q}$

Also see

• Definition:Gaussian Field

• Equivalence of Definitions of Gaussian Rational

• Results about Gaussian rationals can be found here.

Source of Name

This entry was named for Carl Friedrich Gauss.