Definition:Gaussian Rational

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

The set of all Gaussian rationals can be denoted $\Q \sqbrk i$, and hence can be defined as:
 * $\Q \sqbrk i = \set {a + b i: a, b \in \Q}$

Formal Definition
The field $\sqbrk {\Q \sqbrk i, +, \times}$ 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.