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 $\Q \left[{i}\right]$, and hence can be defined as:
 * $\Q \left[{i}\right] = \left\{{a + b i: a, b \in \Q}\right\}$

Formal Definition
The field $(\Q[i],+,\times)$ of Gaussian rationals is the quotient field of the integral domain $(\Z[i],+,\times)$ of Gaussian integers.

This is shown to exist in Existence of Quotient Field.

In view of Quotient Field is Unique, we construct the quotient field of $\Z[i]$, give it a label $\Q[i]$ and call its elements Gaussian rationals.