# Definition:Gaussian Rational

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## 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 $\sqbrk {\Q \sqbrk i, +, \times}$ 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**.

### 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

- Results about
**Gaussian rationals**can be found**here**.

## Source of Name

This entry was named for Carl Friedrich Gauss.