Gaussian rational

In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(i), obtained by adjoining the imaginary number i to the field of rationals Q.

Properties of the field

The field of Gaussian rationals provides an example of an

conductor 4.^[1]

As with cyclotomic fields more generally, the field of Gaussian rationals is neither

complete (as a metric space). The Gaussian integers Z[i] form the ring of integers of Q(i). The set of all Gaussian rationals is countably infinite

.

The field of Gaussian rationals is also a two-dimensional vector space over Q with natural basis $\{1,i\}$ .

Ford spheres

The concept of Ford circles can be generalized from the rational numbers to the Gaussian rationals, giving Ford spheres. In this construction, the complex numbers are embedded as a plane in a three-dimensional Euclidean space, and for each Gaussian rational point in this plane one constructs a sphere tangent to the plane at that point. For a Gaussian rational represented in lowest terms as $p/q$ (i.e. $p$ and $q$ are relatively prime), the radius of this sphere should be $1/2|q|^{2}$ where $|q|^{2}=q{\bar {q}}$ is the squared modulus, and ${\bar {q}}$ is the complex conjugate. The resulting spheres are tangent for pairs of Gaussian rationals $P/Q$ and $p/q$ with $|Pq-pQ|=1$ , and otherwise they do not intersect each other.^[2]^[3]