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 algebraic number field that is both a quadratic field and a cyclotomic field (since i is a 4th root of unity). Like all quadratic fields it is a Galois extension of Q with Galois group cyclic of order two, in this case generated by complex conjugation, and is thus an abelian extension of Q, with conductor 4.^[1]

As with cyclotomic fields more generally, the field of Gaussian rationals is neither ordered nor 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 ${\displaystyle \{1,i\}}$.

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 ${\displaystyle p/q}$ (i.e. ${\displaystyle p}$ and ${\displaystyle q}$ are relatively prime), the radius of this sphere should be ${\displaystyle 1/2|q|^{2}}$ where ${\displaystyle |q|^{2}=q{\bar {q}}}$ is the squared modulus, and ${\displaystyle {\bar {q}}}$ is the complex conjugate. The resulting spheres are tangent for pairs of Gaussian rationals ${\displaystyle P/Q}$ and ${\displaystyle p/q}$ with ${\displaystyle |Pq-pQ|=1}$, and otherwise they do not intersect each other.^[2][3]

References