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, which 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  .

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  , the radius of this sphere should be   where   represents the complex conjugate of  . The resulting spheres are tangent for pairs of Gaussian rationals   and   with  , and otherwise they do not intersect each other.[2][3]


  1. ^ Ian Stewart, David O. Tall, Algebraic Number Theory, Chapman and Hall, 1979, ISBN 0-412-13840-9. Chap.3.
  2. ^ Pickover, Clifford A. (2001), "Chapter 103. Beauty and Gaussian Rational Numbers", Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, Oxford University Press, pp. 243–246, ISBN 9780195348002.
  3. ^ Northshield, Sam (2015), Ford Circles and Spheres, arXiv:1503.00813, Bibcode:2015arXiv150300813N.