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.

## 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.

## 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]}

## References

**^**Ian Stewart, David O. Tall,*Algebraic Number Theory*, Chapman and Hall, 1979, ISBN 0-412-13840-9. Chap.3.**^**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.**^**Northshield, Sam (2015),*Ford Circles and Spheres*, arXiv:1503.00813, Bibcode:2015arXiv150300813N.

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