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.

Properties of the field[edit]

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 uncountably infinite.

Ford spheres[edit]

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 $p/q$ , the radius of this sphere should be $1/q{\bar {q}}$ where ${\bar {q}}$ represents the complex conjugate of $q$ . 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]

References[edit]

^ 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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[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.

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