cyclotomic-0.1: A subfield of the complex numbers for exact calculation

Data.Complex.Cyclotomic

Description

The cyclotomic numbers are a subset of the complex numbers with the following properties:

1. The cyclotomic numbers are represented exactly, enabling exact computations and equality comparisons.
2. The cyclotomic numbers contain the Gaussian rationals (complex numbers of the form `p` + `q` `i` with `p` and `q` rational). As a consequence, the cyclotomic numbers are a dense subset of the complex numbers.
3. The cyclotomic numbers contain the square roots of all rational numbers.
4. The cyclotomic numbers form a field: they are closed under addition, subtraction, multiplication, and division.
5. The cyclotomic numbers contain the sine and cosine of all rational multiples of pi.
6. The cyclotomic numbers can be thought of as the rational field extended with `n`th roots of unity for arbitrarily large integers `n`.

This algorithm for cyclotomic numbers is adapted from code by Martin Schoenert and Thomas Breuer in the GAP project http://www.gap-system.org/ . See in particular source files gap4r4/src/cyclotom.c and gap4r4/lib/cyclotom.gi .

Synopsis

# Documentation

data Cyclotomic Source

A cyclotomic number.

Instances

 Eq Cyclotomic Fractional Cyclotomic Num Cyclotomic Show Cyclotomic

The square root of -1.

The primitive `n`th root of unity. For example, `e`(4) = `i` is the primitive 4th root of unity, and `e`(5) = exp(2*pi*i/5) is the primitive 5th root of unity. In general, `e` `n` = exp(2*pi*i/`n`).

The square root of an `Integer`.

The square root of a `Rational` number.

Sine function with argument in degrees.

Cosine function with argument in degrees.

Complex conjugate.

Real part of the cyclotomic number.

Imaginary part of the cyclotomic number.

Modulus squared.

Export as an inexact complex number.

Is the cyclotomic a real number?

Is the cyclotomic a rational?

Is the cyclotomic a Gaussian rational?

Return Just rational if the cyclotomic is rational, Nothing otherwise.