The Field of Cyclotomic Numbers.

cyclotomic

The field of cyclotomic numbers.

The set of cyclotomic numbers is a field obtained by extending the set of rational numbers with the complex roots of unity. The main function used to construct a cyclotomic number in this package is zeta, it returns a primitive root of the unity. For example zeta(4) is the primitive fourth root of unity, that is the imaginary unit.

library(cyclotomic)
im <- zeta(4)
im^2
## -1

Arithmetic on cyclotomic numbers can be performed with this package. It is exact. In particular it allows to deal with the Gaussian rational numbers: the complex numbers whose both real and imaginary part are rational.

a <- as.cyclotomic(5)
b <- as.cyclotomic("3/2")
(a + im * b)^2
## 91/4 + 15*zeta(4)

Note that while zeta(4) is printed as zeta(4), this is not the case for all roots of unity:

zeta(9)
## -zeta(9)^4 - zeta(9)^7

The set of cyclotomic numbers contains all the square roots of rational numbers, and therefore the package allows exact calculations on such square roots. For example, using float numbers, the following equality does not hold true:

sqrt(5/3) == sqrt(5) / sqrt(3)
## [1] FALSE

But it holds true using the cyclotomic arithmetic:

cycSqrt("5/3") == cycSqrt(5) / cycSqrt(3)
## [1] TRUE

The set of cyclotomic numbers also contains the cosine and the sine of the rational multiples of pi. In particular, it contains the cosine and the sine of any rational number when this number represents an angle given in degrees.

cosDeg(60)
## 1/2
sinDeg(60) == cycSqrt(3) / 2
## [1] TRUE

This package is a port of the Haskell library cyclotomic, written by Scott N. Walck.