The cyclotomic numbers are a subset of the complex numbers that are represented exactly, enabling exact computations and equality comparisons. They contain the Gaussian rationals (complex numbers of the form p + q i with p and q rational), as well as all complex roots of unity. The cyclotomic numbers contain the square roots of all rational numbers. They contain the sine and cosine of all rational multiples of pi. The cyclotomic numbers form a field, being closed under addition, subtraction, mutiplication, and division.