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cos(pi/(32)) = 1/2sqrt(2+sqrt(2+sqrt(2+sqrt(2)))) (1) cos((3pi)/(32)) = 1/2sqrt(2+sqrt(2+sqrt(2-sqrt(2)))) (2) cos((5pi)/(32)) = 1/2sqrt(2+sqrt(2-sqrt(2-sqrt(2)))) (3) ...

Einstein summation is a notational convention for simplifying expressions including summations of vectors, matrices, and general tensors. There are essentially three rules of ...

The term "indeterminate" is sometimes used as a synonym for unknown or variable (Becker and Weispfenning 1993, p. 188). A mathematical expression can also be said to be ...

The Spencer-Brown form is a simple mathematical concept that formalizes what a mathematical object is formally identical to what it is not (Spencer-Brown 1997, pp. ix and ...

Given any real number theta and any positive integer N, there exist integers h and k with 0<k<=N such that |ktheta-h|<1/N. A slightly weaker form of the theorem states that ...

A phenomenon in which a system being forced at an irrational period undergoes rational, periodic motion which persists for a finite range of forcing values. It may occur for ...

The notation Q^_ denotes the algebraic closure of the rational numbers Q. This is equivalent to the set of algebraic numbers, sometimes denoted A.

For any real number r>=0, an irrational number alpha can be approximated by infinitely many rational fractions p/q in such a way that ...

Consider the circle map. If K is nonzero, then the motion is periodic in some finite region surrounding each rational Omega. This execution of periodic motion in response to ...

A number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. Such numbers ...