Group of rational points on the unit circle

The set of such points turns out to be closely related to primitive Pythagorean triples.

Conversely, if (x, y) is a rational point on the unit circle in the 1st quadrant of the coordinate system (i.e. x > 0, y > 0), then there exists a primitive right triangle with sides xc, yc, c, with c being the least common multiple of the denominators of x and y.

The set of rational points on the unit circle, shortened G in this article, forms an infinite abelian group under rotations.

The group operation, or "product" is (x, y) * (t, u) = (xt − uy, xu + yt).

This product is angle addition since x = cos(A) and y = sin(A), where A is the angle that the vector (x, y) makes with the vector (1,0), measured counter-clockwise.

So with (x, y) and (t, u) forming angles A and B with (1, 0) respectively, their product (xt − uy, xu + yt) is just the rational point on the unit circle forming the angle A + B with (1, 0).

The group operation is expressed more easily with complex numbers: identifying the points (x, y) and (t, u) with x + iy and t + iu respectively, the group product above is just the ordinary complex number multiplication (x + iy)(t + iu) = xt − yu + i(xu + yt), which corresponds to the point (xt − uy, xu + yt) as above.

3/5 + 4/5i and 5/13 + 12/13i (which correspond to the two most famous Pythagorean triples (3,4,5) and (5,12,13)) are rational points on the unit circle in the complex plane, and thus are elements of G. Their group product is −33/65 + 56/65i, which corresponds to the Pythagorean triple (33,56,65).

The set of all 2×2 rotation matrices with rational entries coincides with G. This follows from the fact that the circle group

The structure of G is an infinite sum of cyclic groups.

For a prime p of form 4k + 1, let Gp denote the subgroup of elements with denominator pn where n is a non-negative integer.

Furthermore, by factoring the denominators of an element of G, it can be shown that G is a direct sum of G2 and the Gp.

Viewing G as an infinite direct sum, consider the element ({0}; 2, 0, 1, 0, 0, ..., 0, ...) where the first coordinate 0 is in C4 and the other coordinates give the powers of (a2 − b2)/p(r) + i2ab/p(r), where p(r) is the rth prime number of form 4k + 1.

In this group there is a close connection with the hyperbolic cosine and hyperbolic sine, which parallels the connection with cosine and sine in the unit circle group above.

There are isomorphic copies of both groups, as subgroups (and as geometric objects) of the group of the rational points on the abelian variety in four-dimensional space given by the equation

Note that this variety is the set of points with Minkowski metric relative to the origin equal to 0.

The unit hyperbola group corresponds to points of form (1, 0, y, z), with

(Of course, since they are subgroups of the larger group, they both must have the same identity element.)