Concept
In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2 = 1. The set of such points turns out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integer side lengths a, b, c, with c the hypotenuse, such that the sides have no common factor larger than 1. Then on the unit circle there exists the rational point (a/c, b/c), which, in the complex plane, is just a/c + ib/c, where i is the imaginary unit. 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. There is a correspondence between points (a, b) in the x-y plane and points a + ib in the complex plane which is used below. The set of rational points on the unit circle, shortened G in this article, forms an infinite abelian group under rotations. The identity element is the point (1, 0) = 1 + i0 = 1. 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.
