Group of rational points on the unit circle
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 integral 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 x and y denominators. There is a correspondence between points (x,y) in the x-y plane and points x + iy in the complex plane which will be used below, with (a, b) taken as equal to a + ib.
The set of rational points forms an infinite abelian group under rotations, which shall be called G in this article. 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 = cosine(A) and y = sine(A), where A is the angle the radius vector (x, y) makes with the radius vector (1,0), measured counter clockwise. So with (x, y) and (t, u) forming angles A and B, respectively, with (1, 0), their product (xt − uy, xu + yt) is just the rational point on the unit circle with angle A + B. But we can do these group operations in a way that may be easier, with complex numbers: Write the point (x, y) as x + iy and write (t, u) as t + iu. Then the product above is just the ordinary multiplication (x + iy)(t + iu) = xt − yu + i(xu + yt), which corresponds to the (xt − uy, xu + yt) above.
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