Dividing a circle into areas lines (there are two points on one side and one on the other). So the recurrence can be expressed as which can be easily reduced to using the sums of the first formula_3 natural numbers and the first formula_3 squares, this combines to Finally which yields The lemma asserts that the number of regions is maximal if all "inner" intersections of chords are simple (exactly two chords pass through each point of intersection in the interior). This will be the case if the points on the circle are chosen "in general position". Under this assumption of "generic intersection", the number
Group of rational points on the unit circle to the origin, equal to 0. The identity in this larger group is (1,0,1,0), and the group operation is formula_9 For the group on the unit circle, the appropriate subgroup is points of form ("w","x",1,0), with formula_10 and its identity element is (1,0,1,0). The unit hyperbola group corresponds to points of form (1,0,"y","z"), with formula_11 and the identity is again (1,0,1,0). (Of course, since they are subgroups of the larger group, they both must have the same identity element.) Group of rational points on the unit circle In mathematics, the rational points on the unit circle are those points ("x",