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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 ''x''2 + ''y''2 = 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''.
==Group operation==
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 + ''i''0 = 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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