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| In mathematics, the '''[[rational point]]s on the [[unit circle]]''' are those points (''x'', ''y'') such that both ''x'' and ''y'' are [[rational number]]s ("fractions") and satisfy ''x''<sup>2</sup> + ''y''<sup>2</sup> = 1. The set of such points turns out to be closely related to primitive [[Pythagorean triple]]s. 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 1<sup>st</sup> [[Cartesian coordinate system#Quadrants and octants|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''.
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| ==Group operation==
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| The set of rational points forms an [[Abelian group#Infinite abelian groups|infinite abelian group]], 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.
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| ===Example===
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| The points on the unit circle: 3/5 + ''i''4/5 and 5/13 + ''i''12/13 (corresponding to the two most famous Pythagorean right triangles:3,4,5 and 5,12,13) are elements of ''G'', and their group product is (−33/65 + i56/65), which corresponds to a 33,56,65 Pythagorean right triangle. The sum of the squares of the numerators 33 and 56 is 1089 + 3136 = 4225, which is the square of the denominator 65.
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| ===Other ways to describe the group===
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| ::<math>G \cong \mathrm{SO}(2, \mathbb{Q}).</math>
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| The set of all 2×2 [[orthogonal|rotation matrices]] with rational entries coincides with G.This follows from the fact that the [[circle group]] <math>S^1</math> is isomorphic to <math>\mathrm{SO}(2, \mathbb{R})</math>, and the fact that their rational points coincide.
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| ==Group structure==
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| The structure of ''G'' is an infinite sum of [[cyclic group]]s. Let ''G''<sub>2</sub> denote the subgroup of ''G'' generated by the point {{nowrap|0 + 1''i''}}. ''G''<sub>2</sub> is a [[cyclic subgroup]] of order 4. For a prime ''p'' of form 4''k'' + 1, let ''G''<sub>''p''</sub> denote the subgroup of elements with denominator ''p''<sup>''n''</sup>, ''n'' a nonnegative integer. ''G''<sub>''p''</sub> is an infinite cyclic group. The point (''a''<sup>2</sup> − ''b''<sup>2</sup>)/''p'' + (2''ab''/''p'')''i'' is a generator of ''G''<sub>''p''</sub>. Furthermore, by factoring the denominators of an element of ''G'', it can be shown that ''G'' is a direct sum of ''G''<sub>2</sub> and the ''G''<sub>''p''</sub>. That is:
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| ::<math>G \cong G_2 \oplus \bigoplus_{p \equiv 1 \bmod 4} G_p.</math> | |
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| Since it is a [[direct sum]] rather than [[direct product]], only finitely many of the values in the ''Gp''s differ from zero.
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| ===Example===
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| Suppose we take the element in ''G'' corresponding to ({''0''};2,0,1,0,0,...0,...) where the first coordinate ''0'' is in ''C''<sub>4</sub> and the other coordinates give the powers of (''a''<sup>2</sup> − ''b''<sup>2</sup>)/''p''(''r'') + ''i''2''ab''/''p''(''r'') where ''p''(''r'') is the rth prime of form 4''k'' + 1. Then this corresponds to, in ''G'', the rational point (3/5 + ''i''4/5)<sup>2</sup> · (8/17 + ''i''15/17)<sup>1</sup> = −416/425 + i87/425). The denominator 425 is the product of the denominator 5 twice, and the denominator 17 once, and as in the previous example, the square of the numerator −416 plus the square of the numerator 87 is equal to the square of the denominator 425. It should also be noted, as a connection to help retain understanding, that the denominator 5 = ''p''(1) is the 1st prime of form 4''k'' + 1, and the denominator 17 = ''p''(3) is the 3rd prime of form 4''k'' + 1.
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| ==The ''unit hyperbola's'' group of rational points==
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| There is a close connection between this group on the [[unit hyperbola]] and the group discussed above. If <math>\frac {a + ib}{c}</math> is a rational point on the unit circle, where ''a''/''c'' and ''b''/''c'' are [[reduced fraction]]s, then (''c''/''a'', ''b''/''a'') is a rational point on the unit hyperbola, since <math>(c/a)^2-(b/a)^2=1,</math> satisfying the equation for the unit hyperbola. The group operation here is <math>(x,y) \times (u,v)=(xu+yv ,xv+yu),</math> and the group identity is the same point (1,0) as above. In this group there is a close connection with [[hyperbolic cosine]] and [[hyperbolic sine]], which parallels the connection with [[cosine]] and [[sine]] in the unit circle group above.
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| ===Copies inside a larger group===
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| :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 <math>w^2+x^2+y^2-z^2=1.</math> Note that this variety is the set of points with [[Minkowski metric]], relative to the origin, equal to 1. The identity in this larger group is (0,1,0,1), and the group operation is <math>(a,b,c,d)\times (w,x,y,z)=(aw-bx,ax+bw,cy+dz,cz+dy).</math>
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| :For the group on the unit circle, the appropriate subgroup is points of form (''w'',''x'',0,1), with <math>w^2+x^2=1,</math> and its identity element is (0,1,0, 1). The unit hyperbola group corresponds to points of form (0,1,''y'',''z''), with <math>y^2-z^2=1,</math> and the identity is again (0,1,0,1). (Of course, since they are subgroups of the larger group, they both must have the same identity element.) | |
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| ==See also==
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| {{Portal|Mathematics}}
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| *[[circle group]]
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| ==References==
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| {{reflist}}
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| *''The Group of Rational Points on the Unit Circle''[http://mathdl.maa.org/images/upload_library/22/Allendoerfer/1997/0025570x.di021195.02p0087x.pdf], Lin Tan, ''[[Mathematics Magazine]]'' Vol. 69, No. 3 (June, 1996), pp. 163–171
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| *''The Group of Primitive Pythagorean Triangles''[http://www.jstor.org/pss/2690291], Ernest J. Eckert, ''Mathematics Magazine'' Vol 57 No. 1 (January, 1984), pp 22–26
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| [[Category:Abelian group theory]]
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