Editing Quotient group (section)

=== Real numbers modulo the integers ===
Consider the group of [[real number]]s <math>\R</math> under addition, and the subgroup <math>\Z</math> of integers. Each coset of <math>\Z</math> in <math>\R</math> is a set of the form {{tmath|1= a+\Z }}, where <math>a</math> is a real number. Since <math>a_1+\Z</math> and <math>a_2+\Z</math> are identical sets when the non-[[integer part]]s of <math>a_1</math> and <math>a_2</math> are equal, one may impose the restriction <math>0 \leq a < 1</math> without change of meaning. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The quotient group <math>\R\,/\,\Z</math> is isomorphic to the [[circle group]], the group of [[complex number]]s of [[absolute value]] 1 under multiplication, or correspondingly, the group of [[rotation]]s in 2D about the origin, that is, the special [[orthogonal group]] {{tmath|1= \mathrm{SO}(2) }}. An isomorphism is given by <math>f(a+\Z) = \exp(2\pi ia)</math> (see [[Euler's identity]]).