Congruence (Number Theory)/Examples/Modulo 2 pi as Angular Measurement

Example of Congruence Modulo an Integer
Let $\RR$ denote the relation on the real numbers $\R$ defined as:


 * $\forall x, y \in \R: \tuple {x, y} \in \RR \iff \text {$x$ and $y$}$ measure the same angle in radians

Then $\RR$ is the congruence relation modulo $2 \pi$.

The equivalence classes of this equivalence relation are of the form:


 * $\eqclass \theta {2 \pi} = \set {\theta + 2 k \pi: k \in \Z}$


 * Measurement-of-Angle.png

Hence for example:
 * $\eqclass 0 {2 \pi} = \set {2 k \pi: k \in \Z}$

and:
 * $\eqclass {\dfrac \pi 2} {2 \pi} = \set {\dfrac {\paren {4 k + 1} \pi} 2: k \in \Z}$

Each equivalence class has exactly one representative in the half-open real interval:
 * $\hointr 0 {2 \pi} = \set {x \in \R: 0 \le x < 2 \pi}$

and have a one-to-one correspondence with the points on the circumference of a circle.