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

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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:

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


Hence for example:

$\eqclass 0 {2 \pi} = \set {2 k \pi: k \in \Z}$


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

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.