1.8: Reals modulo 2π

Consider three half-lines starting from the same point, $[OA)$, $[OB)$, and $[OC)$. They make three angles $AOB$, $BOC$, and $AOC$, so the value $\measuredangle AOC$ should coincide with the sum $\measuredangle AOB + \measuredangle BOC$ up to full rotation. This property will be expressed by the formula

$$\measuredangle AOB + \measuredangle BOC \equiv \measuredangle AOC,$$

where "$\equiv$" is a new notation which we are about to introduce. The last identity will become a part of the axioms.

We will write $\alpha \equiv \beta$ (mod $2 \cdot \pi$), or briefly

$$\alpha \equiv \beta$$

if $\alpha = \beta + 2 \cdot \pi \cdot n$ for some integer $n$. In this case we say

"$\alpha$ is equal to $\beta$ modulo $2 \cdot \pi$".

For example

$-\pi \equiv \pi \equiv 3 \cdot \pi$ and $\dfrac{1}{2} \cdot \pi \equiv -\dfrac{3}{2} \cdot \pi$.

The introduced relation "$\equiv$" behaves as an equality sign, but

$\cdots \equiv \alpha - 2\cdot \pi \cdots \alpha \cdots \alpha + 2 \cdot \pi \equiv \alpha + 4 \cdot \pi \equiv \cdots$;

that is, if the angle measures differ by full turn, then they are considered to be the same.

With "$\equiv$", we can do addition, subtraction, and multiplication with integer numbers without getting into trouble. That is, if

$\alpha \equiv \beta$ and $\alpha' \equiv \beta'$,

then

$\alpha + \alpha' \equiv \beta + \beta'$, $\alpha - \alpha' \equiv \beta - \beta'$ and $n \cdot \alpha \equiv n \cdot \beta$

for any integer $n$. But "$\equiv$" does not in general respect multiplication with non-integer numbers; for example

$\pi \equiv -\pi$ but $\dfrac{1}{2} \cdot \pi \not \equiv -\dfrac{1}{2} \cdot \pi$.

Exercise $\PageIndex{1}$

Show that $2 \cdot \alpha \equiv 0$ if and only if $\alpha \equiv 0$ or $\alpha \equiv \pi$.

Hint

The quation $2 \cdot \alpha \equiv 0$ means that $2 \cdot \alpha = 2 \cdot k \cdot \pi$ for some integer $k$. Therefore, $a = k\cdot \pi$ for some integer $k$.

Equivalently, $\alpha = 2 \cdot n \cdot \pi$ or $\alpha = (2 \cdot n + 1) \cdot \pi$ for some integer $n$. The first identity means that $\alpha \equiv 0$ and the second means that $\alpha \equiv \pi$.