
# 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 $$\alpha \equiv \pi$$.