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 \(\alpha \equiv \pi\).