Let $X=[0,1)$. For every $x,y\in X$ we define $$ x\dot{+}y:= \begin{cases} x+y & \text{if $x+y<1$} \\ x+y-1 & \text{if $x+y\ge1$} \end{cases} $$ I do not understand the following statement:
This operation can be displayed as an sum of angles $\text{mod}\;2\pi$: the sum ($\text{mod}\;1$) of $y$ corresponds to the rotation ($\text{mod}\;2\pi$) of an angle $2\pi y.$
Could someone help me to understand this? Thanks!
Let $R_\theta$ be the rotation by angle $\theta$ around the origin.
Verify that it's a linear transformation of the plane $\Bbb R^2$.
Let $GL(\Bbb R^2)$ denote the group of invertible linear transformations of $\Bbb R^2$.
Now (one version of) the precise statement is that $(X,+) \to GL(\Bbb R^2),\ \ t\mapsto R_{2\pi t}$ is an (injective) group homomorphism.
Note also that $X$ is (isomorphic to) the quotient group $\Bbb R/\Bbb Z$.