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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!

Jack J.
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    I think the statement is trying to compare this addition operation with the way we might take the sum of some angles in radians and reduce the sum to an equivalent angle in the range $[0,2\pi)$. But the language is very unclear. – David K Jan 27 '19 at 14:47

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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 $\Bbb R/\Bbb Z$.

Berci
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