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The cosets of $\mathbb{Z}$ in $\mathbb{R}$ are all sets of the form $a+\mathbb{Z}$, with $0 ≤ a < 1$ a real number. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. -- Examples of Quotient Group, Wiki

I cannot figure out the differences between $\mathbb{R}/ \mathbb{Z}$ and $\mathbb{R}$. Besides, "subtracting 1 if the result is greater than or equal to 1", what does "the result" mean here? Why do we need to subtract 1? I was wondering what is the background of $\mathbb{R}/ \mathbb{Z}$.

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    There is a non-zero element in the quotient which is its own inverse.2012-02-02
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    So many high quality answers I don't feel the need to post another one!!2012-02-02
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    A useful property is the following: If $G$ is a finite subgroup of $\mathbb{R}/\mathbb{Z}$ then $G$ is cyclic. The easiest way to see this is through the correspondence provided by Dylan below.2012-02-02
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    Now $\mathbb{R}/\mathbb{Z}$ has non-trivial finite subgroups, but how many does $\mathbb{R}$ have?2012-02-02
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    $\mathbb{R}^n/\mathbb{Z}^n$ is an $n-$torus!2012-05-15

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