TITLE: Understanding quotient groups
QUESTION [6 upvotes]: Admittedly this will be probably be a naive question, but here it goes: 
Is it possible to flesh out in simple terms, for someone with little background in group theory, what it means to take the quotient group of $\mathbb{R}$ or $\mathbb{C}$ by a lattice $\Lambda$ or by $\mathbb Z$? How to evaluate this quotient? As in what comes out?
More precise examples: 

If $\Lambda$ is a lattice spanned by linearly independent primitive vectors $\mathbf{a}_1$ and $\mathbf{a}_2,$ then how to interpret the quotient: $\mathbb C /\Lambda?$
Another example: the circle $S^1$ can be identified with the quotient $\mathbb R/2\pi\mathbb Z,$ but I fail to make this identification myself. Any help is much appreciated here.

REPLY [2 votes]: It may help you to consider some examples of quotient groups that you are already comfortable in working with:

$\mathbb Z/n\mathbb Z$

The set of integers $\mathbb Z$ forms a group under addition. It contains a subgroup $n\mathbb Z=\{nk:k\in\mathbb Z\}$ consisting of the elements of $\mathbb Z$ that are divisible by $n$.
The quotient group is the set of cosets
$$\{n\mathbb Z, 1+n\mathbb Z, 2+n\mathbb Z,\ldots,(n-1)+n\mathbb Z\}$$
where $m+n\mathbb Z =\{m+nk:k\in\mathbb Z\}$, which is a group under the operation inherited from $\mathbb Z$:
$$(a+n\mathbb Z)+(b+n\mathbb Z) = (a+b)+n\mathbb Z$$
where you should notice that if $a+b\in n\mathbb Z$, then $(a+b)+n\mathbb Z = n\mathbb Z$.
This group is exactly the group of integers modulo $n$: write
$$a\equiv b\pmod n$$
if and only if $n\mid(b-a)$. Then the map sending $a+n\mathbb Z$ to $a\pmod n$ is an isomorphism.

$\mathbb R/\mathbb Z$

Let $[\cdot]:\mathbb R\to [0,1)$ be the map which sends a real number $x$ to its decimal part (e.g. $231.35642\mapsto 0.34642$).
If $x,y\in \mathbb R$, then it is easy to check that $$[x]+[y]=[x+y]$$
so that in particular, $[\cdot]$ defines a group structure on $[0,1)$. 
You should check that with this group structure, $[0,1)\cong \mathbb R/\mathbb Z$. $\mathbb R/\mathbb Z$ is therefore the group of real numbers where we consider two numbers as the same if they differ by an integer.
An important observation is that there is an isomorphism between $[0,1)$ and the unit circle in $S^1\subset\mathbb C^\times$ (both are groups under multiplication) given by
$$[0,1)\to S^1 \\ x\mapsto e^{2\pi ix}$$
So $\mathbb R/\mathbb Z\cong S^1$, and can be viewed as a unit line in $\mathbb R$, where the two ends are glued together.
The quotient $\mathbb C/(\mathbb Z+\mathbb Zi)$ is a two-dimensional analgoue of this: it is a unit square in $\mathbb R^2$ with opposite sides glued together - or a torus $S^1\times S^1$

and $\mathbb C/\mathbb Z$ can be defined similarly for a general lattice.