understanding $\mathbb{R}$/$\mathbb{Z}$ I am having trouble understanding the factor group, $\mathbb{R}$/$\mathbb{Z}$, or maybe i'm not.  Here's what I am thinking.
Okay, so i have a group $G=(\mathbb{R},+)$, and I have a subgroup $N=(\mathbb{Z},+)$.  Then I form $G/N$.  So this thing identifies any real number $x$ with the integers that are exactly 1 unit step away.  So if $x=\frac{3}{4}$, then $[x]=({...,\frac{-5}{4},\frac{-1}{4},\frac{3}{4},\frac{7}{4},...})$ and i can do this for any real number.  So therefore, my cosets are unit intervals $[0,1)+k$, for integers $k$.  Herstein calls this thing a circle and I was not sure why, but here's my intuition.  The unit interval is essentially closed and since every real number plus an integer identifies with itself, these "circles" keep piling up on top of each other as if its one closed interval.  Since it's closed it is a circle.  Does that make sense?  
Now how do I extend this intuition to this?
$G'=[(a,b)|a,b\in{\mathbb{R}}], N'=[(a,b)|a,b\in{\mathbb{Z}}].$  What is $G'/N'$?  How is this a torus?  I can't get an intuitive picture in my head...
EDIT:  Actually, are the cosets just simply $[x]=[x\in{\mathbb{R}}|x+k,k\in{\mathbb{Z}}]?$

You can also use the following nice facts. I hope you are inspired by them.
$$\mathbb R/\mathbb Z\cong T\cong\prod_p\mathbb Z(p^{\infty})\cong\mathbb R\oplus(\mathbb Q/\mathbb Z)\cong\mathbb C^{\times}$$
