Limit of $\mathbb{Z}/n\mathbb{Z}$ This is a super weird question coming from someone who is just starting out in mathematics.
If $n$ is a positive integer, one can notice that as $n$ grows, $\mathbb{Z}/n\mathbb{Z}$ starts to look more and more like $\mathbb{N}$. If I were a small integer $k$, the entire group would begin to look more and more indistinguishable from $\mathbb{N}$ as $n$ grows - my additive inverse is getting further and further away. Is this idea something that is studied or formalized anywhere?
 A: This can be formalized to some extent in a few different ways depending on what you want to do. In additive number theory it's common to model problems in $\mathbb{Z}$ as problems in $\mathbb{Z}/n$ for large $n$; the idea is that e.g. if all you're doing is adding up a bounded number of bounded elements it doesn't matter which of the two you do it in, if $n$ is large enough. This can be formalized using concepts like local groups or Freiman homomorphisms; this is not at all my area though so I don't know much more than that.
In the areas I'm more familiar with it is possible to rigorously define a particular kind of "limit" of the groups $\mathbb{Z}/n$, namely their categorical limit. This produces a very interesting group called the profinite integers $\widehat{\mathbb{Z}}$, which show up in various places, for example as the absolute Galois group of a finite field. These are a kind of "completion" of the integers; in particular they're uncountable, but $\mathbb{Z}$ still sits inside as a dense subgroup.
