TITLE: What are the generators of $(\mathbb{R},+)$?
QUESTION [11 upvotes]: The real numbers as a group under addition are an infinitely-generated group. I'm not sure what those generators are though (or if it even makes sense to ask the question).
For example, could we say that the generators are $1, 0.1,0.01,\dots$ as a real number can be defined as a limit $\sum_i^n w_i 10^{-i}$ for some $w$?

REPLY [14 votes]: When you talk about $\mathbb{R}$ as an abstract group, you've forgotten its topology, so you're no longer allowed to say the word "limit." The subgroup generated by the numbers $\frac{1}{10^n}, n \in \mathbb{N}$ is the subgroup of rational numbers with terminating decimal expansions, which is much smaller than the real numbers.
However, if you remember the topology, then $\mathbb{R}$ becomes a topological group, and then it's true that $\mathbb{R}$ is topologically generated by the numbers $\frac{1}{10^n}$ in the sense that the closure of the subgroup they generate is all of $\mathbb{R}$. 
In practice, though, once we start talking about topological groups it no longer becomes particularly useful to work with generators. Instead, in the special case of Lie groups a reasonable substitute is to work with their Lie algebras, which you can think of as their "infinitesimal generators."

REPLY [3 votes]: This hint is a good point but need you to know Divisible groups. It is not finitely generated because any nontrivial divisible group cannot be finitely generated.