# How would one notate the subset of the rationals with terminating decimal expansions?

Is there a convention for notating a subset of the rationals with restrictions on the denominators? I'd prefer there to be a relatively intuitive and concise notation for the set $$\{\frac{n}{10^m}:n,m\in\mathbb{Z}\}$$ in the ballpark of something along the lines of the following made-up notation: $$\mathbb{Q}_{10}$$

• You could call it $\frac{\mathbb Z}{10^{\mathbb N_0}}$. There isn't a specific convention for that (that I know of), but there's a general convention that applying operations to sets that are usually applied to its elements yields the set of all elements you could get with any elements from the sets, e.g. $A+B=\{a+b\mid a\in A\land b\in B\}$. (Note that you shouldn't use $\mathbb Z/10^{\mathbb N_0}$, since the quotient is defined for rings over ideals and thus might be misunderstood (though in this case that wouldn't make sense since $10^{\mathbb N_0}$ is not an ideal of $\mathbb Z$).) Apr 30, 2020 at 8:45

I think $${\Bbb Z}[\frac{1}{10}]$$ is quite standard. It is used for instance in the definition of the Prüfer group:
... where $${\Bbb Z}[1/p]$$ denotes the group of all rational numbers whose denominator is a power of p, using addition of rational numbers as group operation