What is the standard notation for the set of rationals with finite fractional part?

Question

As the title says. What is the standard notation for the set of rationals with finite fractional part, when written in base $n$ with a radix point? I expected $ℚ_n$, but that's taken for n-adic numbers.

Answer 1

The set of all rationals with eventually-terminating base-$k$ representation has a snappy algebraic description: it's the smallest ring containing every integer as well as ${1\over k}$. The standard notation for this is $$\mathbb{Z}[k^{-1}]$$ (or $\mathbb{Z}[{1\over k}]$), which in my experience is usually pronounced "$\mathbb{Z}$ adjoin $1\over k$" or "$\mathbb{Z}$ adjoin $k^{-1}$."

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The square brackets, incidentally, are important: when working with fields, round brackets refer to the smallest field extension containing the given element. So e.g. ${1\over\pi}\not\in\mathbb{Q}[\pi]$ but ${1\over\pi}\in \mathbb{Q}(\pi)$. That's not an issue here, but it's worth noting.