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The set of rational numbers is customarily denoted $\mathbb{Q}$. I need a symbol for the set of rational numbers having denominator $D$. For example, for $D = 3$, this set would be $$\{ \cdots, -4/3, -3/3, -2/3, -1/3, 0/3, 1/3, 2/3, 3/3, 4/3, \cdots \}.$$

I am considering adopting the notation $\mathbb{Q}_D$, but I wanted to ask if there is a standard or more widely recognized symbol for this set.

  • If so, what is it?
  • If not, what symbol should I use?
  • Is it necessary to clarify explicitly that my set includes rational numbers in "unsimplified" form (e.g. $3/3$ in the example above)?
Max
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  • There is a notation for the rational numbers whose denominator is a power of $3$, that notation is $\mathbb Z[1/3]$. The idea is that $\mathbb Z[1/3]$ is the smallest subring of $\mathbb Q$ that contains $\mathbb Z$ and $1/3$. But your subset, while it is an additive subgroup, is not a subring; I don't think there's any standard notation for it. – Lee Mosher Jun 28 '22 at 02:12
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    There is the notation $,3\mathbb Z,$ for multiples of $,3,$, so maybe $,\frac{1}{3} \mathbb Z,$ or $,\mathbb Z/3,$ could be used for your set. However, that notation has potential for confusion, and would have to be clearly defined before being used. – dxiv Jun 28 '22 at 02:25
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    Thanks. $\mathbb{Z}/3$ was also suggested to me by a colleague, and I like this notation because it makes it possible to put a more complicated expression in the place of $3$ without destroying the legibility. (You should consider turning your comment into an answer. There is no rule against short answers.) – Max Jun 28 '22 at 03:10
  • @Max Posted as an answer, with some additional comments. – dxiv Jun 28 '22 at 04:36

1 Answers1

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The notation $\,n\mathbb Z\,$ used for integer multiples of $\,n\,$ could suggest a symmetrical notation $\,\mathbb Z/n\,$ for rational fractions with denominator $\,n\,$. This would also make it clear that the numerator is unrestricted, so the set would include fractions where a common factor might cancel out, for example $\,\frac{n}{n} = 1 \in \mathbb Z /n\,$.

One problem with this notation is that, in certain contexts at least, it would conflict with the traditional use of $\,\mathbb Z / I\,$ for quotient groups (or rings). In simple cases, the ambiguity could be resolved from the context alone, for example $\,\mathbb Z / 3\,$ could only mean the set of fractions with denominator $\,3\,$, while $\,\mathbb Z / 3\mathbb Z\,$ could only mean the ring of integers $\bmod 3\,$. However, this wouldn't always work in more complex cases, for example $\,\mathbb Z[x]/(x^2+1)\,$ could mean either the set of rational functions with denominator $\,x^2+1\,$ and the numerator a polynomial with integer coefficients, or the ring of polynomials $\,\bmod (x^2+1)\,$.

The notation could maybe use a symbol other than "$/$" to disambiguate, for example $\,\frac{\mathbb Z}{3}\,$ or $\,\mathbb Z \div 3\,$, though neither look too "natural".

dxiv
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