Can I define the quotient group $\mathbb{Q}/\mathbb{Z}$ under addition as $\{\frac{a}{b}|a,b\in\mathbb{Z}\}$ or is that wrong? How would you define it?
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I meant the quotient group under addition – charles_xavier Feb 18 '15 at 12:36
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1So you didn't mean all rational numbers except for integers? – barak manos Feb 18 '15 at 12:38
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i edited the question – charles_xavier Feb 18 '15 at 12:39
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Can you please describe this set in plain English? – barak manos Feb 18 '15 at 12:41
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@barakmanos the quotient group of the rational numbers (with addition as operation) by the subgroup of integers. See the defintion of Quotient Group if necessary. – quid Feb 18 '15 at 12:46
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2What you have written is wrong. The set of elements of ${\mathbb Q}/{\mathbb Z}$ is equal to the set of cosets ${\frac{a}{b} + {\mathbb Z} \mid a,b \in {\mathbb Z}, b \ne 0 }$. – Derek Holt Feb 18 '15 at 13:01
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$p/q$ ~ $p/q + z, \forall z \in \mathbb{Z}$ – Vinícius Ferraz Feb 18 '15 at 14:40
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It is the rational circle group. – Bill Dubuque Feb 18 '15 at 15:05
2 Answers
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The set under consideration is as you say the quotient group of the rationals with addition by the integers.
Taken strictly its elements are equivalence classes of rational numbers, so $$\mathbb{Q}/\mathbb{Z} = \{[q] \colon q \in \mathbb{Q} \}$$ where $[q] = \{q' \in \mathbb{Q} \colon q - q' \in \mathbb{Z}\}$.
Having said this what you mean is perhaps a set of representative of the classes, here you could take all rationals $q$ such that $0 \le q <1$, or staying with your notation all $a/b$ with a nonzero natural number $b$ and $0 \le a < b$ also natural (and $\gcd(a,b)=1$ if you want to eliminate duplicates).
However if you do this you need to be carful how you define your operation: you could say for $0 \le q,q' < 1$ you have $q\oplus q' = q + q'$ if $q+q'<1$ and $q\oplus q' = q + q' - 1$ if $1\le q+q'$ (note that $q + q'$ is always less than $2$); here I denote by $\oplus$ the operation on your set of representatives and by $+$ the usual addition in the rationals.
quid
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The correct definition of $\mathbb{Q}/\mathbb{Z}$ is given by taking $\mathbb{Q}$ and identifying elements $a,b\in\mathbb{Q}$ whenever $a-b\in\mathbb{Z}$.
You can make the identification $$\mathbb{Q}/\mathbb{Z} = \{e^{2i\pi t}\in\mathbb{C}|t\in\mathbb{Q}\}\subset S^1$$ the "rational points" in the (complex) unit circle.
Daniel Robert-Nicoud
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