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The additive group Z of integers is generator by 1 but additive group of rational numbers is not why?

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    $\begingroup$ Try expressing some non integral number as a sum of $1$'s and $-1$'s. $\endgroup$
    – b00n heT
    Commented Nov 22, 2016 at 7:54
  • $\begingroup$ How would you get from one rational number to the next using the number 1? $\endgroup$
    – Triatticus
    Commented Nov 22, 2016 at 7:55

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For a finite number of rationals $q_1,\ldots,q_n \in \mathbb{Q}$, consider the group generated by these rationals. What can you say about the denominator of any number (when written in reduced form) in this group?

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