Exercise · ZW-DENSITY-OF-THE-RATIONALS-004 · ★☆☆ Basic
Is there a positive rational smaller than all the others?
Choose one option.
Show answer
- D. No: given \(q>0\), \(q/2\) is smaller
Explanation
If such a \(q\) existed, density would give a rational between \(0\) and \(q\) (for instance \(q/2\)), a contradiction. Also \(0\) is not positive.
Review the concept: Density of the rationals
Property of the rationals \(\mathbb Q\) inside the reals: between any two distinct reals \(a<b\) there is always a rational \(q\).
\[a
More practice problems: density of the rationals
- ZW-DENSITY-OF-THE-RATIONALS-001 Between any two distinct reals there is always:
- ZW-DENSITY-OF-THE-RATIONALS-002 Which rational lies between tfrac13 and tfrac12?
- ZW-DENSITY-OF-THE-RATIONALS-003 Which rational lies between √2 and √3?
- ZW-DENSITY-OF-THE-RATIONALS-005 Which number lies between 0.999 and 1?
- ZW-DENSITY-OF-THE-RATIONALS-006 The rational midpoint of tfrac27 and tfrac37 is:
- ZW-DENSITY-OF-THE-RATIONALS-007 In the density proof with a=0.3 and b=0.4, take n=11 with tfrac1n<b-a and m=lfloor narfloor+1. Whic…
- ZW-DENSITY-OF-THE-RATIONALS-008 Which of these numbers between 1 and 2 is irrational?
- ZW-DENSITY-OF-THE-RATIONALS-009 Between two reals a<b there are:
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