Exercise · ZW-DENSITY-OF-THE-RATIONALS-007 · ★★☆ Intermediate

In the density proof with \(a=0.3\) and \(b=0.4\), take \(n=11\) with \(\tfrac1n<b-a\) and \(m=\lfloor na\rfloor+1\). Which rational \(m/n\) results?

Choose one option.

Show answer

D. \(\tfrac4{11}\)

Explanation

\(na=3.3\), so \(m=4\) and \(q=\tfrac4{11}\approx0.364\), which lies between \(0.3\) and \(0.4\). Neither \(\tfrac3{11}\) nor \(\tfrac5{11}\) falls in the interval.

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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 \(q\).

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