Density of Rationals and Irrationals in the Reals
Abide by Reason reviews density properties for viewers who know that rationals “are everywhere” but haven’t seen how this interacts with measure.
Covering Rationals with Shrinking Intervals to Show Measure Zero
This argument is aimed at students of real analysis and probability who want a concrete proof that the rationals in ([0,1]) have Lebesgue measure zero.
Measure Zero of Rationals and Full Measure of Irrationals
This note targets learners reconciling the fact that rationals are dense yet “occupy” zero measure, while irrationals almost completely fill the interval.
Zero-Probability Events That Still Occur and the Dart on the Unit Interval
Abide by Reason addresses the common confusion around events that have probability zero but are not impossible, using the dart-on-the-interval thought experiment.
Cardinality Differences and Intuition for Rationals vs Irrationals Measure
This note is for viewers trying to build an intuitive bridge between the set-theoretic difference (countable vs uncountable) and the measure difference between rationals and irrationals.