Under what circumstances is it possible to represent irrational numbers that go on forever—like pi—with simple fractions, like 22/7? The proof establishes that the answer to this very general question turns on the outcome of a single calculation. The Duffin-Schaeffer conjecture has you add up the measures of the sets of irrational numbers captured by each approximating fraction. It represents this number as a large arithmetic sum. Then it makes its key prediction: If that sum goes off to infinity, then you have approximated virtually all irrational numbers; if that sum instead stops at a finite value, no matter how many measures you sum together, then you’ve approximated virtually no irrational numbers.

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