Continued fraction convergent — where it appears
Also named here as rational approximation — the same set of essays touches all of them, so they are one junction rather than several.
A fraction that never closes
Euclid's algorithm throws away everything except the number of squares it peeled at each step. Those counts are a second name for the number it started from — one that terminates exactly when the ratio is a ratio.
How close a fraction can get
Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.
Named alongside it
The objects these essays reach for when they reach for this one.
Golden ratioRational approximationContinued fractionsCounting argumentDirichletEuclidean algorithmExistence proofFibonacciGreatest common divisorIncommensurabilityLiouville numberNonconstructive