The best you can do with a small denominator
Pick a number that is not a fraction. Ask for the nearest fraction with a denominator under a hundred, a thousand, a million. The answers are not a scattered list — they are rungs on one ladder, and the ladder is the same algorithm whether the number is π, the length of the year, or a perfect fifth. Everything below is computed on this page, and every rung is then handed to a brute-force search over every denominator up to two hundred thousand, which has never heard of the algorithm and gets a vote.
The ladder
The record board
Every denominator from 1 to Q, each with its own nearest numerator: one dot per fraction, height is how far it missed. Almost all of them are useless. The record holders — each one nearer than every fraction with a smaller denominator — are joined into the staircase, and the question is whether the ladder above predicted them.
The good fraction nobody quotes
The famous list is the convergents, and it is the right list for a
different question: which fraction wins on |qx − p|, the miss
per unit of denominator. On the plainer question — which fraction is simply
nearest — the winners are the convergents and the
intermediate fractions, the ones you pass through while climbing a single
rung. A textbook rarely names them, and one of them is always sitting
between two famous ones, quietly better than the first.
Four names for one number
A fraction is chosen, the world runs on it, and what is left over gets a
name and a literature. Every row below is the same quantity —
|qx − p|, the miss — wearing local clothes.
The worst number to approximate
Scale each miss by q² and every number gets a score you can compare: the
smallest value q²·miss keeps coming back to. It is a score for
how badly a number can be approximated, and Hurwitz proved in 1891
that no number scores above 1/√5 — every real number has fractions
that get at least that close, infinitely many times. One number sits exactly
on the bound, and it is the one whose ladder has no big rungs anywhere: all
ones, forever, so nothing is ever a windfall.
A sunflower puts the golden ratio between successive seeds for this reason
and no other: a seed angle that is well approximated by p/q
makes q spokes and wastes the space between them. The worst-approximable
number is the one that never lines up.
Where the ladder stops
A continued fraction is a merciless amplifier of what you do not know. Nudge the number by one part in the last digit you trust and the early terms do not move — but the late ones are pure invention. So the page climbs the ladder twice, once from each end of the interval it actually knows, and keeps only the rungs where the two agree.