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Concept

Rational approximation

A fraction standing in for a number that is not one, and the question of how much accuracy a given denominator buys. How well a number can be approximated is measured against the denominator, and the golden ratio is the worst case.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

φ as a continued fraction. The nested fraction, one quotient per step, descending to the right.

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.

number · Euclidean algorithm 8 multiples of φ in 7 boxes. The fractional parts of the first multiples of a number, dropped into equal boxes along the unit interval.

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.

number · Pigeonhole How closely a fraction can come, and the barrier that says no closer. Two panels at very different scales: the approximations to √2, which stay above the barrier a degree-two number obeys, and the truncations of a constructed number, which fall below every barrier drawn.

Approached too fast to be algebraic

An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.

number · Irrationality 13 record approximations in 26 turns. The distance from π to each fraction the descent passes, against its denominator, on logarithmic axes. 13 of them beat every fraction with a smaller denominator.

The fractions that beat every smaller one

Walking down the tree towards a number produces a sequence of fractions closing in on it. Most of them are steps along the way; a few are the best approximations there are — closer than every fraction with a smaller denominator — and which few is decided by where the turns change direction.

number · Stern brocot

Named alongside it

The objects these essays reach for when they reach for this one.

Continued fractionsContinued fraction convergentDiophantine approximationExistence proofGolden ratioLiouville numberTranscendenceAlgebraic numberConstructionCounting argumentDegreeDirichlet

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