Concept

Continued fraction convergent

A fraction obtained by cutting a continued fraction short, which is the best approximation available for the size of its denominator. The convergents of an irrational number are exactly the fractions that beat every fraction with a smaller denominator.

Named by 2 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

Named alongside it

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

Continued fractions Golden ratio Rational approximation Counting argument Dirichlet Euclidean algorithm Existence proof Fibonacci Greatest common divisor Incommensurability Liouville number Nonconstructive

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