Christoffer De Geer March 21, 2026

Continued Fractions

pi = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))

written [3; 7, 15, 1, 292, 1, 1, 1, 2, ...]

A continued fraction expresses a number as an integer plus the reciprocal of another continued fraction. Every real number has a unique continued fraction expansion. Rational numbers terminate; quadratic irrationals repeat periodically; transcendentals like pi have no pattern. The convergents (rational approximations formed by truncating) are provably the best approximations of any rational with that size denominator.

Famous continued fractions compared: periodic = quadratic irrational

Table comparing continued fractions of phi sqrt2 e and pi showing which are periodic and which are irregular
CONSTANT	CF NOTATION	TYPE
phi	[1; 1, 1, 1, 1, ...]	periodic
sqrt(2)	[1; 2, 2, 2, 2, ...]	periodic
sqrt(3)	[1; 1, 2, 1, 2, ...]	periodic
e	[2; 1, 2, 1, 1, 4, 1, 1, 6...]	pattern
pi	[3; 7, 15, 1, 292, 1, ...]	no pattern
Theorem: a CF is periodic if and only if the number is a quadratic irrational (Lagrange, 1770)
phi is the "hardest" to approximate: its CF of all 1s is the worst possible convergence

Convergents of pi: best rational approximations

Table of convergents of pi showing increasingly accurate rational approximations with small denominators
CONVERGENT	DECIMAL	ERROR
3/1	3.000000	0.14159
22/7	3.142857	0.00126
333/106	3.141509	0.000083
355/113	3.141592…	0.0000003
103993/33102	3.14159265…	2.7e−10
355/113 is correct to 6 decimal places with only a 3-digit denominator

Successive convergents of π alternate above and below

Convergents 3, 22/7, 333/106, 355/113, 103993/33102 alternate above and below π. Each is the best rational approximation with that denominator or smaller.

Golden Ratio φ → Square Root of 2 → Khinchin's Constant → Lévy's Constant → Euler's Number e →

Related topics

Phi Sqrt2 Khinchin

Key facts about Continued Fractions

Every real number has a unique continued fraction expansion. Rational numbers have finite expansions. Quadratic irrationals (like sqrt(2) and phi) have eventually periodic expansions. Transcendentals like pi have no pattern. The convergents of a continued fraction are the best rational approximations: 22/7 and 355/113 are convergents of pi, matching it to 2 and 6 decimal places respectively. Phi = [1; 1, 1, 1, ...] is the hardest number to approximate, making it the most irrational in a precise sense.

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#memoryAll memory gamesHold a sequence, grid, or list in working memory and reproduce it under pressure.#numbersAll numbers gamesArithmetic, ratios, units, and number sense in short timed rounds.#sequenceAll sequence gamesReproduce or extend an ordered string of items in the right order.