An atlas of continued fractions · 22 constants · up to 2,000 steps each
Every number hides a staircase.
Take a number, split off its whole part, turn what’s left upside down, and repeat forever. The whole parts you collect are called partial quotients, and together they form the number’s fingerprint. Some fingerprints repeat, some count upward, most look like noise. Ramanujan read them like sentences. Put them side by side and the differences start to explain themselves.
Pick any constant in the bar above. Every chart on the page follows your choice.
Fingerprints · the first 40 partial quotients
Twenty-two fingerprints, side by side
Each row is a constant, each cell one partial quotient an. Brighter means bigger. Rows are grouped by what’s known about them. Hover a cell to see the fraction you get by stopping there, or click a row to focus it.
Best approximations · Hurwitz 1891
The golden ratio is the hardest number to approximate
Every stopping point gives a fraction p/q. The error shrinks roughly like 1/q², so multiply it back up: q²·|x − p/q|. Low dots are unusually good fractions. High dots mean the number resists being approximated.
The Gauss map · x ↦ 1/x − ⌊1/x⌋
One rule generates every staircase
Flipping the remainder is a function: x ↦ {1/x}, the fractional part of 1/x. Its graph is the fan of curves below, and each branch you land on is the next partial quotient. Follow a number’s path (a cobweb) and its whole staircase unrolls.
Khinchin 1935 · Lévy 1936
Random-looking numbers agree on one average
For almost every real number, the geometric mean of the partial quotients settles on the same value, Khinchin’s constant, 2.685452…. The denominators qn also grow at a universal rate. The numbers with structure are exactly the ones that break the rule.
Gauss–Kuzmin 1928 · P(a = k) = log₂(1 + 1/(k(k+2)))
Ones turn up 41.5% of the time
Gauss predicted, and Kuzmin proved, how often each partial quotient should appear in a typical number: 1 about 41.5% of the time, 2 about 17%, 3 about 9.3%, and so on down. Compare each constant’s actual counts with that prediction.
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Generalized continued fractions · Brouncker, Euler, Apéry, Ramanujan
Let the numerators grow too, and π and e turn orderly
Allow numbers other than 1 on top of each step and chaos turns into order. π’s simple staircase looks random, but 4/π has one with odd squares climbing the numerators. Ramanujan filled notebooks with fractions like these. In 2021 a computer search called the Ramanujan Machine started finding new ones. Pick a formula to see how fast it homes in on its constant.
The Rogers–Ramanujan continued fraction · 1894 / 1913