円周率の桁

Archimedes gave the first rigorous bounds in Measurement of a Circle, around 250 BCE: by inscribing and circumscribing 96-sided polygons he pinned the ratio between 223/71 and 22/7. For the next eighteen centuries, progress mostly meant more sides. Ludolph van Ceulen reached 35 decimal places by polygon arithmetic around 1600, a feat that got π called the Ludolphine number in German writing for generations.

Infinite series broke the deadlock. Madhava of Sangamagrama and the Kerala school found the arctangent power series in the fourteenth century — including the expansion usually credited to Leibniz, who arrived at it independently in 1673 — and Madhava used twenty-one terms plus a correction factor to get eleven correct decimals. William Jones wrote the bare symbol π in 1706, and Euler's adoption of it in 1736 and 1748 settled the notation. Lambert proved π irrational in a proof presented in 1761 and printed in 1768. Lindemann proved it transcendental in 1882, which answered the ancient problem of squaring the circle: it cannot be done.

Hand computation produced one famous casualty. William Shanks published 707 decimal places in 1873 and was believed for seventy years, until D. F. Ferguson, working with a mechanical desk calculator in 1944–45, found that Shanks had gone wrong at the 528th place and that every digit after it was wrong too.

Legislatures fared worse. House Bill 246 of the 1897 Indiana General Assembly, drafted from Edward J. Goodwin's circle-squaring, implied π = 3.2. The House passed it without dissent; the Senate shelved it indefinitely after C. A. Waldo of Purdue, in town to ask for funding, explained the problem to the senators. It never became law.

Modern records use the Chudnovsky brothers' 1988 series — the one this page runs. Emma Haruka Iwao and Google Cloud reached 100 trillion decimal places in 2022, and the record passed 300 trillion in 2025.

主な性質

• π is irrational, proved by Johann Heinrich Lambert in work presented in 1761 and published in 1768, so its decimal expansion never terminates and never settles into a repeating block.
• π is transcendental, proved by Ferdinand von Lindemann in 1882: it is not a root of any polynomial with rational coefficients, which is exactly why a circle cannot be squared with compass and straightedge.
• Archimedes proved 223/71 < π < 22/7, so the familiar 22/7 ≈ 3.142857 is an overestimate.
• 355/113 = 3.14159292… matches π through the first six decimal places, a relative error of roughly 8 × 10⁻⁸.
• Six consecutive 9s begin at the 762nd decimal place. The nickname "Feynman point" is apocryphal — the story is absent from Feynman's memoirs, and the earliest known version is Douglas Hofstadter's in 1985.
• Whether π is normal, meaning every block of digits appears with its expected frequency, is an open problem; the digits computed so far pass the usual statistical tests but that proves nothing.
• The Bailey–Borwein–Plouffe formula, found by Simon Plouffe in 1995, yields the nth hexadecimal digit of π without computing the earlier ones. No comparable formula is known for base 10.
• Each term of the Chudnovsky series adds about 14.18 decimal digits, so the 5,000 digits offered here need fewer than 360 terms.

登場する場面

• Every formula for a round thing: circumference 2πr, area πr², the period of a pendulum, and the normalising constant of the normal distribution.
• Buffon's needle problem estimates π from the fraction of randomly dropped needles that cross a set of ruled lines — one of the earliest Monte Carlo methods.
• y-cruncher, the program behind the modern digit records, is widely used to stress-test CPUs and memory, because an arithmetic slip anywhere shows up when the final result fails its checksum.
• Pi Day, 14 March, from the US date order 3/14; Emma Haruka Iwao's 31.4-trillion-digit record was dated to Pi Day 2019.
• Digit-memorisation contests, where reciting tens of thousands of places is a competitive sport — a feat of mnemonics rather than of mathematics.