Pi

c. 1900 BC – present

Mathematics — constants and geometry

The ratio of a circle's circumference to its diameter, the same for every circle, and approximately 3.14159. Denoted by the sixteenth letter of the Greek alphabet, it is irrational — proved so by Johann Lambert in 1761 — and transcendental, proved by Ferdinand von Lindemann in 1882, which settled once and for all that a circle cannot be squared with ruler and compasses. It appears far beyond geometry, in probability, Fourier analysis and the Gaussian integral, and its decimal expansion has become the standard showpiece for supercomputing records.

How to spot them

▸The ratio of a circle's circumference to its diameter, roughly 3.14159
▸Proved irrational by Lambert in 1761 and transcendental by Lindemann in 1882
▸Because it is transcendental, the ancient problem of squaring the circle is impossible
▸The symbol was popularised by Euler in 1737 after William Jones used it in 1706
▸Celebrated on 14 March, and approximated by the fraction 22/7

Key facts

Archimedes trapped it between 223/71 and 22/7 using 96-sided polygons (c. 250 BC)

Measurement of a Circle — the first rigorous bounds, by the method of exhaustion

Lindemann's 1882 proof that it is transcendental, which killed squaring the circle (1882)

Ferdinand von Lindemann showed no polynomial with rational coefficients has it as a root

The fraction 355/113, accurate to six decimal places, found by Zu Chongzhi (c. 480)

Milü — the best simple rational approximation until the 15th century

An 1897 bill in the Indiana General Assembly that would have legislated its value (1897)

The Indiana bill, which implied a value of about 3.2; it was quietly shelved in the state Senate

A 2025 record run reaching 314 trillion decimal places on a single server (2025)

StorageReview's 110-day calculation — the digit count itself a pun on 3.14

Celebrated worldwide on 14 March, a date chosen for how Americans write it (1988)

Larry Shaw founded the celebration at the San Francisco Exploratorium

Buffon's needle: drop a needle on ruled lines and the drop rate estimates it (1777)

The Comte de Buffon's needle problem — the first geometric probability problem

Key ideas

Irrational means it cannot be written as a fraction; transcendental means it is not the root of any polynomial with rational coefficients. Transcendence is the stronger property, and is what makes squaring the circle impossible.
It appears where no circle is visible: the Gaussian integral, the Basel problem, the Wallis product and the Leibniz series all produce it.

Quick-fire facts

•Approximately 3.14159265358979; the fraction 22/7 is accurate to two decimal places, 355/113 to six
•The Greek letter was first used for the ratio by the Welsh mathematician William Jones in 1706, and made standard by Leonhard Euler from 1737
•Johann Heinrich Lambert proved it irrational in 1761; Ferdinand von Lindemann proved it transcendental in 1882
•Archimedes bounded it between 3 10/71 and 3 1/7 by inscribing and circumscribing 96-sided polygons
•The Rhind Papyrus implies a value of about 3.1605; the Babylonians used 25/8
•The Basel problem, solved by Euler in 1735, gives the sum of the reciprocals of the squares as its square divided by six
•The Leibniz–Gregory series 1 − 1/3 + 1/5 − 1/7 + … converges to a quarter of it, but agonisingly slowly
•Modern records use the Chudnovsky algorithm; the 2025 record stood at 314 trillion digits
•A 'piem' is a mnemonic in which each word's letter-count gives the next digit, e.g. 'How I wish I could calculate…'
•The Feynman point is the run of six consecutive 9s beginning at the 762nd decimal place
•It is said that only about 40 digits are needed to compute the circumference of the observable universe to the width of a hydrogen atom

Drill it — 8 cards

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Sources

MacTutor History of Mathematics Archive, University of St Andrews — Pi through the ages
Tom's Hardware — Pi calculation world record shattered at 314 trillion digits