Pi to N decimals

All about π: the most famous constant in mathematics

π is the ratio between the circumference and the diameter of any circle — a constant whose first few digits, 3.14159265358979323846…, every math student memorizes at some point. It was proven irrational by Johann Heinrich Lambert in 1761 (no finite or repeating decimal expansion) and transcendental by Ferdinand von Lindemann in 1882 (it is not a root of any polynomial with rational coefficients, which finally killed the ancient problem of "squaring the circle").

A short history of computing π

• Ancient Egypt — the Rhind papyrus uses (16/9)² ≈ 3.1605; ancient Babylon used 3.125.
• Archimedes (~250 BCE) — inscribed and circumscribed 96-sided polygons in a circle to prove 3 + 10/71 < π < 3 + 1/7, i.e. about 3.1418.
• Madhava–Leibniz series — π = 4·∑ (-1)ⁿ/(2n+1). Beautiful but painfully slow: roughly one correct digit per ten terms.
• Machin-like formulas (1706) — combinations of arctan that converge much faster and dominated record attempts for two centuries.
• Ramanujan (1914) — series that produce ~8 correct digits per term; foundation of modern record arithmetic.
• Gauss–Legendre — quadratic convergence (roughly quadruples the number of correct digits per iteration).
• Chudnovsky brothers (1989) — currently the fastest known algorithm in practice, giving about 14 correct digits per iteration; the engine behind every recent world record.
• Spigot / BBP (Bailey–Borwein–Plouffe, 1995) — lets you compute the n-th hexadecimal digit of π without computing the previous ones.

World records

In 2011, Shigeru Kondo reached 10 trillion digits on a home PC using Alexander Yee's y-cruncher. In 2022, Emma Haruka Iwao at Google Cloud hit 100 trillion digits. In 2024, StorageReview.com pushed it to 105 trillion digits. Beyond a few dozen digits, the value is mostly a benchmark for hardware and numerical libraries: with 39 digits of π you can compute the circumference of the observable universe to the precision of a single hydrogen atom.

Memorizing π and "Pi-em"

A popular memorization technique is the Pi-em: a poem in which the number of letters of each word equals a digit of π. The classic English one starts: "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics" — 3 . 1 4 1 5 9 2 6 5 3 5 8 9. The current record for memorized digits is held by Suresh Kumar Sharma (70,030 digits, 2015).

Pi Day, Tau Day and pop culture

Pi Day is celebrated on March 14 (3/14 in US date format) and was formally recognized by the US House of Representatives in 2009; it is also Albert Einstein's birthday. Tau Day (June 28, 6/28) celebrates the rival constant τ = 2π championed by Michael Hartl and Bob Palais. The digits of π also famously appear in the Indiana Pi Bill of 1897 (an attempt to legislate π = 3.2 that never became law).

FAQ

How many digits does this tool generate? The number of decimals is configurable. Behind the scenes the digits are produced using arbitrary-precision integer arithmetic (BigInt) so that you do not hit the 15–17 significant-digit ceiling of IEEE-754 double precision.

Why use BigInt instead of a floating-point loop? Standard JavaScript Number values only carry about 15 reliable decimal digits. Anything past that requires BigInt (integers of unbounded size) and a spigot- or Machin-like algorithm working on those integers.

Can I look up a specific sequence inside π? Yes — projects like pi.delivery and angio.net/pi let you search for birthdays or arbitrary strings in the first few hundred billion digits. Your phone number almost certainly appears somewhere.

Are π's digits really random? Statistically they pass every standard test of randomness, but π is not a random number — every digit is fully determined. Whether it is a normal number (every digit pattern appears with the expected frequency) is still an open mathematical question.