What is Pi (π)?
Pi is the ratio of any circle's circumference to its diameter. No matter the size of the circle, this ratio is always exactly the same: π = 3.14159265358979... The definition is geometric but pi appears in physics, probability, engineering, and every branch of mathematics.
Pi cannot be written as a fraction of two integers (proved by Johann Heinrich Lambert in 1761). It is also transcendental: not the solution to any polynomial with integer coefficients (proved by Ferdinand von Lindemann in 1882). This means it is impossible to square a circle using compass and straightedge. Its decimal expansion never ends and never repeats.
Archimedes of Syracuse (~250 BCE) was the first to rigorously bound pi, showing it lies between 3+10/71 and 3+1/7 using inscribed and circumscribed polygons with 96 sides. The Babylonians used 3.125, and the Egyptians 3.1605. The symbol π was introduced by Welsh mathematician William Jones in 1706 and popularised by Euler. As of 2024, pi has been computed to over 100 trillion decimal digits.
Pi appears far beyond circles: in the normal distribution (the bell curve contains √(2π)), in Euler's identity e^(iπ) + 1 = 0, in the probability that two random integers share no common factor (6/π²), in Stirling's factorial approximation n! ≈ √(2πn)(n/e)ⁿ, in quantum mechanics, and in the formula for the volume of a sphere (4πr³/3).
π ≈ 3.14159265358979323846. Irrational (Lambert, 1761). Transcendental (Lindemann, 1882). Pi Day is March 14 (3/14 in US date format). The fraction 22/7 overestimates pi by 0.04%. The better approximation 355/113 is accurate to 6 decimal places. Whether pi is a normal number (every digit sequence appearing with equal frequency) is unknown but widely believed.
Starting with 6-sided hexagons (giving 3.000 < π < 3.464) and doubling up to 96-sided polygons, Archimedes proved 3 + 10/71 < π < 3 + 1/7, giving 3.1408 < π < 3.1429. He never computed π, he trapped it. The method works because the circle's perimeter lies between the two polygon perimeters.
Pi is irrational. Its decimal expansion never ends and never repeats. The digits shown below are verified against the machin formula.
FAQ
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Who discovered \u03C0 and when?
Ancient Egyptians and Babylonians approximated \u03C0 around 1900-1600 BC. Archimedes of Syracuse (~250 BC) gave the first rigorous method to bound it. Welsh mathematician William Jones introduced the symbol \u03C0 in 1706, and Leonhard Euler popularized it globally. -
Why is March 14 called Pi Day?
In the month/day format used in the US, 3/14 matches the first three digits of \u03C0 (3.14...). Pi Day was first organized by physicist Larry Shaw at the San Francisco Exploratorium in 1988, and the US Congress officially recognized it in 2009. -
When was \u03C0 proven to be irrational?
Johann Heinrich Lambert proved \u03C0 is irrational in 1761, meaning it cannot be written as a simple fraction. Mathematicians had used approximations like 22/7 for centuries without knowing whether an exact fraction existed. -
When was \u03C0 proven to be transcendental?
Ferdinand von Lindemann proved \u03C0 is transcendental in 1882. This settled the ancient Greek problem of squaring the circle - constructing a square with the same area as a circle using only compass and straightedge - proving it is impossible. -
What formula is used to compute \u03C0 to billions of digits?
Modern computations use the Chudnovsky formula and the Bailey-Borwein-Plouffe (BBP) formula. The BBP formula (published 1995) allows any hexadecimal digit of \u03C0 to be computed directly without calculating the preceding digits.
Pi
Memorize pi, e, and 38 mathematical constants using the numpad path method
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