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.
Archimedes used 96-sided polygons to prove 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. Every digit shown below is computed from the leibniz formula.