π

The ratio of a circle's circumference to its diameter has been approximated since antiquity. The Babylonians used 3.125 around 1900 BCE. The Egyptians used 3.1605 in the Rhind Papyrus (~1650 BCE). Archimedes of Syracuse, around 250 BCE, calculated the ratio to be between 3.1408 and 3.1429 by inscribing and circumscribing 96-sided polygons around a circle. None of these civilizations used a single symbol for the ratio.

William Jones, a Welsh mathematician, introduced the Greek letter π for the ratio in his 1706 book Synopsis Palmariorum Matheseos. He chose π as the first letter of perimetros (περίμετρος, circumference) or periphereia (περιφέρεια, periphery). Leonhard Euler adopted the symbol in 1737, and Euler's influence made it universal. The number existed for millennia. The name is three centuries old.

Johann Heinrich Lambert proved in 1761 that π is irrational — its decimal expansion never terminates and never repeats. Ferdinand von Lindemann proved in 1882 that π is transcendental — it is not the root of any polynomial equation with rational coefficients. This meant that squaring the circle (constructing a square with the same area as a given circle using compass and straightedge) is impossible. A problem that had frustrated mathematicians for two thousand years was not merely unsolved but unsolvable.

Modern computers have calculated π to over 100 trillion decimal digits. The computation is used to test supercomputers and algorithms, not to learn more about π — for any practical purpose, 40 digits of π would suffice to calculate the circumference of the observable universe to within the width of a hydrogen atom. The pursuit of more digits is pure persistence. The number does not end. Neither does the calculation.