What is pi, and why does it matter?

Pi is the ratio of a circle's circumference to its diameter, and the remarkable part is that this ratio is identical for every circle in flat geometry, whether the circle is the size of a coin or a galaxy. Its value is approximately 3.14159, and it is irrational, meaning it cannot be expressed as a fraction of two integers and its decimal expansion continues forever without repeating, a fact proven in 1761. It is also transcendental, proven in 1882, which settled the ancient problem of squaring the circle by showing the construction is impossible with compass and straightedge.

Pi is not confined to circles. It appears throughout mathematics and physics in places with no obvious circular geometry: in the normal distribution that governs statistics, in Fourier analysis underlying signal processing and audio compression, in Heisenberg's uncertainty principle, in Einstein's field equations, and in the behavior of waves and oscillations generally, essentially wherever periodicity or rotation is involved. Practical work rarely needs precision: NASA uses around 15 decimal places for interplanetary navigation, and roughly 40 digits would compute the circumference of the observable universe to within the width of a hydrogen atom, which makes the ongoing computation of trillions of digits a test of computing hardware and algorithms rather than a mathematical necessity.