π is fixed by a circle; some constants are just famous
A mathematical constant is a number locked by a clear definition and often nicknamed with a letter or a mathematician’s surname. Some, like π, fall out of geometry itself. Others stay famous mainly because history kept computing them to absurd precision.
Constants show up across fields; e and π wander from geometry into number theory, statistics and calculus with stubborn regularity. Popular ones have been chased for centuries into ever longer decimal expansions. Every named constant is definable, and nearly all can also be computed digit by digit, Chaitin's constant being the celebrated exception.
School curricula meet a first tier early. The principal square root of two, sometimes called Pythagoras' constant, is the positive number that squares to two; geometrically it is the diagonal of a unit square, courtesy of the Pythagorean theorem. It is algebraic and irrational, possibly the earliest number recognized as such, and begins 1.41421356…, catalogued as OEIS sequence A002193. Before calculators, people leaned on 99/70, which misses by less than one ten-thousandth despite its small denominator, and its continued fraction is simply 1 followed by an endless string of 2s.
Pi is defined in Euclidean geometry as a circle's circumference divided by its diameter, yet it also appears in the Gaussian integral, the complex roots of unity and the Cauchy distribution in probability. Physics is full of it too, from formulas where constants are defined with π factored out to the ground-state wave function of hydrogen. It is irrational and transcendental; 22/7 and 355/113 approximate it unusually well, and memorizing or computing ever more digits has become a record-chasing pursuit.
Euler's number e, the constant of exponential growth, can be defined as the value that (1 + 1/n) raised to the power n approaches as n grows without bound. Jacob Bernoulli met it through compound interest: start with one dollar at annual rate R, compound ever more often, and the balance after a year closes in on e to the power R.
Source: Mathematical constant
