Almost every number is transcendental, yet proving any single one is hard
Pi and e are the celebrities, but transcendental numbers, those that are not the root of any polynomial with integer coefficients, make up almost all real numbers. The strange part is how few can actually be proved transcendental, and how long it took mathematicians to confirm that any existed at all.
Every transcendental real number is irrational, but not the other way round. The square root of 2 is irrational yet solves x squared minus 2 equals zero, so it counts as algebraic. The word itself, from Latin for climbing beyond, entered mathematics in a 1682 paper by Leibniz, and Euler in the 18th century probably first defined these numbers in the modern sense. Lambert suspected in 1768 that both e and pi qualified.
Proof came slowly. Joseph Liouville showed in 1844 that transcendental numbers exist, and in 1851 built an example by hand: a decimal whose digits are 1 only at positions 1, 2, 6, 24 and so on, the factorials, and 0 everywhere else. Charles Hermite proved e transcendental in 1873, the first number shown to be so without being built for the purpose. In 1882 Ferdinand von Lindemann did the same for pi, which settled an ancient puzzle: a circle cannot be squared using only compass and straightedge.
Georg Cantor explained in 1874 why such numbers must be everywhere. Algebraic numbers can be listed one by one, since there are countably many polynomials each with finitely many roots, but the real numbers cannot be listed. So transcendental numbers vastly outnumber algebraic ones.
Plenty remains open. In 1900 David Hilbert asked whether an algebraic number raised to an irrational algebraic power must be transcendental; the Gelfond–Schneider theorem answered yes in 1934. Yet nobody knows whether e plus pi is transcendental, though at least one of e plus pi and e times pi must be. Kurt Mahler showed in 1953 that pi is not one of Liouville's special numbers.
Source: Transcendental number
