Transcendental number

A number that is not a root of any nonzero polynomial with rational coefficients. Every transcendental number is irrational, but not every irrational number is transcendental.

Algebraic number: A number that is a root of a nonzero polynomial with rational coefficients. Transcendental numbers are precisely the numbers outside this class.
Pi: The ratio of a circle's circumference to its diameter in Euclidean geometry. Its transcendence was proved by Ferdinand von Lindemann in 1882.
Liouville's theorem on diophantine approximation: A theorem bounding how closely algebraic numbers can be approximated by rational numbers. Its bounds let sufficiently accurate rational approximations prove a number transcendental.
Joseph Liouville: A French mathematician who proved the existence of transcendental numbers and constructed examples. His approximation theorem and explicit construction established the subject's first rigorous examples.
Squaring the circle: The classical problem of constructing a square equal in area to a given circle using straightedge and compass. Pi's transcendence proves this construction impossible under the classical rules.
Polynomial: An expression formed from constants and variables using addition, subtraction, multiplication, and nonnegative integer powers. Transcendence asks whether a number makes some rational-coefficient polynomial equal zero.
Euler's number: The base of the natural logarithm, approximately 2.71828. Its transcendence follows from Charles Hermite's 1873 proof.
Hermite's proof of the transcendence of e: Charles Hermite's 1873 proof that Euler's number is not algebraic. It established a landmark constant's transcendence using auxiliary functions and estimates.
Charles Hermite: A French mathematician who proved that Euler's number is transcendental. His 1873 result was the first proof for a major named mathematical constant.
Straightedge and compass construction: A geometric construction using only an unmarked straightedge and a compass. Constructible lengths are algebraic, sharply limiting which constants such constructions can produce.

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