transcendental number
Numbers split into two camps. Algebraic numbers are those captured by a polynomial equation with whole-number (or rational) coefficients — like sqrt(2), which solves x^2 - 2 = 0. A transcendental number is one that escapes every such net: no polynomial with rational coefficients, however high its degree, has it as a root. It “transcends” all algebra of this kind.
Precisely, a real or complex number is transcendental if it is not algebraic over the rational numbers — that is, it satisfies no nonzero polynomial equation with rational coefficients. The two most famous transcendental numbers are pi (proven transcendental by Lindemann in 1882) and e (proven by Hermite in 1873).
Although they are hard to identify individually, transcendental numbers are overwhelmingly common: the algebraic numbers form a countable set, while the real numbers are uncountable, so almost every real number is transcendental. Yet proving any specific number transcendental is often very difficult, and the status of many natural constants is still unknown.
The number e (about 2.71828) is transcendental: no equation like c0 + c1·e + c2·e^2 + ... = 0 with rational coefficients (not all zero) can ever hold. By contrast sqrt(2) is algebraic, since x^2 - 2 = 0.
e is transcendental; sqrt(2) is algebraic.
Because pi is transcendental, the ancient problem of “squaring the circle” with straightedge and compass is impossible — that construction would require pi to be algebraic of a special kind. Not all irrational numbers are transcendental: sqrt(2) is irrational yet algebraic.