The Algebraic and Transcendental Numbers

A real or complex number z is called algebraic if it is the root of a polynomial equation zⁿ + a_{n - 1}z^{n - 1} + … + a₁z + a₀ = 0, where the coefficients a₀, a₁, … a_{n - 1} are all rational; if z cannot be a root of such an equation, it is said to be transcendental. The number is algebraic because it is a root of the equation z² + 2 = 0; similarly, i, a root of z² + 1 = 0, is also algebraic. However, F. Lindemann showed (1882) that π is transcendental, and using this fact he proved the impossibility of “squaring the circle” by straight edge and compass alone (see geometric problems of antiquity). The number e has also been found to be transcendental, although it still remains unknown whether e + π is transcendental.

The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2007, Columbia University Press. All rights reserved.

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