number

The Algebraic and Transcendental Numbers

A real or complex number z is called algebraic if it is the root of a polynomial equation z ⁿ  +  a _{n   -  1} z n   -  1 + … +  a 1 z  +  a 0  =  0, where the coefficients a 0, a 1, …  a n   -  1 are all rational; if z cannot be a root of such an equation, it is said to be transcendental. The number 2 is algebraic because it is a root of the equation z 2 + 2  =  0; similarly, i, a root of z 2 + 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 © 2012, Columbia University Press. All rights reserved.