# 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 * 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.

- Introduction
- The Real Numbers
- The Integers and Rational Numbers
- The Complex Numbers
- The Algebraic and Transcendental Numbers
- The Natural Numbers
- Bibliography

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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