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.

• 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.

See more Encyclopedia articles on: Mathematics