number: 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 − 1z^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 2 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.

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

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

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