The following sections of this BookRags Literature Study Guide is offprint from Gale's For Students Series: Presenting Analysis, Context, and Criticism on Commonly Studied Works: Introduction, Author Biography, Plot Summary, Characters, Themes, Style, Historical Context, Critical Overview, Criticism and Critical Essays, Media Adaptations, Topics for Further Study, Compare & Contrast, What Do I Read Next?, For Further Study, and Sources.
(c)1998-2002; (c)2002 by Gale. Gale is an imprint of The Gale Group, Inc., a division of Thomson Learning, Inc. Gale and Design and Thomson Learning are trademarks used herein under license.
The following sections, if they exist, are offprint from Beacham's Encyclopedia of Popular Fiction: "Social Concerns", "Thematic Overview", "Techniques", "Literary Precedents", "Key Questions", "Related Titles", "Adaptations", "Related Web Sites". (c)1994-2005, by Walton Beacham.
The following sections, if they exist, are offprint from Beacham's Guide to Literature for Young Adults: "About the Author", "Overview", "Setting", "Literary Qualities", "Social Sensitivity", "Topics for Discussion", "Ideas for Reports and Papers". (c)1994-2005, by Walton Beacham.
All other sections in this Literature Study Guide are owned and copyrighted by BookRags, Inc.
Transcendental numbers, named after the Latin expression meaning to climb beyond, are numbers which exist beyond the realm of algebraic numbers. Mathematicians have defined algebraic numbers as those which can function as a solution to polynomial equations consisting of x and powers of x. In 1744, the Swiss mathematician Leonhard Euler (1707-1783) established that a large variety of numbers (for example, whole numbers, fractions, imaginary numbers, irrational numbers, negative numbers, etc.) can function as a solution to a polynomial equation, thereby earning the attribute algebraic. However, Euler pointed to the existence of certain irrational numbers which cannot be defined as algebraic. Thus,2, , and e are all irrationals, but they are nevertheless divided into two fundamentally different classes. The first number is algebraic, which means that it can be a solution to a polynomial equation. For example, 2 is the solution of x2 - 2 = 0. But and e cannot solve a polynomial equation, and are therefore defined as transcendental. While , which represents the ratio of the circumference of a circle to its diameter, had been known since antiquity, its transcendence took many centuries to prove: in 1882, Ferdinand Lindemann (1852-1939) finally solved the problem of "squaring the circle" by establishing that there was no solution. There are infinitely many transcendental numbers, as there are infinitely many algebraic numbers. However, in 1874, Georg Cantor (1845-1919) showed that the former are more numerous than the latter, suggesting that there is more than one kind of infinity.