Transcendental number

From Citizendium
Revision as of 12:27, 8 May 2008 by imported>Barry R. Smith (irrationality vs. transcendence, pi, e)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search
This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In mathematics, a transcendental number is any complex number that is not algebraic, i.e. it is not a root of any polynomial whose coefficients are integers, or, equivalently, it is not a root of any polynomial whose coefficients are rational.

Transcendental numbers are necessarily irrational, but there are many irrational numbers that are not transcendental. For instance, is irrational. However it is algebraic, since it is a root of the polynomial . It is thus irrational but not transcendental.

Proving a number to be transcendental is generally much more difficult than just proving it is irrational. Examples of real numbers known to be transcendental are and .