Transcendental number

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A transcendental number is any real or complex number that is not an algebraic number, i.e., it is not the solution of any polynomial equation of the form

anxn + an-1xn-1 + ... + a1x1 + a0 = 0

where n >= 1 and the coefficients ai are integers (or, equivalently, rationals), not all 0.

The set of algebraic numbers is countable while the set of transcendental numbers is uncountable, so in a very real sense there are many more transcendental numbers than algebraic ones. However, only few classes of transcendental numbers are known and proving that a given number is transcendental can be extrememely difficult.

The first numbers to be proved transcendental were the Liouville numbers, by Joseph Liouville in 1844. This was also the first proof that transcendental numbers exist. The first important number to be proved transcendental was e, by Charles Hermite in 1873. Other known transcendental numbers include:

  • ea if a is algebraic and nonzero
  • π
  • eΠ
  • 2√2 or more generally ab where a ≠ 0,1 is algebraic and b is algebraic but not rational. The general case of Hilbert's seventh problem, namely to determine whether ab is transcendental whenever a ≠ 0,1 is algebraic and b is irrational, remains unresolved.
  • sin(1) (see trigonometric function)
  • ln(a) if a is positive, rational and ≠ 1 (see natural logarithm)
  • Γ(1/3) and Γ(1/4) (see Gamma function).