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Transcendental number

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

<math>a_n x^n + a_{n-1} x^{n-1}+ ... + a_1 x^1 + a_0 = 0</math>
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 all real numbers is uncountable; this implies that the set of all transcendental numbers is also uncountable, so in a very real sense there are many more transcendental numbers than algebraic ones. However, only a few classes of transcendental numbers are known and proving that a given number is transcendental can be extremely difficult. Another property of the normality of one number might also help to distinguish it to be transcendental.

The existence of transcendental numbers was first proved in 1844 by Joseph Liouville[?], who exhibited examples, including the Liouville constant:

<math>
\sum_{k=0}^\infty 10^{-k!} = 0.110001000000000000000001000....
                               
</math> in which the nth digit after the decimal point is 1 if n is a factorial (i.e., 1, 2, 6, 24, 120, 720, ...., etc.) and 0 otherwise. The first number to be proved transcendental without having been specifically constructed to achieve this was e, by Charles Hermite[?] in 1873. In 1882, Carl Louis Ferdinand von Lindemann[?] published a proof that the number π is transcendental. In 1874, Georg Cantor found the argument described above establishing the ubiquity of transcendental numbers.

Here is a list of some numbers known to be transcendental:

  • 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)
  • ln(a) if a is positive, rational and ≠ 1
  • Γ(1/3) and Γ(1/4) (see Gamma function).
  • Ω, Chaitin's constant.
  • <math>\sum_{k=0}^\infty 10^{-\lfloor \beta^{k} \rfloor};\qquad \beta > 1\; , </math>
where <math>\beta\mapsto\lfloor \beta \rfloor</math> is the floor function. For example if β = 2 then this number is 0.1010001000000010000000000000001000...

The discovery of transcendental numbers allowed the proof of the impossibility of several ancient geometric problems involving ruler and compass construction; the most famous one, squaring the circle, is impossible because π is transcendental.



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