The Online Encyclopedia and Dictionary






Transcendental number

In mathematics, a transcendental number is any irrational number that is not an algebraic number, i.e., it is not the solution of any polynomial equation of the form

a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_1 x^1 + a_0 = 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 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:

\sum_{k=1}^\infty 10^{-k!} = 0.110001000000000000000001000....

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, 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.

See also Lindemann-Weierstrass theorem.

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

  • ea if a is algebraic and nonzero. In particular, e itself is transcendental.
  • 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.
  • ln(a) if a is positive, rational and ≠ 1
  • \sum_{k=0}^\infty 10^{-\lfloor \beta^{k} \rfloor};\qquad \beta > 1\; ,
where \beta\mapsto\lfloor \beta \rfloor is the floor function. For example if β = 2 then this number is 0.11010001000000010000000000000001000...

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.

Last updated: 02-10-2005 21:30:15
Last updated: 05-03-2005 17:50:55