A transcendental number is a real or complex number that is not algebraic — in other words it cannot satisfy any nonzero polynomial equation with integer coefficients. Concretely, there is no finite list of integers a_n, ..., a_0 for which a_n x^n + ... + a_1 x + a_0 = 0 holds for a transcendental value of x. This definition separates transcendental numbers from algebraic numbers (roots of such polynomials) and places them strictly among the irrational numbers: every transcendental number is irrational, but most irrationals are not transcendental.

Characterization and basic properties

Formally, a complex number z is transcendental if there exists no nonzero polynomial P with integer (equivalently rational) coefficients such that P(z)=0. The set of algebraic numbers is countable because polynomials with integer coefficients can be enumerated; consequently the complement — the set of transcendental numbers — is uncountable and in fact has full Lebesgue measure on the real line. That is, from the point of view of measure theory, "almost all" real numbers are transcendental.

Historical development

Observers such as Gottfried Wilhelm Leibniz and Leonhard Euler recognized the possibility of numbers that were not roots of algebraic equations, but an explicit proof that such numbers exist came later. Joseph Liouville produced the first explicit examples in 1844 by constructing numbers now called Liouville numbers, which can be approximated exceptionally well by rationals; his work established the existence of transcendental numbers rigorously. Subsequent milestones include Charles Hermite's 1873 proof that the base of natural logarithms e is transcendental and Ferdinand von Lindemann's 1882 proof that π is transcendental. These results settled classical questions — for example, Lindemann's theorem implies the impossibility of squaring the circle using straightedge and compass.

Examples and notable facts

  • Liouville's constant, constructed by placing ones in rapidly increasing decimal positions, was the first explicitly exhibited transcendental number and motivated greater study of Diophantine approximation.
  • Euler's number e and pi (π) are well-known transcendental numbers; their transcendence has deep consequences in geometry and analysis.
  • The Gelfond–Schneider theorem (a 20th‑century result) gives a rich source of transcendental numbers: if a and b are algebraic with a ≠ 0,1 and b irrational algebraic, then any value of a^b is transcendental. This theorem explains why many exponentials and powers are transcendental.

Methods of proof and applications

Proving a specific number is transcendental is typically difficult and requires tools from analysis and number theory. Techniques include constructing very good rational approximations (Liouville's approach), applying linear forms in logarithms, and using transcendence measures that quantify how closely algebraic numbers can approximate a given number. Transcendence results are important in number theory and algebraic geometry and have concrete corollaries in classical construction problems and the theory of special functions.

Distinctions and further remarks

It is useful to contrast algebraic and transcendental numbers: algebraic numbers have finite algebraic degree (the degree of their minimal polynomial), while transcendental numbers have no such degree. The algebraic integers form a related class studied in algebraic number theory. Readers seeking formal definitions and proofs can consult introductory texts in algebraic number theory and transcendence theory; for background on real and complex numbers see real numbers and complex numbers, while discussion of integer coefficients and related polynomial notions appears at integer coefficients. For the relation to irrationality see irrational numbers. Historical accounts can be found in treatments mentioning Leibniz and Euler and the first rigorous constructions such as Liouville's 1844 proof.