Joseph Liouville

Joseph Liouville (1809–1882) was a French mathematician whose work ranged across analysis, number theory, and mathematical physics. He proved a theorem on rational approximation that led him to construct the first known transcendental numbers.

Liouville number: A real number that can be approximated by rationals unusually well, beyond every fixed power of the denominator. Liouville constructed such numbers to demonstrate that transcendental numbers exist.
Diophantine approximation: The study of how closely real numbers can be approximated by rational numbers. Liouville’s theorem is an early landmark in this broader subject.
Complex analysis: The study of functions of complex variables and their analytic properties. Liouville’s theorem in complex analysis is a separate, influential result bearing his name.
Évariste Galois: A French mathematician whose work on polynomial equations founded group theory and modern Galois theory. Liouville edited and published Galois’s mathematical papers after his death.
Fundamental theorem of algebra: The theorem that every nonconstant polynomial with complex coefficients has a complex root. Liouville’s complex-analysis theorem offers a classic route to proving this result.
Transcendental number: A number that is not a root of any nonzero polynomial with integer coefficients. His construction gave the first explicit examples of numbers beyond the algebraic numbers.
Algebraic number: A complex number that is a root of a nonzero polynomial with rational coefficients. The approximation bound distinguishes algebraic irrational numbers from the numbers Liouville constructed.
Entire function: A complex-valued function that is holomorphic at every point of the complex plane. Entire functions are the objects governed by his boundedness theorem.
Joseph Fourier: A French mathematician and physicist known for Fourier analysis and the theory of heat. Liouville studied at the École Polytechnique, where Fourier was among his influential teachers.
Liouville-Arnold theorem: A theorem describing the structure of regular motion in integrable Hamiltonian systems. It develops the integrability framework associated with Liouville’s work in mechanics.

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