What is an irrationality measure?

It quantifies how closely a number can be approximated by rationals: a larger measure means better approximations are possible. Rationals have measure one, algebraic irrationals exactly two (by Roth), and Liouville numbers infinite measure.

How does approximation prove a number is transcendental?

If a number can be approximated by rationals faster than Liouville's bound allows for any algebraic number, it cannot be algebraic, so it must be transcendental.

Diophantine Approximation

Diophantine approximation measures how closely irrational numbers can be approached by fractions; the answer depends delicately on the number, separating rationals, algebraic irrationals, and transcendentals.

Definition

Diophantine approximation is the study of how well real numbers can be approximated by rational numbers, quantified by how small the difference between a number and a fraction can be relative to the size of the fraction's denominator.

Scope

This topic covers Dirichlet's approximation theorem and the pigeonhole principle, continued fractions as best approximations, the irrationality measure of a number, Liouville's theorem and the construction of Liouville (transcendental) numbers, the Thue-Siegel-Roth theorem on approximation of algebraic numbers, and applications to bounding solutions of Diophantine equations and to transcendence proofs.

Core questions

How well can every irrational number be approximated by rationals, as guaranteed by Dirichlet's theorem?
Why are continued fraction convergents the best rational approximations?
How does Liouville's theorem limit the approximability of algebraic numbers and thereby exhibit transcendental numbers?
What sharper limit does the Thue-Siegel-Roth theorem impose, and how does it bound solutions of Diophantine equations?

Key theories

Dirichlet's approximation theorem: For any irrational number there are infinitely many fractions approximating it to within one over the square of the denominator, a bound proved by the pigeonhole principle and essentially achieved by continued fractions.
Liouville's theorem and transcendence: Algebraic numbers cannot be approximated by rationals faster than a power depending on their degree; numbers approximable faster, such as Liouville's constant, must be transcendental.
Thue-Siegel-Roth theorem: An irrational algebraic number cannot be approximated to an exponent essentially greater than two; this best-possible bound implies finiteness of solutions for broad classes of Diophantine equations.

Clinical relevance

Approximation quality controls the stability of numerical algorithms involving irrational ratios and underlies lattice reduction (the basis of attacks and constructions in lattice cryptography) and the design of low-discrepancy sequences used in quasi-Monte Carlo integration.

History

Continued fraction approximations were studied by Euler and Lagrange. Liouville constructed the first explicit transcendental numbers in 1844 using his approximation bound; Thue, Siegel, and finally Roth in 1955 sharpened the bound for algebraic numbers, a result for which Roth received the Fields Medal.

Key figures

Peter Gustav Lejeune Dirichlet
Joseph Liouville
Axel Thue
Klaus Roth

Frequently asked questions

What is an irrationality measure?: It quantifies how closely a number can be approximated by rationals: a larger measure means better approximations are possible. Rationals have measure one, algebraic irrationals exactly two (by Roth), and Liouville numbers infinite measure.
How does approximation prove a number is transcendental?: If a number can be approximated by rationals faster than Liouville's bound allows for any algebraic number, it cannot be algebraic, so it must be transcendental.