Liouville's Approximation Theorem states that an algebraic irrational number of a given degree cannot be approximated too well by rational numbers, giving an explicit bound, depending on that degree, on how close a rational number with a given denominator can come to it. Named for Joseph Liouville, it was used to construct the first numbers proven transcendental, by exhibiting numbers that violate the bound and therefore cannot be algebraic of any degree.
Facts
StatementIf a real number is an irrational root of an irreducible polynomial of degree n greater than 1 with integer coefficients, then there exists a positive real number A such that for every pair of integers p and q with q greater than zero, the distance between the number and p over q is greater than A divided by q to the power of n. 1 Classification
Statement Form Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Liouville number, Wikipedia
Proved By
Source Liouville number, Wikipedia
Sources
1. Liouville number, Wikipedia
Liouville numbers and transcendence section, theorem sentence
If a is an irrational root of an irreducible polynomial of degree n > 1 with integer coefficients, then there exists a real number A > 0 such that for all integers p, q with q > 0, the absolute value of a - p over q is greater than A over q to the n.
Liouville numbers and transcendence section, historical sentence
In 1844, Joseph Liouville proved a bound showing that there is a limit to how well algebraic numbers can be approximated by rational numbers
In Branch: Number Theory, Lead sentence
In number theory, a Liouville number is a real number x with the property that, for every positive integer n , there exists a pair
Proved By: Joseph Liouville, Lead paragraph
In number theory, a Liouville number is a real number
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