Every algebraic irrational number has an irrationality measure of exactly two, meaning it cannot be approximated unusually well by rational numbers. Proved by Klaus Roth, extending earlier work of Axel Thue and Carl Ludwig Siegel, it earned Roth a Fields Medal.
Facts
StatementEvery irrational algebraic number has irrationality measure exactly two: for any e greater than 0 it admits only finitely many rational approximations p/q with error less than q to the power of -(2+e), and this exponent cannot be improved. 1 Classification
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.
In Branch
Sources
1. Roth's Theorem (Thue-Siegel-Roth) (Wikipedia)
Wikimedia Foundationlead paragraph, second sentence
It is of a qualitative type, stating that algebraic numbers cannot have many rational approximations that are 'very good'.
lead paragraph, third sentence, historical refinement chronology
Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and culminating with Klaus Roth (1955).
View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.