Dirichlet's Approximation Theorem states that for any real number and any positive integer bound on the denominator, a rational number exists with denominator no larger than that bound whose distance from the given real number is smaller than one over the denominator times the bound, which in turn guarantees infinitely many such close rational approximations to any irrational number. Named for Peter Gustav Lejeune Dirichlet, it is a foundational result of Diophantine approximation proved using the pigeonhole principle.
Facts
Classification
Statement Form Statement Form Statement Form StatementFor any real number and any positive integer bound N on the denominator, there exist integers p and q with q no larger than N such that the distance between q times the real number and p is at most one divided by N plus one, which is smaller than one divided by N. 1 Connections
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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.
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.
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Source Dirichlet's approximation theorem, Wikipedia
Proved By
Source Dirichlet's approximation theorem, Wikipedia
Sources
1. Dirichlet's approximation theorem, Wikipedia
Lead section, definition sentence
for any real numbers a and N, with 1 less than or equal to N, there exist integers p and q such that 1 less than or equal to q less than or equal to N and the absolute value of qa - p is less than or equal to 1 divided by the floor of N plus 1, which is less than 1 divided by N.
In Branch: Number Theory, Lead sentence
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for
Proved By: Peter Gustav Lejeune Dirichlet, Lead paragraph
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers
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