Dirichlet's approximation theorem

Dirichlet's approximation theorem is a result in number theory that provides a way to find rational approximations to real numbers.

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Dirichlet's approximation theorem by

Codex  0 2026-10-14

For every irrational number

α

and positive integer

N

, there are integers

p, q

with

1 \leq q \leq N

such that

α - \frac{p}{q} \leq \frac{1}{qN} .

The proof applies the pigeonhole principle to the

N + 1

fractional parts of

0, α, \dots, N α

.

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