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  3. 017 · The irrationality exponent of π is 2
  5. The irrationality exponent of pi is 2
  7. Irrationality exponent of pi is 2
  9. Pi is irrational
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Pi is irrational

Selected node

claimroutineinferred_dep

Pi is irrational

Shows π\piπ is irrational: a rational value would be an exact approximation
makes approximation errors nonzero for the pigeonhole step
Claim·§4 Choice of parameters and proof of Theorem~thm:main·inside thmIrrationality exponent of pi is 2

Statement (paper)

The number π\piπ is irrational: if π\piπ were rational, then exact representations π=p/q\pi=p/qπ=p/q with arbitrarily large denominators qqq (unreduced fractions, zero error) would satisfy , which has been excluded for arbitrarily large qqq, a contradiction.

In plain English

The number π\piπ is irrational, since a rational value π=p/q\pi=p/qπ=p/q would give exact approximations with arbitrarily large denominators, which the preceding argument rules out.

Why it's here

Exact representations (unreduced, with zero error) satisfy the same approximation condition that the previous claim showed cannot hold for arbitrarily large qqq, so rationality leads to a contradiction. This lets the later pigeonhole step assume π\piπ is irrational, so the approximation errors are nonzero.
Classified routine:
This follows directly from claim-thm-main-4: if π were rational, unreduced fractions would give exact approximations with arbitrarily large denominators, contradicting the preceding exclusion.