irrationality measure

Let α. Let

M(α)={μ>0q0=q0(α,μ)>0 such that |α-pq|>1qμp,q,q>q0}.

The irrationality measure of α, denoted by μ(α), is defined by


If M(α)=, we set μ(α)=.

This definition is (loosely) a measure of the extent to which α can be approximated by rational numbersPlanetmathPlanetmathPlanetmath. Of course, by the fact that is dense in , we can make arbitrarily good approximations to real numbers by rationals. Thus this definition was made to represent a stronger statement: it is the ability of rational numbers to approximate α given a fixed growth bound on the denominators of those rational numbers.

By the Dirichlet’s Lemma, μ(α)2. Roth [6, 7] proved in 1955 that μ(α)=2 for every algebraic real number. It is well known also that μ(e)=2. For almost all real numbers the irrationality measure is 2. However, for special constants, only some upper bounds are known:

Constant Upper bound Reference
π 8.0161 Hata (1993) [3]
π/3 4.6016 Hata (1993) [3]
e 2 Davis (1978) [1]
π2 5.4413 Rhin and Viola (1996) [4]
log2 3.8914 Rukhadze (1987) [8], Hata (1990) [2]
ζ(3) 5.5139 Rhin and Viola (2001) [5]

It is worth noting that the last column of the above table is simply a list of references, not a collectionMathworldPlanetmath of discoverers. For example that fact that the irrationality measure of e is 2 was known to Euler.


  • 1 Davis, C.S., Rational approximations to e, J. Austral. Math. Soc. Ser. A 25 (1978), 497–502.
  • 2 Hata, M. ‘Legendre Type Polynomials and Irrationality Measures’, J. reine angew. Math. 407, 99–125, 1990.
  • 3 Hata, M., ‘Rational approximations to π and some other numbers’, Acta Arith. 63, 335–349 (1993).
  • 4 Rhin, G. and Viola, C. ‘On a permutation group related to zeta(2)’, Acta Arith. 77 (1996), 23–56.
  • 5 Rhin, G. and Viola, C. ‘The group structureMathworldPlanetmath for ζ(3), Acta Arith. 97 (2001), 269–293.
  • 6 Roth, K.F., ‘Rational Approximations to Algebraic NumbersMathworldPlanetmath, Mathematika 2 (1955), 1–20.
  • 7 Roth, K.F. ‘Corrigendum to ’Rational Approximations to Algebraic Numbers” Mathematika 2 (1955), 168.
  • 8 Rukhadze, E.A. ‘A Lower Bound for the Rational Approximation of by Rational Numbers’ Vestnik Moskov Univ. Ser. I Math. Mekh., 6 (1987), 25-29 and 97.
Title irrationality measure
Canonical name IrrationalityMeasure
Date of creation 2013-03-22 14:12:22
Last modified on 2013-03-22 14:12:22
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Definition
Classification msc 11J82
Related topic DirichletsApproximationTheorem