Let be a real number, and let be the set of positive real numbers for which

(1)

has (at most) finitely many solutions for and integers. Then the irrationality measure, sometimes called the Liouville-Roth constant or irrationality exponent, is defined as the threshold at which Liouville's approximation theorem kicks in and is no longer approximable by rational numbers,

(2)

where is the infimum. If the set is empty, then is defined to be , and is called a Liouville number. There are three possible regimes for nonempty :

(3)

where the transitional case can correspond to being either algebraic of degree or being transcendental. Showing that for an algebraic number is a difficult result for which Roth was awarded the Fields medal.

The definition of irrationality measure is equivalent to the statement that if has irrationality measure , then is the smallest number such that the inequality

(4)

holds for any and all integers and with sufficiently large.

The irrationality measure of an irrational number can be given in terms of its simple continued fraction expansion and its convergents as

			(5)
			(6)

(Sondow 2003). For example, the golden ratio has

(7)

which follows immediately from (6) and the simple continued fraction expansion .

Exact values include for Liouville's constant and (Borwein and Borwein 1987, pp. 364-365). The best known upper bounds for other common constants are summarized in the following table, where is Apéry's constant, and are q-harmonic series, and the lower bounds are 2.

constant	upper bound	reference
	8.0161	Hata (1992)
	5.441243	Rhin and Viola (1996)
	3.8913998	Rukhadze (1987), Hata (1990)
	5.513891	Rhin and Viola (2001)
	4.80	Amdeberhan and Zeilberger (1998)
	4.80	Amdeberhan and Zeilberger (1998)

Amdeberhan, T. and Zeilberger, D. "q-Apéry Irrationality Proofs by q-WZ Pairs." Adv. Appl. Math. 20, 275-283, 1998.

Beukers, F. "A Rational Approach to Pi." Nieuw Arch. Wiskunde 5, 372-379, 2000.

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 3-4, 2004.

Borwein, J. M. and Borwein, P. B. "Irrationality Measures." §11.3 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 362-386, 1987.

Finch, S. R. "Liouville-Roth Constants." §2.22 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 171-174, 2003.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford: Clarendon Press, 1979.

Hata, M. "Legendre Type Polynomials and Irrationality Measures." J. reine angew. Math. 407, 99-125, 1990.

Hata, M. "Improvement in the Irrationality Measures of and ." Proc. Japan. Acad. Ser. A Math. Sci. 68, 283-286, 1992.

Hata, M. "Rational Approximations to and Some Other Numbers." Acta Arith. 63, 335-349, 1993.

Hata, M. "A Note on Beuker's Integral." J. Austral. Math. Soc. 58, 143-153, 1995.

Hata, M. "A New Irrationality Measure for ." Acta Arith. 92, 47-57, 2000.

Rhin, G. and Viola, C. "On a Permutation Group Related to ." Acta Arith. 77, 23-56, 1996.

Rhin, G. and Viola, C. "The Group Structure for ." Acta Arith. 97, 269-293, 2001.

Rukhadze, E. A. "A Lower Bound for the Rational Approximation of by Rational Numbers." [In Russian]. Vestnik Moskov Univ. Ser. I Math. Mekh., No. 6, 25-29 and 97, 1987.

Sondow, J. "Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik." Submitted to the proceedings of Journées Arithmétiques, Graz 2003 in the Journal du Theorie des Nombres Bordeaux. http://arXiv.org/abs/math.NT/0406300/.

Stark, H. M. An Introduction to Number Theory. Cambridge, MA: MIT Press, 1994.

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Weisstein, Eric W. "Irrationality Measure." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IrrationalityMeasure.html