As Lagrange showed, any Irrational Number has an infinity of rational approximations
which satisfy
(1) |
(2) |
(3) |
(4) |
See also Hurwitz's Irrational Number Theorem, Liouville's Rational Approximation Theorem, Liouville-Roth Constant, Markov Number, Roth's Theorem, Segre's Theorem, Thue-Siegel-Roth Theorem
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 40, 1987.
Chandrasekharan, K. An Introduction to Analytic Number Theory. Berlin: Springer-Verlag, p. 23, 1968.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 187-189, 1996.