Approximation Of Irrationals By Rationals
G. H. Hardy, Elizabeth Mary Wright · 2008
Abstract Statement of the problem. The problem considered in this chapter is that of the approximation of a given number ξ ,usually irrational, by a rational fraction Since the rationals are dense in the continuum, there are rationals as near as we please to any ξ .Given ξand any positive number E, there is an r= p/qsuch that any number can be approximated by1a ratio1nal with any assigned degree of accuracy. We ask now how simplyor, what is essentially the same thing, how rapidlycan we approximate to ξ? Given ξand E, how complex must p/qbe (i.e. how large q)to secure an approximation with the measure of accuracy E? Given ξand q, or some upper bound for q, how small can we make E? and in Ch. X we proved a num1ber of s1imilar theorems by the use of continued fractions.‡ The inequality (11.1.1), or stronger inequalities of the same type, will recur continually throughout this chapter.