ating it to a certain degree of accuracy that only finitely many such rational approximations exist for rational numbers .Specifically, he proved that for any number  there are infinitely many pairs of relatively prime numbers  such that  if and only if  is irrational.51 years later, Hurwitz further improved Dirichlet's approximation theorem by a factor of , improving the right-hand side from  to  for irrational numbers: The above result is best possible since the golden ratio  is irrational but if we replace  by any larger number in the above expression then we will only be able to find finitely many rational numbers that satisfy the inequality for .Furthermore, he showed that among the irrational numbers, the least readily approximable numbers are those of the form  where  is the golden ratio,  and .(These number