constant e = 2.718281828459045235... has the (regular) continued fraction representation Its best approximations for the second definition are while, for the first definition, they are Measure of the accuracy of approximations The obvious measure of the accuracy of a Diophantine approximation of a real number by a rational number is However, this quantity can always be made arbitrarily small by increasing the absolute values of and ; thus the accuracy of the approximation is usually estimated by comparing this quantity to some function of the denominator , typically a negative power of it. For such a comparison, one may want upper bounds or lower bounds of the accuracy. A lower bound is typically described by a theorem like "for every element of some subset of the real numbers and every rational number , we have ". In some cases, "every rational number" may be replaced by "all rational numbers except a finite number of them", which amounts to multiplying by some constant depending on . For upper bounds, one has to take into account that not all the "best" Diophantine approximations provided by the convergents may have the desired accuracy. Therefore, the theorems take the form "for every element of some subset of the real numbers, there are infinitely many rational numbers such that ". Badly approximable numbers A badly approximable number is an x for which there is a positive constant c such that for all rational p/q we have The badly approximable numbers are precisely those with bounded partial quotients. Equivalently, a number is badly approximable if and only if its Markov constant is bounded. Lower bounds for Diophantine approximations Approximation of a rational by other rationals A rational number may be obviously and perfectly approximated by for every positive integer i. If we have because is a positive integer and is thus not lower than 1. Thus the accuracy of the approximation is bad relative to irrational numbers (see next sections). It may be remarked that the preceding proof uses a variant of the pigeonhole principle: a non-negative integer that is not 0 is not smaller than 1. This apparently trivial remark is used in almost every proof of lower bounds for Diophantine approximations, even the most sophisticated ones. In summary, a rational number is perfectly approximated by itself, but is badly approximated by any other rational number. Approximation of algebraic numbers, Liouville's result In the 1840s,