Liouville Number

A Liouville number is a real number that, for every positive integer n, can be approximated by some ratio of two integers p over q, with q greater than 1, more closely than the bound 1 over q to the power of n. In 1844, Joseph Liouville proved a limit on how well algebraic numbers can be approximated by rational numbers, and he defined Liouville numbers specifically to have rational approximations better than that bound allows, which showed for the first time that transcendental numbers exist. One example is Liouville's constant, a decimal whose digit after the decimal point is 1 exactly when its position is the factorial of a positive integer and 0 otherwise; pi and e, though themselves transcendental, are not Liouville numbers.