how complicated is a rational number? its size is not a very good indicator for this. for instance, 1987985792837/1987985792836 is approximately 1, but so much more complicated than 1. we'll explain how to measure the complexity of a rational number using various notions of height. we'll then see how heights are used to prove some basic finiteness theorems in number theory. one example will be the mordell-weil theorem: that on any rational elliptic curve, the group of rational points is finitely generated.