 The rational numbers are dense: for any two rational numbers \(q_1\) and \(q_2\) such that \(q_1 &lt; q_2\), there is at least one rational number \(q_3\) such that \(q_1 &lt; q_3 &lt; q_2\)—for example, the arithmetic mean of \(q_1\) and \(q_2\), \((q_1+ q_2)/2\), lies between them. Indeed, for any two rational numbers \(q_1\) and \(q_2\) such that \(q_1\) is strictly less than \(q_2\), for any natural number \(n\), there are more than \(n\) distinct rational numbers that lie between \(q_1\) and \(q_2\). Where the integers spread infinitely ‘outward’ in both directions, the rationals also divide infinitely ‘inward’.