Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but such a definition would have been meaningless to Euclid. In modern notation, Euclid's definition of equality is that given quantities "p", "q", "r" and "s", "p"∶"q"∷"r"∶"s" if and only if, for any positive integers "m" and "n", "np"<"mq", "np"="mq", or "np">"mq" according as "nr"<"ms", "nr"="ms", or "nr">"ms", respectively. This definition has affinities with Dedekind cuts as, with "n" and "q" both positive, "np" stands to "mq" as stands to the rational number (dividing both terms by "nq").