In fact, the rational number are rational because they are a fractions of two numbers. it may be different with positive integers or the sum of egyptians fractions which are link by a rule. $1/2 = 1/3 + 1/6$ $1/3 = 1/4 + 1/12$ $1/4 = 1/5 + 1/20$ $1/5 = 1/6 + 1/30$ $1/6 = 1/7 + 1/42$ so we can say $1/U(1)=1/U(0)-1/U(0)U(1)$ $U(1)/U(0)-U(1)/U(0)U(1)=1 $ if $ U(1)=U(0)+1$ it is in french, but you can study the egyptian fraction (and the code) in clouds: http://jeux-et-mathematiques.davalan.org/arit/egy/index.html#table2 There are the same question here. http://math.stackexchange.com/questions/238496/a-question-about-rational/238511#238511