Rational Numbers. Rational numbers are fractions of integers, numbers like 69 or7222 . But they have the curious property, different from integers, that two fractions,like 69 and 72108 , are really equal:69 = 72108,even though they look different. So any definition of rational numbers must dealwith the problem that the same rational number can be expressed as a fraction oftwo integers in many different ways.This problem was solved by the ancient Greeks (500-300 B.C.) in their development of the ratio of two numbers. For numbers a and b, the ratio of a to b is thesame as the ratio of c to d if ad = bc. Thus the ratio of 1 to 2 is the same as the ratioof 3 to 6. The ratio of 2 to 1+ ?7 is the same as the ratio of ?7?1 to 3.From this idea of ratio we can give a set-theoretic definition of the rational numbers as follows.Take the set S of all ordered pairs (a,b) of integers, where b = 0. Partition the setS into subsets, by the rule:two pairs (a,b) and (c,d) are in the same subset if the ratio of a to b is the sameas the ratio of c to d, that is, if and only if ad = bc (or, in fractional notation, ifa/b = c/d Β but using fractional notation assumes that we have defined fractionsalready!)We give the name a/b to the set of ordered pairs containing the pair (a,b). Thusthe symbol a/b is a lab
el for the set of all ordered pairs (m,n) such that the ratio ofm to n is the same as the ratio of a to b.For example, 6/9 is a label for the set of all pairs (m,n) with 6n = 9m: in setnotation,69 = {(m,n)|6n = 9m},and72108 = {(m,n)|72n = 108m},Since the set {(m,n)|6n = 9m} = {(m,n)|72n = 108m}, it follows that 6/9 =72/108. In fact, if (c,d) is any ordered pair in the set {(m,n)|6n = 9m}, then{(m,n)|6n = 9m} = {(m,n)|cn = dm}(intuitively, because if 6/9 = c/d then 6/9 = m/n if and only if c/d = m/n) and sowe can name the set {(m,n)|6n = 9m} by c/d.Thus we define equality of fractions by:ab = cd if and only if {(m,n)|an = bm} = {(m,n)|cn = dm}.So 6/9 = 72/108 because 6/9 and 72/108 are names for the same set. On theother hand, the fractions 1/2 and 3/4 are not equal, because the sets they are namesfor,{(m,n)|n = 2m} and {(m,n)|3n = 4m},6 1 Numbersare differentΒin fact, are disjoint: if (m,n) is a pair with n = 2m, then 3m = 4m.To summarize, elements of Q may be defined as certain subsets of the set ofpairs (m,n) of integers with b = 0. The subset containing the pair (a,b) we label bythe fraction a/b; then the subset containing the pair (a,b) is {(m,n)|an = bm}. Twofractions a/b and c/d are equal if and only if {(m,n)|an = bm} = {(m,n)|cn = dm},which is the case if and only if ad = bc.