Counting the Positive Rationals: a Brief Survey

We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Gödel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian fractions, and the sequence of ratios of successive hyperbinary representation numbers. Ginsberg [7] gave an injective mapping of the rational numbers Q into the positive integers Z by interpreting the fraction −a/b as a positive integer written in base 12, with the division slash and the minus sign as symbols for 10 and 11, respectively. This idea was previously publicized by Campbell [2], who encountered it in the early 1970s, and also used it to establish the countability of the ring Q[x] of polynomials with rational coefficients. Citing Campbell and using essentially the same arguments, Touhey [21] gave injections of Q and Z[x] into Z. More recently, Kantrowitz [14] formulated the main idea as a general principle, namely that the set of all words of finite length that may be formed from the letters of a countable alphabet is countable. If we identify the set Q of positive rational numbers with the subset of Z × Z consisting of the ordered pairs of coprime positive integers, then any injective mapping φ : Z × Z → Z restricts to an injection of Q into Z. Additional examples follow. All maps φ have domain Z × Z and codomain Z. 1. An extremely simple bijection is given by φ(n,m) = (2n − 1)2. Bijectivity of φ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a power of 2. 2. Let φ(n,m) = (2 − 1)2. It is easy to verify that φ is injective using elementary divisibility properties of the positive integers. However, if (2 − 1)2 is written in binary (base 2), then φ can be regarded as the map which encodes the ordered pair (n,m) as the string {1}{0} of n consecutive ones followed by m consecutive zeros. From this viewpoint, injectivity is obvious. Grant and Priest [9] offered the surjection g : Z → Q defined by g(x) = (k + 1)/(m + 1), where k and m are the number of fours and sevens, respectively, in the Date: February 2, 2008. 1991 Mathematics Subject Classification. Primary: 26A03; Secondary: 11A51, 11A55, 11A63.