A beautiful construction due to Calkin and Wilf, foreshadowed a century earlier, provides an elegant and deep way to list all fractions. Beginning with 1 / 1, each fraction in this construction gives birth to two new ones.
The mathematician Leopold Kronecker (1823–1891) is known for having written, "God made the integers; all else is the work of man." In fact, 1 is enough, and some would even say that the empty set suffices to construct all numbers. Here, however, we shall focus on positive fractions, using a tree-like construction to view them as one large family whose Eve is 1/1.
The large family of fractions
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In this construction, each fraction a/b gives rise to two child fractions: the larger, (a + b)/b, and the smaller, a/(a + b). Thus, for example, the children of 22/7 are 22/29 and 29/7.
