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Two Enumerations of the Rationals

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Here are two methods for enumerating the rational numbers. The Calkin–Wilf method starts with corresponding to the binary number 1. If the binary number corresponds to the fraction , then the binary number corresponding to is 0 appended to , and the one corresponding to is 1 appended to . According to this scheme, the Calkin–Wilf enumeration begins with , where semicolons separate the rows of an array.

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The Stern–Brocat method is more complicated, but it winds up reversing the Calkin–Wilf binary number except for the leading 1. The corresponding enumeration then reads .

Since either of the above arrays can be put into one-to-one correspondence with the set of natural numbers (positive integers) ℕ, the rationals must also comprise a set of cardinality .

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Contributed by: Ed Pegg Jr (September 2015)
Open content licensed under CC BY-NC-SA

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