Deconstructing Bases: Fair, Fitting, and Fast Bases

Abstract

Elementary school students wrestling with decimals quickly realize that not all fractions are created equal. While the relatively awkward fraction 17/32 turns into the modestly nice finite decimal 0.53125, other seemingly simple ones, like 2/3 = 0.66666666... or 1/7 = 0.14285714..., confront us with the sophisticated notion of infinite repeating decimals. Can we find a finitely fair base, one in which all fractions have finite representations? It is natural to start our hunt for a finitely fair base by changing from the familiar base 10 to base b, where b is any integer greater than 1. Unfortunately, the following review of representations base b reveals that for any b some fractions must have infinite repeating representations, while other fractions have finite representations. Recall that O.aa2a3 .. .b = alb a/b + a3/b2 = a3/b3 l n/bn, where the subscripted b indicates the base and an is an integer satisfying 0 < an < b 1. A fraction p/q in reduced form has a k-place representation in base b exactly when q divides bk but doesn't divide bk-l. For example, 32 = 25 divides 105 but not 104, so base 10 uses five decimal places for 17/32. If q has a prime factor not in b, the reduced fraction p/q has an infinite repeating representation base b. Since no fixed b has every prime factor, every base has some fractions with infinite repeating representations. Clearly, a finitely fair base requires something new, a mathematical of the idea of a base. The postmodern term deconstruction describes something mathematicians have done for two centuries: probe a familiar concept more deeply to expose new interpretations and understandings. An initial deconstruction of base in the next section leads to a finitely fair base, called base {n!}. A further deconstruction in the middle section leads more generally to variable bases, which we use to find bases that fit a given real number with a specified representation. The final section critiques bases in yet another way, leading to competing measures for finitely fair bases. The Prime Number Theorem enables us to approximate one of these measures in terms of the other one and so resolve the competition between them. For ease we consider only representations of real numbers between 0 and 1, although the reader is invited to extend these ideas to the integer parts.