In "A Mathematicians Apology" Godfrey Hardy remarked that questions involving the decimal digits of numbers constitute non-serious mathematics. This has long been conventional wisdom. For instance, discussions of the decimal digits of PI very often elicit disdainful comments pointing out that one might just as well consider the digits of pi in some other base, and hence the problem is arbitrary and of no mathematical significance. Just for fun, let me argue the case that this conventional wisdom is wrong. First, I would suggest that some of the disdain for base-related problems in mathematics is carried over, perhaps subliminally, from UNITS-dependent questions in physics, which are undeniably naive and pointless. For example, there is a certain kind of person who will spend years studying the numerical properties of, say, the number 186282, because he has heard that this is the Speed of Light - in units of miles per sec. This is certainly misguided activity, but we should be clear about WHY it's misguided. The person may actually turn up some intersting and non-trivial mathematical properties related to the decimal number 186282, so if he were engaged in the study purely for its mathematical content, the worst we could say is that it's a rather highly specialized choice of subjects. However, the study is flat-out misguided if the person is laboring under the impression that the numerical properties of the integer 186282 are related to the physical phenomenon of light - unless one is trying to make some anthropomorphic point about the significance of our choice of units. In contrast, consider a problem concerning the decimal digits of pi. It's tempting to think that these digits are as unrelated to the mathematics of PI as the digits 186282 are unrelated to the physics of light propagation...but of course that's not true. Although the choice of base 10 is not unique, the integer 10 is one of the smallest integers, the first square-free composite number, etc., and the decimal digits of pi represent mathematically meaningful information about the relationship between the integer 10 and the transcendental number pi. For any given natural number B greater than 1 we have the UNIQUE sequence of integers c0,c1,... in the range 0 to B-1 such that pi = c0 + c1/B + c2/B^2 + c3/B^3 + ... This is a perfectly respectable mathematical construction, and the digits express a meaningful ANALYSIS of the transcendental number pi. We can liken it to the Fourier expansion of a signal into its constituient harmonics, or the expansion of a function into a sequence of orthogonal functions. It's true that we're free to select the basis of our expansion (e.g., Legendre polynomials, simple sine functions, or any of infinitely many other possible bases), but this does not imply that these expansions are mathematically meaningless. Each possible well-defined expansion represents an analysis of the source (e.g., the number pi) relative to a particular basis. It is not necessarily obvious, apriori, what basis of analysis might reveal some interesting mathematical structure. For example, it has recently been shown that an algorithm exists to compute the nth hexidecimal (base 16) digit of pi without computing the preceding digits. This particular construction does not work for arbitrary bases, only for certain bases like 16. It is not known whether any analogous special attributes characterize the base-10 digits of pi, but the point is that the base-B expansion of pi has actually turned out to exhibit non-trivial structure for certain values of B. In general, for any given B, the base-B representation defines an extremely useful mapping between the reals and the set of integer sequences [c0,c1..] where the ci are restricted to the range 0 to B-1. In fact, this mapping is by far the most widely used instrument of mathematics for practical (not to mention impractical) purposes. It's perfectly legitimate to ask if there are any interesting number-theoretic properties for certain specific sequences [c0,c1..]. For example, setting B=2, we know that numbers of the form 111..111 (which are called Mersenne numbers) can be tested for primality (using the Lucas-Lehmer test) much faster than the general case because of their special structure. Thus, the real number 1.11111.... (base 2) can be construed as a sequence of integers 1, 11, 111, ... whose special structure can be exploited for non-trivial number-theoretic purposes. Do the base B representations of any other real numbers (e.g., PI or e) yield sequences of integers with exploitable structure? To be candid, another reason that base-related problems are usually avoided is that they tend to be extremely difficult. The difficulty is due to the non-linearity imposed by the constraint that the coefficients must all be in the range 0 to B-1. This is what distinguishes such questions from (and makes them much harder than) questions about ordinary polynomials. Consider the reverse-sum palidrome problem, i.e., beginning with a number 196, add the "reverse" 691, then iterate. Will this necessarily lead to a palindrome (a number that equals its reverse)? The answer seems to be no, but no one can prove it. There is a tendancy on the part of mathematicians, when they can see no way of even approaching a problem, to declare that the problem is simply insignificant and/or uninteresting. For example, recall Gauss' famous comment, when he was urged to tackle Fermat's Last Theorem, that "Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of". Even today it it commonly said that Fermat's Last Theorem, as an isolated proposition, is of little interest, but the fact remains that important parts of modern number theory (e.g., Kummer's ideal theory) were developed specifically in efforts to solve this "uninteresting problem". It seems clear that the problem represents one particular consequence (and an easily stated one) of some deep and profound mathematical facts, so it is disengenuous for mathematicians to call it uninteresting or unimportant. Does anyone doubt that if Gauss had seen a way of proving this particular proposition, he would have done so? He says the proposition has little interest for him, but this lack of interest is mainly due to the fact that it is one of the (infintely many) propositions that he feels he can neither prove nor disprove. It's perfectly sensible for people to choose to focus their attention on problems that they believe they have some chance of solving, but we shouldn't make the mistake of thinking that a problem is mathematically insignificant simply because we have no clue how to solve it. There is, however, validity in the argument that some problems are "natural" whereas others are mere contrivances. For example, it might be argued that the "digital reversal" of a number is not a natural or meaningful concept, but consider a number N with the base-B representation c0 + c1 B + c2 B^2 + ... + ck B^k What are the "roots" of this number? In other words, given the coefficients c0,c1,...ck, for what values of B does the above expression equal zero? Now consider the digit reversal of N: ck + ... + c2 B^(k-2) + c1 B^(k-1) + c0 B^k What are the roots of this number? If we normalize these two polynomials the roots of N are simply the inverses of the roots of rev(N). Thus, rev(N) can be expressed naturally in terms of the roots of N (and vice versa). For other examples of non-trivial mathematical structure that explicitly involves base-b representations of numbers, see the notes Cyclic Divisibility Geometric Dot Products and Digit Reversals Powers of Primes Dividing Factorials Fibonacci, 1/89, And All That Generating Functions for Point Set Distances Least Significant Non-Zero Digit of n! It's conceivable that all genuine mathematical structure has meaning and significance for reality. In other words, it may be that there is no excess (unused) potential structure in the universe. In any case, the structure of analytical expansions (including radix representations of real numbers) can certainly not be assumed to be meaningless or insignificant.

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