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A Dictionary of Real Numbers

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A Dictionary of Real Numbers, 9780534128401 (0534128408), Springer, 1990
How do we recognize that the number .93371663... is actually 21og1 0(e + 7r)/2 ? Gauss observed that the number 1.85407467... is (essentially) a rational value of an elliptic integral—an observation that was critical in the development of nineteenth century analysis. How do we decide that such a number is actually a special value of a familiar function without the tools Gauss had at his disposal, which were, presumably, phenomenal insight and a prodigious memory? Part of the answer, we hope, lies in this volume.

This book is structured like a reverse telephone book, or more accurately, like a reverse handbook of special function values. It is a list of just over 100,000 eight-digit real numbers in the interval [0,1) that arise as the first eight digits of special values of familiar functions. It is designed for people, like ourselves,who encounter various numbers computationally and want to know if these numbers have some simple form. This is not a particularly well-defined endeavor—every eight-digit number is rational and this is not interesting. However, the chances of an eight digit number agreeing with a small rational, say with numerator and denominator less than twenty-five, is small. Thus the list is comprised primarily of special function evaluations at various algebraic and simple transcendental values. The exact numbers included are described below.

Each entry consists of the first eight digits after the decimal point of the number in question. The values are truncated not rounded. The next part of the entry specifies the function, and the final part of the entry is the value at which the function is evaluated. So -4.828313737... is entered as "8283 1373 In : 5~3 " .

The abbreviations are also described below. There are two exceptions to this format. One is to describe certain combinations of two functions, for instance "8064 9591 exp(\/3) + exp(e)" the meaning of which is self-evident. The other is for real roots of cubic polynomials, i.e. "2027 1481 r t : (5,8, - 8 , - 3 ) " means that the polynomial 5 z3 + Sx2 - 8x - 3 has a real root, the fractional part of which is .20271481 Repeats have in general been excluded except for cases where two genuinely different numbers agree through at least eight digits. These latter coincidences account for the fewer than fifty repeat entries.

If the number you are checking is not in the list try checking one divided by the number and perhaps a few other variants such as one minus the number. Of course, finding the number in the list, in general, only indicates that this is a good candidate for the number. Proving agreement or checking to further accuracy is then appropriate.
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