Celebratio Mathematica

D. B. Fuchs and M. B. Fuchs: “The arithmetic of binomial coefficients,” pp. 1–12 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1970:6 (1970). MR 1727594 incollection

Every student knows the formulas \begin{align*} (1+x)^2 &= 1 + 2x + x^2, \\ (1+x)^3 &= 1 + 3x + 3x^2 + x^3. \end{align*} The numbers \( (1,\,2,\,1) \), \( (1,\,3,\,3,\,1) \), as well as numbers obtained in an analogous way by raising \( (1+x) \) to the fourth power, the fifth power, and so on, are called binomial coefficients. This article deals with various properties of binomial coefficients. In the first section we lay out the “general theory”: Many of the theorems we prove here used to be part of the school curriculum. In the second section we will show very easy way to find the remainder when a binomial coefficient is divided by a prime number. The third, and concluding, section deals with certain remarkable properties of binomial coefficients. The main assertions in this section are formulated as hypotheses.

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs and M. B. Fuchs: “On best approximations, I,” pp. 27–35 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1971:6 (1971). MR 1727597 incollection

How can one find the best rational approximation for an irrational number? For example, which approximation of the number \( \sqrt{2} \) is the best: \( \frac{3}{2} \), \( \frac{7}{5} \) or \( 1.41 \)? The answer seems to be fairly simple: The smaller the error, the better the approximation. But this is not the whole story, since we write \( \pi \approx 3.14 \) even though we might know five or more decimal digits. It stands to reason that when choosing an approximation we want to decrease not only the error but also the denominator: The smaller the denominator, the less awkward the fraction and the easier to manage it (to store, substitute in formulas, etc.).

In this article we examine the problem of how to meet these two contradictory demands, of finding a rational approximation for a given irrational number that is as far as possible simultaneously the most precise and least awkward.

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs and M. B. Fuchs: “On best approximations, II,” pp. 37–47 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1971:11 (1971). MR 1727598 incollection

It is inconvenient to work with irrational numbers. Rational numbers are much easier to work with. Hence the problem of rational approximations to irrational numbers and their efficiency is most important. In Part I of our article we compared different approximations to the number \( \alpha \) and proved that for any irrational \( \alpha \) and arbitrarily large \( N \) there exist infinitely many rational approximations \( p/q \) such that \[ q\Bigl|\alpha - \frac{p}{q}\Bigr| < \frac{1}{N} .\] This, second part of the article deals with the subtler question of whether for any \( \alpha \) there exist infinitely many approximations \( p/q \) such that \[ q^2\Bigl|\alpha - \frac{p}{q}\Bigr| \] is less than a given number. It turns out that this number cannot be arbitrary, in accordance with the Hurwitz–Borel theorem, which states that for any \( \alpha \) there exist infinitely many different approximations \( p/q \) such that \[ q^2\Bigl|\alpha - \frac{p}{q}\Bigr| < \frac{1}{\sqrt{5}} .\] Note that the number \( \sqrt{5} \) cannot be replaced by a larger number. Before reading any further, the reader is advised to have another look at the previous part of the article and also to read the article by N. M. Baskin in Kvant 1970, no. 8.

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs and M. B. Fuchs: “Rational approximations and transcendence,” pp. 65–69 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1973:12 (1973). MR 1727601 incollection

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