Fedora web server - 184 ARITHMETIC 4.1 Ultimately, an article by G.
184 ARITHMETIC 4.1 Ultimately, an article by G. W. Leibniz [Memoires de 1 Academie Royale des Sciences (Paris: 1703), 110-1161, which illustrated binary addition, subtraction, multiplication, and division, really brought binary notation into the limelight, and this article is usually referred to as the birth of radix-2 arithmetic. Leibniz later referred to the binary system quite frequently. He did not recommend it for practical calculations, but he stressed its importance in number-theoretical inves- tigations, since patterns in number sequences are often more apparent in binary notation than they are in decimal; he also saw a mystical significance in the fact that everything is expressible in terms of zero and one. Leibniz s unpublished manuscripts show that he had been interested in binary notation as early as 1679, when he referred to it as a bimal system (analogous to decimal ). A careful study of Leibniz s early work with binary numbers has been made by Hans J. Zacher, Die Hauptschriften zur Dyadik von G. W. Leibnia (Frankfurt am Main: Klostermann, 1973). Zacher points out that Leibniz was familiar with John Napier s so-called local arithmetic, a way for calculating with stones that amounts to using a radix-2 abacus. [Napier had published the idea of local arithmetic as an appendix to his little book Rhabdologia in 1617; it may be called the world s first binary computer, and it is surely the world s cheapest, although Napier felt that it was more amusing than practical. See Martin Gardner s discussion in Scientific American 228 (April 1973), 106-111.1 It is interesting to note that the important concept of negative powers to the right of the radix point was not yet well understood at that time. Leibniz asked James Bernoulli to calculate r in the binary system, and Bernoulli solved the problem by taking a 35-digit approximation to 7r, multiplying it by 1035, and then expressing this integer in the binary system as his answer. On a smaller scale this would be like saying that r = 3.14, and (314)10 = (100111010)~; hence T in binary is 100111010! [See Leibniz, Math. Schriften, ed. by K. Gehrhardt, 3 (Halle: 1855), 97; two of the 118 bits in the answer are incorrect, due to computational errors.] The motive for Bernoulli s calculation was apparently to see whether any simple pattern could be observed in this representation of 7r. Charles XII of Sweden, whose talent for mathematics perhaps exceeded that of all other kings in the history of the world, hit on the idea of radix-8 arithmetic about 1717. This was probably his own invention, although he had met Leibniz briefly in 1707. Charles felt that radix 8 or 64 would be more convenient for calculation than the decimal system, and he considered introdu.cing octal arithmetic into Sweden; but he died in battle before decreeing such a change. [See The Works of Voltaire 21 (Paris: E. R. DuMont, 1901), 49; E. Swedenborg, Gentleman s Magazine 24 (1754), 423-424.1 Octal notation was proposed also in colonial America before 1750, by the Rev. Hugh Jones, rector of a parish in Maryland [cf. Gentleman s Magazine 15 (1745), 377-379; H. R. Phalen, AMM 56 (1949), 461-4651. More than a century later, a prominent Swedish-American civil engineer named John W. Nystrom decided to carry Charles XII s plans a step further, by devising a complete system of numeration, weights, and measures based on radix-16 arithmetic. He wrote, I am not afraid, or do not hesitate, to advocate a