188 ARITHMETIC 4.1 right, (Web hosting bandwidth) or as fractions with
188 ARITHMETIC 4.1 right, or as fractions with the radix point at the extreme left, or as some mixture of these two extremes; the rules for the appearance of the radix point in each result are straightforward. It is easy to see that there is a simple relation between radix b and radix bk: (. . . U~UQU~UO.U-1U-2.. . )* = (. . .A3A2A1&.A-1A-z.. . )* , (5) where A3 = (Uk,+k-1 . . . Ukj+lUkj)b; see exercise 8. Thus we have simple techniques for converting at sight between, say, binary and octal notation. Many interesting variations on positional number systems are possible be- sides the standard b-ary systems discussed so far. For example, we might have numbers in base (-lo), so that (. . . U3U2UlUo.UplU-2.. . j-l, = . . . + u3(-10)3 + u2(-10)2 + u&loy + a0 + . . . ..- = . 1000u3 + loo@ -10Ul + a0 -&U-l + &ju–2 -. . . . Here the individual digits satisfy 0 < ak 5 9 just as in the decimal system. The number 12345 67890 appears in the negadecimal system as (193755 73910)-10, (6) since the latter represents 10305070900 -9070503010. It is interesting to note that the negative of this number, -12345 67890, would be written (28466 48290)-10, (7) and, in fact, every real number whether positive or negative can be represented without a sign in the -10 system. Negative-base systems were first considered by Vittorio Griinwald [Giornale di matematiche di Battaglini 23 (1885), 203-221, 3671, who explained how to perform the four arithmetic operations in such systems; Griinwald also discussed root extraction, divisibility tests, and radix conversion. However, since his work was published in a rather obscure journal, it seems to have had no effect on other research, and it was soon forgotten. The next publication about negative- base systems was apparently by A. J. Kempner [AMM 43 (1936), 610-6171, who discussed the properties of non-integer radices and remarked in a footnote that negative radices would be feasible too. After twenty more years the idea was rediscovered again, this time by Z. Pawlak and A. Wakulicz [Bulletin de 1 Academie Polonaise des Sciences, Classe III, 5 (1957), 233-236; %rie des sciences techniques 7 (1959), 713-7211, and also by L. Wade1 [IRE Transactions EC-6