Java web server - 4.1 POSITIONAL NUMBER SYSTEMS 189 (1957), 1231. For
4.1 POSITIONAL NUMBER SYSTEMS 189 (1957), 1231. For further references see IEEE Transactions EC-12 (1963), 274- 276; Computer Design 6 (May 1967), 52-63. There is evidence that the idea of negative bases occurred independently to quite a few people. For example, D. E. Knuth had discussed negative-base systems in 1955, together with a further generalization to complex-valued bases, in a short paper submitted to a science talent search contest for high-school seniors. The base 2i gives rise to a system called the quater-imaginary number system (by analogy with quaternary ), which has the unusual feature that every complex number can be represented with the digits 0, 1, 2, and 3 without a sign. [See D. E. Knuth, CACA4 3 (1960), 245-247.1 For example, (11210.31)s, = 1 . 16 + 1 . (-8i) + 2. (-4) + 1 . (2i) + 3 * (-fi) + 1(-a) = 7$ -73i. Here the number (uzn . . . alao.a-1 . . . u-zk)zi is equal to (a2n.. . u2u()*u-2.. . u-2&4 + 2i(u2n–1.. .U3Ul.U-1.. .u–2k+l)-& so conversion to and from quater-imaginary notation reduces to conversion to and from negative quaternary representation of the real and imaginary parts. The interesting property of this system is that it allows multiplication and division of complex numbers to be done in a fairly unified manner without treating real and imaginary parts separately. For example, we can multiply two numbers in this system much as we do with any base, merely using a different carry rule: whenever a digit exceeds 3 ,we subtract 4 and carry -1 two columns to the left; when a digit is negative, we add 4 to it and carry +l two columns to the left. A study of the following example shows this peculiar carry rule at work: 12231 [9 - lOi] 12231 [9 - lOi] 12231 10320213 13022 13022 12231 021333121 [-19 -18Oi] A similar system that uses just the digits 0 and 1 may be based on &i, but this requires an infinite nonrepeating expansion for the simple number i itself. Vittorio Griinwald proposed using the digits 0 and l/d in odd-numbered positions, to avoid such a problem, but this actually spoils the whole system [cf. Commentari dell Ateneo di Brescia (1886), 43-541. Another binary complex number system may be obtained by using the base i -1, as suggested by W. Penney [JACM 12 (1965), 247-2481: ( . . . a4a3a2alao.a-l . . . )2-l = . ..- 4a4 + (2+2i)a3 -2ia2 + (i-l)aI + uo -$(i+l)a-I + + . +.