190 ARITHMETIC 4.1 I (Medical web site) -l+i +i -l+i -1-i
190 ARITHMETIC 4.1 I -l+i +i -l+i -1-i -i Fig. 1. The set S. (Illustration by P. M. Farmwald, R. W. Gosper, and R. E. Maas.) In this system, only the digits 0 and 1 are needed. One way to demonstrate that every complex number has such a representation is to consider the interesting set S shown in Fig. 1; this set is, by definition, all points that can be written as Ck,l ak(i -1)-k, for an infinite sequence al, a2, us, . . . of zeros and ones. Figure-l shows that S can be decomposed into 256 pieces congruent to &S; note that if tjhe diagram of S is rotated counterclockwise by 135 , we obtain two adjacent sets congruent to (l/&?)S (since (i -1)s = S U (S + 1)). For details of a proof that S contains all complex numbers that are of sufficiently small magnitude, see exercise 18. Perhaps the prettiest number system of all is the balanced ternary notation, which consists of base-3 representation using -1, 0, and +l as trits (ternary digits) instead of 0, 1, and 2. If we use the symbol i to stand for -1, we have the following examples of balanced ternary numbers: