4.1 POSITIONAL NUMBER SYSTEMS 191 Balanced (Business web hosting) ternary Decimal
4.1 POSITIONAL NUMBER SYSTEMS 191 Balanced ternary Decimal 10T 8 1 lTO.11 328 1 1 10.1 1 -325 1110 -33 0.1 1 1 1 1.. . 4 One way to find the representation of a number in the balanced ternary system is to start by representing it in ternary notation; for example, 208.3 = (21201.022002200220.. .)3. (A very simple pencil-and-paper method for converting to ternary notation is given in exercise 4.4-12.) Now add the infinite number . . .lllll.lllll.. . in ternary notation; we obtain, in the above example, the infinite number (. . .11111210012.210121012101.. .)s. Finally, subtract . ..11111.11111… by decrementing each digit; we get 208.3 = (lOllOl.lOiOlOiOlOiO.. . )s. (8) This process may clearly be made rigorous if we replace the artificial infinite number . . . 11111.11111.. . by a number with suitably many ones. The balanced ternary number system has many pleasant properties: a) The negative of a number is obtained by interchanging 1 and i. b) The sign of a number is given by its most significant nonzero trit, and in general we can compare any two numbers by reading them from left to right and using lexicographic order, as in the decimal system. c) The operation of rounding to the nearest integer is identical to truncation (i.e., deleting everything to the right of the radix point). Addition in the balanced ternary system is quite simple, using the table iiiiiiiiioooooooooiiiiiill~ iiioooiiiiiioooiiiiiiooo~~l ioiio~ioiioiioiioiiolTolTolio~ Toil 111 i 0 i 0 iii i 0 i 0 i 0 iii i 0 10 iii iii10 (The three inputs to the addition are the digits of the numbers to be added and the carry digits.) Subtraction is negation followed by addition; and multiplication also reduces to negation and addition, as in the following example: iioi [I71iioi 1171 ii01 ii010 iioi 0111101 P891