12-3 Other Bases

The base - 1 - i has essentially the same properties as the base - 1 + i discussed above. If a certain bit pattern represents the number a + bi in one of these bases, then the same bit pattern represents the number a - bi in the other base.

The bases 1 + i and 1 - i can also represent all the complex integers, using only 0 and 1 for digits. These two bases have the same complex-conjugate relationship to each other, as do the bases -1 ?i. In bases 1 ?i, the representation of some integers has an infinite string of 1's on the left, similar to the two's-complement representation of negative integers. This arises naturally by using uniform rules for addition and subtraction, as in the case of two's-complement. One such integer is 2, which (in either base) is written ?1101100. Thus, these bases have the rather complex addition rule 1 + 1 = ?1101100.

By grouping into pairs the bits in the base -2 representation of an integer, one obtains a base 4 representation for the positive and negative numbers, using the digits -2, -1, 0, and 1. For example,

Similarly, by grouping into pairs the bits in the base - 1 + i representation of a complex integer, we obtain a base -2i representation for the complex integers using the digits 0, 1, - 1 + i, and i. This is a bit too complicated to be interesting.

The "quater-imaginary" system (Knu2) is similar. It represents the complex integers using 2i as a base, and the digits 0, 1, 2, and 3 (with no sign). To represent some integers, namely those with an odd imaginary component, it is necessary to use a digit to the right of the radix point. For example, i is written 10.2 in base 2i.