### 12-2 Base -1 + i

By using - 1 + i as the base, where i is all complex integers (complex numbers with integral real and imaginary parts) can be expressed as a single "number" without an explicit sign or other irregularity. Surprisingly, this can be done using only 0 and 1 for digits, and all integers are represented uniquely. We will not prove this or much else about this number system, but will just describe it very briefly.

It is not entirely trivial to discover how to write the integer 2. [1] However, this can be determined algorithmically by successively dividing 2 by the base and recording the remainders. What does a "remainder" mean in this context? We want the remainder after dividing by - 1 + i to be 0 or 1, if possible (so that the digits will be 0 or 1). To see that it is always possible, assume that we are to divide an arbitrary complex integer a + bi by - 1 + i. Then, we wish to find q and r such that q is a complex integer, r = 0 or 1, and

[1] The interested reader might warm up to this challenge.

where qr and qi denote the real and imaginary parts of q, respectively. Equating real and imaginary parts and solving the two simultaneous equations for q gives

Clearly, if a and b are both even or are both odd, then by choosing r = 0, q is a complex integer. Furthermore, if one of a and b is even and the other is odd, then by choosing r = 1, q is a complex integer.

Thus, the integer 2 can be converted to base - 1 + i by the plan illustrated below.

Because the real and imaginary parts of the integer 2 are both even, we simply do the division, knowing that the remainder will be 0:

Because the real and imaginary parts of - 1 - i are both odd, again we simply divide, knowing that the remainder is 0:

Because the real and imaginary parts of i are even and odd, respectively, the remainder will be 1. It is simplest to account for this at the beginning by subtracting 1 from the dividend.

Because the real and imaginary parts of 1 are odd and even, the next remainder will be 1. Subtracting this from the dividend gives

Because we have reached a 0 quotient, the process terminates, and the base - 1 + i representation for 2 is seen to be 1100 (reading the remainders upwards).

Table 12-3 shows how each bit pattern from 0000 to 1111 is interpreted in base - 1 + i, and how the real integers in the range -15 to +15 are represented.

The addition rules for base - 1 + i (in addition to the trivial ones involving a 0 bit) are as follows:

##### Table 12-3. Conversions between Decimal and Base -1 + I

n (base -1 + i)

n (decimal)

n (decimal)

n (base -1 + i)

-n (base -1 + i)

0

0

0

0

0

1

1

1

1

11101

10

-1 + i

2

1100

11100

11

i

3

1101

10001

100

-2i

4

111010000

10000

101

1 - 2i

5

111010001

11001101

110

-1 - i

6

111011100

11001100

111

-i

7

111011101

11000001

1000

2 + 2i

8

111000000

11000000

1001

3 + 2i

9

111000001

11011101

1010

1+ 3i

10

111001100

11011100

1011

2 + 3i

11

111001101

11010001

1100

2

12

100010000

11010000

1101

3

13

100010001

1110100001101

1110

1 + i

14

100011100

1110100001100

1111

2 + i

15

100011101

1110100000001

When adding two numbers, the largest number of carries that occurs in one column is six, so the largest sum of a column is 8 (111000000). This makes for a rather complicated adder. If one were to build a complex arithmetic machine, it would no doubt be best to keep the real and imaginary parts separate, [2] with each represented in some sensible way such as two's-complement.

[2] This is the way it was done at Bell Labs back in 1940 on George Stibitz's Complex Number Calculator [Irvine].