BINARY NUMBERS - Structured Computer Organization - page 677

N
decimal

N
binary

N
signed mag.

− N

1's compl.

− N

2's compl.

− N

excess 128

−

1 00000001 10000001 11111110 11111111 01111111

2 00000010 10000010 11111101 11111110 01111110

3 00000011 10000011 11111100 11111101 01111101

4 00000100 10000100 11111011 11111100 01111100

5 00000101 10000101 11111010 11111011 01111011

6 00000110 10000110 11111001 11111010 01111010

7 00000111 10000111 11111000 11111001 01111001

8 00001000 10001000 11110111 11111000 01111000

9 00001001 10001001 11110110 11110111 01110111

10 00001010 10001010 11110101 11110110 01110110

20 00010100 10010100 11101011 11101100 01101100

30 00011110 10011110 11100001 11100010 01100010

40 00101000 10101000 11010111 11011000 01011000

50 00110010 10110010 11001101 11001110 01001110

60 00111100 10111100 11000011 11000100 01000100

70 01000110 11000110 10111001 10111010 00111010

80 01010000 11010000 10101111 10110000 00110000

90 01011010 11011010 10100101 10100110 00100110

100 01100100 11100100 10011011 10011100 00011100

127 01111111 11111111 10000000 10000001 00000001

128 Nonexistent Nonexistent Nonexistent 10000000 00000000

Figure A-7. Negative 8-bit numbers in four systems.

ferent singularity. The bit pattern consisting of a 1 followed by all 0s is its own
complement. The result is to make the range of positive and negative numbers
unsymmetric; there is one negative number with no positive counterpart.
The reason for these problems is not hard to find: we want an encoding system
with two properties:

1. Only one representation for zero.
2. Exactly as many positive numbers as negative numbers.

The problem is that any set of numbers with as many positive as negative numbers
and only one zero has an odd number of members, whereas m bits allow an even
number of bit patterns. There will always be either one bit pattern too many or one
bit pattern too few, no matter what representation is chosen. This extra bit pattern
can be used for 0 or for a large negative number, or for something else, but no

matter what it is used for it will always be a nuisance.

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