Complements are used in digital computers for simplifying the subtraction operation and for logical manipulations. There are two types of complements for each base-r system: (1) the r’s complement and (2) the (r-1)’s complement. When the value of the base is substituted, the two types receive the names 2’s and 1’s complement for binary numbers, or 10’s and 9’s complement for decimal numbers:
The r’s Complement:
Given a positive number N in base r with an integer part of n digits, the r’s complement of N is defined as rn-N for N≠0 and 0 for N=0. The following numerical example will help clarify the definition.
The 10’s complement of (52520)10 is 105 – 52520 = 47480.
The number of digits in the number is n = 5.
The 10’s complement of (0.3267)10 is 1 – 0.3267 = 0.6733.
No integer part, so 10n = 100 = 1.
The 10’s complement of (25.639)10 is 102 – 25.639 = 74.361.
The 2’s complement of (101100)2 is (26)10 – (101100)2 = (1000000 – 101100)2 = 010100.
The 2’s complement of (0.0110)2 is (1 – 0.0110)2 = 0.1010.
The (r – 1)’s Complement:
Given a positive number N in base r with an integer part of n digits and a fraction part of m digits, the (r – 1)’s complement of N is defined as rn – r-m – N. Some numerical examples follow:
The 9’s complement of (52520)10 is (105 – 1 – 52520) = 99999 – 52520 = 47479.
No fraction part, so 10-m = 100 = 1.
The 9’s complement of (0.3267)10 is (1 – 10-4 – 0.3267) = 0.9999 – 0.3267 = 0.6732.
No integer part, so 10n = 100 = 1.
The 9’s complement of (25.639)10 is (102 – 10-3 – 25.639) = 99.999 – 25.639 = 74.360.
The 1’s complement of (101100)2 is (26 – 1)10 – (101100)2 = (111111 – 101100)2 = 010011.
The 1’s complement of (0.0110)2 is (1 – 2-4)10 – (0.0110)2 = (0.1111 – 0.0110)2 = 0.1001.
