Complements
Purpose:
o Used in digital computers to simplify subtraction operations.
o Important for logical manipulations.
Types of Complements:
1. r's Complement (Radix Complement)
2. (r - 1)'s Complement
Complements in Various Number Systems:
• Binary System (Base 2):
o r’s complement = 2’s complement
o (r – 1)’s complement = 1’s complement
• Decimal System (Base 10):
o r’s complement = 10’s complement
o (r – 1)’s complement = 9’s complement
• Octal System (Base 8):
o r’s complement = 8’s complement
o (r – 1)’s complement = 7’s complement
• Hexadecimal System (Base 16):
o r’s complement = 16’s complement
o (r – 1)’s complement = 15’s complement
Radix Complement (r’s complement):
• For a given positive number N in base r, with n digits:
o r's complement = 𝑟𝑛− 𝑁(𝑖𝑓 𝑁 ≠ 0)
o If N = 0, then r's complement is simply 0
Example 1.26. Find the 10’s complement of
(23450)10. Solution:
𝑛 = 5, 𝑁 = 23450, 𝑟 = 10
The 10’s complement of (23450)10 = 𝑟𝑛 − 𝑁 = 105 − 23450 = 𝟕𝟔𝟓𝟓𝟎
Example 1.27. Find the 10’s complement of
(0.3245)10. Solution:
𝑛 = 0, 𝑁 = 0.3245, 𝑟 = 10
The 10’s complement of (0.3245)10 = 𝑟𝑛 − 𝑁 = 100 − 0.3245 = 𝟎.
𝟔𝟕𝟕𝟓𝟓
Example 1.28. Find the 10’s complement of
(23.324)10. Solution:
𝑛 = 2, 𝑁 = 23.324, 𝑟 = 10
The 10’s complement of (23.324)10 = 𝑟𝑛 − 𝑁 = 102 − 23.324 = 𝟕𝟔.
𝟔𝟕𝟔
Example 1.29. Find the 2’s complement of
(10110)2. Solution:
𝑛 = 5, 𝑁 = 10110, 𝑟 = 2
The 2’s complement of (10110)2 = 𝑟𝑛 − 𝑁 = (25)10 − (10110)2
= (32)10 − (10110)2 = (100000)2 −
(10110)2
= (100000)2 − (10110)2 = 𝟎𝟏𝟎𝟏𝟎
