1. Chapter 1 (Part b) Digital Systems and Binary Numbers
Digital Logic Design
Chapter 1 (Part b) Digital Systems and Binary Numbers
Originally by T.Tasniem Nasser Al-Yahya

2. Outline of Chapter 1 (Part b)
1.5 Complements

3. Complements
Complements are used in digital computers to simplify the subtraction operation and for logical manipulation.
Simplifying operations leads to simpler , less expensive circuits to implement the operations.

4. 1.5 Complements
There are two types of complements for each base-r system:
Diminished Radix Complement - (r-1)’s Complement.
Radix Complement - (r)’s Complement.

5. Complements Diminished Radix Complement-(r-1)’s Complement
Given a number N in base r having n digits, the (r–1)’s complement of N is defined as:
(rn –1) – N

6. Complements Diminished Radix Complement-(r-1)’s Complement
Example for 6-digit decimal numbers:
Example for 7-digit binary numbers:
Observation:
Subtraction from (rn – 1) will never require a borrow
Diminished radix complement can be computed digit-by-digit
For binary: 1 – 0 = 1 and 1 – 1 = 0
Subtract each digit from 9
9’s complement of is – =
Subtract each digit from 1 (1-0=1, 1-1=0)
1’s complement of is – =

7. Complements Diminished Radix Complement-(r-1)’s Complement
1’s Complement (Diminished Radix Complement)
All ‘0’s become ‘1’s
All ‘1’s become ‘0’s
Example ( )2
 ( )2
If you add a number and its 1’s complement …

8. Complements Radix Complement-(r)’s Complement
Given a number N in base r having n digits, the (r)’s complement of N is defined as:
Comparing with the (r  1) 's complement, we note that the r's complement is obtained by adding 1 to the (r  1) 's complement, since rn – N = [(rn  1) – N] + 1.

9. Complements Radix Complement-(r)’s Complement
Example: Base-10
Example: Base-2
Observation:
For binary: starting from right to left keep all the zeroes unchanged and toggle all bits to the left of the first ‘1’.
The 10's complement of is ( 9's complement+1)
The 10's complement of is
The 2's complement of is ( 1's complement+1)
The 2's complement of is

10. Complements Subtraction using Complements
When subtraction is implemented with digital hardware it is more efficient to use complements.
Procedure for subtracting two n-digit unsigned numbers M – N in base r :
Line up the numbers from right to left. If one number is shorter extend it by adding leading zeros to the front of the number. (my notes)
Add minuend M to the r's complement of the subtrahend N.
This performs M + (rn - N)

11. Complements subtraction using Complements
If M >= N, the sum will produce an end carry rn which is discarded, and what is left is the result M - N
If M < N, the sum does not produce an end carry. It is equal to the r's complement of (M - N).
The correct answer is generated by
taking the r's complement of the answer
then adding a negative sign to the front
Overflow

12. Complements subtraction using Complements
Example 1.7
Given the two binary numbers X = and Y = , perform the subtraction (a) X – Y ; and (b) Y  X, by using 2's complement.
Discard carry
There is no end carry. To clarify, the answer is Y – X =  (2's complement of ) = 

13. Reading
Chapter 1: 1.5