This document discusses binary operations and computer arithmetic. It covers binary addition and subtraction operations, as well as 1's and 2's complements which allow subtraction to be performed using addition by taking the complement of the subtrahend. The document also provides examples of performing binary addition, subtraction, and using complements to subtract. It notes that early computers had separate functions for operations like addition and multiplication but that today's computers perform computations using adders and can subtract using complements to eliminate the need for separate subtractors.

1. LECTURE - 3
 Binary Addition
 Binary Subtraction
 Complements
1’s Complement
2’s Complement
 Subtraction Using Complements

2. BASIC OPERATIONS
 ADDITION
 SUBTRACTION
 MULTIPLICATION
 DIVISION

3. COMPUTER OPERATIONS
 BINARY ADDITION
 BINARY SUBTRACTION

4. The First Computer
• Covered a whole Room
• Separate Functions for Adding,
Subtracting, Multiplication and Division

5. ELIMINATION OF OPERATIONS
 MULTIPLICATION and DIVISION were eliminated
 MULTIPLICATION :
repeated addition
(2 x 4 = 8) , (2+ 2+ 2 + 2 = 8)
 DIVISION :
repeated subtractions

6. TODAY’S COMPUTER
 Computations
Addition – (ADDERS)
Subtraction – (SUBTRACTORS)

7. BINARY ADDITION
LOGIC:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 (1 Carry)

8. EXAMPLE
1 0 0 1 1 0
+ 1 1 0 1 0 1
1 0 1 1 0 1 1
Carry

9. BINARY SUBTRACTION
LOGIC:
0 - 0 = 0
0 - 1 = 1 ( Borrow 1)
1 - 0 = 1
1 - 1 = 0

10. EXAMPLE
1 1 0 1 0 1
- 0 1 1 0 1 0
0 1 1 0 1 1

11. COMPLEMENTS
2 types:
 1’s Complement
 2’s Complement

12. 1’s COMPLEMENT
PROCEDURE
 Change every 0 to 1 and every 1 to 0
EXAMPLE
Find 1’s Complement of:
( 1 1 0 0 1 0 1 0 )2

13. 2’s COMPLEMENT
PROCEDURE
1. Find 1’s Complement:
Change every 0 to 1 and every 1 to 0
2. Add 1 to L.S.B

14. EXAMPLE
Find 2’s Complement of :
( 1 1 0 1 0 1 0 )2

15. SOLUTION : ( 1 1 0 1 0 1 0 )2
STEP 1
0 0 1 0 1 0 1
STEP 2
0 0 1 0 1 0 1
+ 1
0 0 1 0 1 1 0 LSB

16. SUBTRACTION USING COMPLEMENTS
SUBTRACTION
 MINUEND (1st value)
 SUBTRAHEND (2nd value)

17. SUBTRACTION USING 1’S
COMPLEMENT
STEP 1
Take 1’s Complement of Subtrahend
STEP 2
Add the result to Minuend
STEP 3
Look for end Carry.

18. END CARRY
 If Carry Exists:
Add the carry to LSB of result
 If NO Carry:
Take 1’s complement of result
again and
place a –ve sign

19. EXAMPLE
Subtract: ( 1 1 0 1 0 1 1 - 1 0 1 1 0 1 )2
SOLUTION
Minuend: 1 1 0 1 0 1 1
Subtrahend: 0 1 0 1 1 0 1
STEP 1 1’s Complement of Subtrahend
1 0 1 0 0 1 0
STEP 2 Add to Minuend
STEP 3 Add end carry to LSB of result.

20. SUBTRACTION USING 2’S
COMPLEMENT
 STEP 1
Take 2’s Complement of Subtrahend
 STEP 2
Add the result to Minuend
 STEP 3
Look for end Carry.

21. END CARRY
 If Carry Exists:
Drop it
 If NO Carry:
 Take 2’s complement of result again and
 place a –ve sign

22. EXAMPLE
Subtract: ( 1 1 0 1 0 1 1 - 1 0 1 1 0 1 )2

23. ADVANTAGE OF
COMPLEMENTS
 It Converts Subtraction Process to Addition
 Thus, the need for a Subtractor is Eliminated