The document discusses binary operations in computing including addition, subtraction, multiplication, and division. It explains how early computers performed separate functions for each operation, but that multiplication and division were later eliminated by representing them as repeated addition and subtraction. The document also details binary addition and subtraction logic, 1's and 2's complement representations for negative numbers, and how complements allow subtraction operations to be performed using addition by treating the second operand as a negative number.

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