Unit 2: Data Representation contd.
0 likes948 views
The document provides an educational resource on complement theory, including 1's and 2's complements, along with arithmetic operations such as addition and subtraction in various number systems (binary, decimal, octal, and hexadecimal). It contains examples demonstrating the operations of 1's and 2's complement arithmetic, as well as BCD addition. Additionally, the document outlines methods for converting between different number systems and performing arithmetic calculations using these complements.
![5
1. 1’s Complement Theory
Example 1 : Subtract 1010 from 1111 using 1’s complement theory. (15-10 Small negative)
1 0 1 0 0 1 0 1
1 1 1 1
0 1 0 1+
1] 0 1 0 0
+ 0 0 0 1
0 1 0 1 =(5)
1’s complement
Carry “1” means the answer is positive .
01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow](https://image.slidesharecdn.com/unit2datarepcontd-201105140621/85/Unit-2-Data-Representation-contd-5-320.jpg)
![6
1. 1’s Complement Theory
Example 2 : Subtract 1010 from 1000 using 1’s complement theory. (Large negative 8-10)
1 0 1 0 0 1 0 1
1 0 0 0
0 1 0 1+
0] 1 1 0 1
1’s complement
Carry “0” means the answer is negative and available in 1’s complement form.
1 1 0 1 0 0 1 0 = (2)
01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow](https://image.slidesharecdn.com/unit2datarepcontd-201105140621/85/Unit-2-Data-Representation-contd-6-320.jpg)
Unit 2: Data Representation contd.
- 1. Lectures on Complement Theory (1’s and 2’s, 9’s and 10’s), BCD Addition, Decimal, Octal, and Hexadecimal Addition for Open Educational Resource on Logic Development and Programming (EC221) by Dr. Piyush Charan Assistant Professor Department of Electronics and Communication Engg. Integral University, Lucknow
- 2. 2 1. Complement Theory 2. 1’s and, 2’s complement operation Number System Continued.... 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 3. 3 1.Complement Theory Example 1 Get 1’s complement of 50 Complement Digits 50 = 110010 001101 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 4. 4 1’s Complement Arithmetic (ADD/SUB Method) 1. Read both the operands 2. Negative operand(s) (if any) is converted into 1’s complement form 3. Add both the numbers 4. If carry is generated (i.e. =1) then the resultant number is positive. 5. Add ONE to the output of setp4, to get the final answer. 6. If carry is not generated then the answer is Negative and available in 1’s complement form. 7. Convert output of step 6 into 1’s complement to get final answer. 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 5. 5 1. 1’s Complement Theory Example 1 : Subtract 1010 from 1111 using 1’s complement theory. (15-10 Small negative) 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 1+ 1] 0 1 0 0 + 0 0 0 1 0 1 0 1 =(5) 1’s complement Carry “1” means the answer is positive . 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 6. 6 1. 1’s Complement Theory Example 2 : Subtract 1010 from 1000 using 1’s complement theory. (Large negative 8-10) 1 0 1 0 0 1 0 1 1 0 0 0 0 1 0 1+ 0] 1 1 0 1 1’s complement Carry “0” means the answer is negative and available in 1’s complement form. 1 1 0 1 0 0 1 0 = (2) 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 7. 7 2’s Complement Arithmetic 1. How to get 2’s complement form 2. Arithmetic operation using 2’s complement theory 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 8. 8 2’s Complement Arithmetic (How to get 2’s complement form..?) Example 1 Example 2 Complement Digits Add 1 5 = 00000101 -5 = 11111011 11111010 +1 Complement Digits Add 1 -13 = 11110011 13 = 00001101 00001100 +1 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 9. 9 2’s Complement Arithmetic (Method) 1. Read both the operands 2. Negative operand (if any) is converted into 2’s complement form 3. Add both the numbers (2’s complement of negative operand with the other one). 4. If carry is generated (i.e. =1) then the resultant number is positive and in original form 5. If carry is not generated(when we have negative operand) then the carry is assumed =0. 6. Carry zero means the resultant number is negative and in a 2’s complement form. 7. Convert the 2’s complement form into the original form. 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 10. POS + NEG → POS Answer 10 Take the 2’s complement of the negative number and use regular binary 8-bit addition. 000010019 + (-5) 4 ⎯→ ⎯ 11111011+ 00000101 11111010 +1 11111011 2’s Complement Process 100000100 Last Bit = 1: Answer is Positive Disregard 9th Bit 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 11. POS + NEG → NEG Answer 11 Take the 2’s complement of the negative number and use regular 8-bit binary addition. 11110111(-9) + 5 -4 ⎯→ ⎯ 00000101+ 00001001 11110110 +1 11110111 2’s Complement Process 011111100 Last Bit = 0: Answer is Negative . Discard the last bit 11111100 00000011 +1 00000100 To Check: Perform 2’s Complement On Answer 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 12. Verify the logic using following combinations: 1: (10) –(01) 2: (10) –(02) 3: (10) –(05) 4: (10) –(08) 5: (10) –(09) 6: (10) –(10) 7: (210) –(08) 8: (120) –(55) 9: (52) –(18) 01-10-2020 12Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 13. A+B A B 2’s of B Addition Ans A=10 B=-1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 CY =1 So ans is +ve 1 1 0 0 1 +9 B=-2 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 1 0 1 1 1 0 CY =1 So ans is +ve 1 1 0 0 0 +8 B=-5 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 0 1 CY =1 So ans is +ve +5 B=-8 1 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 CY =1 So ans is +ve +2 B=-9 1 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 0 1 CY =1 So ans is +ve +1 B=-10 1 0 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 0 0 0 CY =1 So ans is +ve +0 2’s Complement Arithmetic (Examples) 01-10-2020 13Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 14. Example: Perform 2’s complement subtraction on 210-08 210 = 1 1 0 1 0 0 1 0 (Subtrahend) 8= 0 0 0 0 1 0 0 0 (Minuend) 2’s complement of 8 is = 1 1 1 1 1 0 0 0 Add both the numbers: 1 1 0 1 0 0 1 0 +1 1 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 Carry = 1 means and is positive +202 01-10-2020 14Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 15. 15 2’s Complement Arithmetic (Examples on varying number of bits) Example: Perform 2’s complement arithmetic for (30)-(50) using 1: 6-bit number system 2: 8-bit number system 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 16. Example: Perform 2’s complement arithmetic for (30)-(50) using: (30)= 0 1 1 1 1 0 (-50)= 1 1 0 0 1 0 2’s complement 0 0 1 1 0 1 0 0 0 0 0 1 0 0 1 1 1 0 Add both the numbers 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 1 1 0 0 Carry =0 means number is negative and in 2’s compl form (30)= 0 0 0 1 1 1 1 0 (-50)= 0 0 1 1 0 0 1 0 2’s complement 1 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 Carry =0 means number is negative and in 2’s compl form 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 = -20 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 = -20 Add both the numbers 1: 6-bit number system 2: 8-bit number system 01-10-2020 16Dr. Piyush Charan, Dept. of ECE, IU Lucknow
- 17. BCD Numbers 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow 17
- 18. 01-10-2020 Dr. Piyush Charan, Dept. of ECE, IU Lucknow 18
- 19. 1. Binary to decimal Conversion 2. BCD Addition 11/2/2020 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 19
- 20. 9’s & 10’s Complement 11/2/2020 20 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
- 21. Decimal(base 10) Addition 11/2/2020 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 21
- 22. Octal (base 8) Addition 11/2/2020 22 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
- 23. Hexadecimal(base 16) Addition 11/2/2020 23 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow