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Unit 4_Part 1_Number System

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Dr Piyush Charan

This document contains lecture notes on number systems for an open educational resource on basic electronics. It covers various number systems including binary, octal, decimal, and hexadecimal. Conversion between these number systems is discussed along with 1's and 2's complement representations. Examples are provided to demonstrate converting between number bases and performing addition and subtraction using 1's and 2's complement representations. The document is authored by Dr. Piyush Charan of Integral University for a course on basic electronics.

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Lecture Notes on Number Systems for Open Educational Resource on Basic Electronics (EC101) by Dr. Piyush Charan Assistant Professor Department of Electronics and Communication Engg. Integral University, Lucknow
Introduction to Number Systems January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 2
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 3
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 4
Binary Number System Base Radix = 2 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 5
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 6
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 7
Conversion Decimal to Binary January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 8
Conversion Decimal to Binary with decimal points January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 9
Hexadecimal to Binary January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 10
Binary to Hexadecimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 11
Decimal to Octal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 12
Decimal to Hexadecimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 13
Decimal to Hexadecimal with Decimal Point January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 14
Hexadecimal to Decimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 15
Hexadecimal to Decimal with Decimal Point January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 16
Octal to Decimal with Decimal Point January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 17
Octal to Decimal with Decimal Point January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 18
Binary to Octal with floating point values January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 19
Octal to Binary with floating point values January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 20
Hexadecimal to Octal Conversion January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 21
Octal to Hexadecimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 22
Octal to Hexadecimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 23
Octal(base=8) Decimal to Octal 8 98 2 8 12 4 8 1 1 0 Octal number system has the radix 8 and will have numbers starting from 0 → 7 (98)10→ (142)8 January 1, 2021 24 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
Octal(base=8) Octal to Decimal 1 4 2 2 1 0 (142)8 →(98)10 1×8^2+4×8^1+2×8^0 =1×64+32+2 =98 January 1, 2021 25 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
Binary to Octal • 0→000 • 1→001 • 2→010 • 3→011 • 4→100 • 5→101 • 6→110 • 7→111 2 7 1 2 3 1 2 1 1 0 January 1, 2021 26 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
Binary to Octal Conversion • (100001110011)2→(4163)8 1 0 0 0 0 1 1 1 0 0 1 1 4 1 6 3 January 1, 2021 27 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
Octal to Hexadecimal • (4163)8 → (?)16 1 0 0 0 0 1 1 1 0 0 1 1 4 1 6 3 1 0 0 0 0 1 1 1 0 0 1 1 8 7 3 January 1, 2021 28 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 29
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 30
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January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 33
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 34
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 35
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 36
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 37
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 38
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42 1. Complement Theory 2. 1’s and, 2’s complement operation Number System Continued.... January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
43 1.Complement Theory Example 1 Get 1’s complement of 50 Complement Digits 50 = 110010  001101 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
44 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. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
45 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 . January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
46 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) January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
47 2’s Complement Arithmetic 1. How to get 2’s complement form 2. Arithmetic operation using 2’s complement theory January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
48 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 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
49 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. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
POS + NEG → POS Answer 50 Take the 2’s complement of the negative number and use regular binary 8-bit addition. 00001001 9 + (-5) 4 ⎯→ ⎯ 11111011 + 00000101  11111010 +1 11111011 2’s Complement Process 100000100 Last Bit = 1: Answer is Positive Disregard 9th Bit January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
POS + NEG → NEG Answer 51 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 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
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) January 1, 2021 52 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
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) January 1, 2021 53 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
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 January 1, 2021 54 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
55 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 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
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 January 1, 2021 56 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
BCD Numbers January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 57
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 58
1. Binary to decimal Conversion 2. BCD Addition January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 59
9’s & 10’s Complement January 1, 2021 60 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
Decimal(base 10) Addition January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 61
Octal (base 8) Addition January 1, 2021 62 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
Hexadecimal(base 16) Addition January 1, 2021 63 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
Hexadecimal Number System January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 64
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 65
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 66
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 67
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 68
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 69
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 70
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 71
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 72
Assignment Question Q. Solve for x: January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 73
Assignment Solution January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 74
January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 75

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Unit 4_Part 1_Number System

  • 1. Lecture Notes on Number Systems for Open Educational Resource on Basic Electronics (EC101) by Dr. Piyush Charan Assistant Professor Department of Electronics and Communication Engg. Integral University, Lucknow
  • 2. Introduction to Number Systems January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 2
  • 3. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 3
  • 4. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 4
  • 5. Binary Number System Base Radix = 2 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 5
  • 6. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 6
  • 7. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 7
  • 8. Conversion Decimal to Binary January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 8
  • 9. Conversion Decimal to Binary with decimal points January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 9
  • 10. Hexadecimal to Binary January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 10
  • 11. Binary to Hexadecimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 11
  • 12. Decimal to Octal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 12
  • 13. Decimal to Hexadecimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 13
  • 14. Decimal to Hexadecimal with Decimal Point January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 14
  • 15. Hexadecimal to Decimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 15
  • 16. Hexadecimal to Decimal with Decimal Point January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 16
  • 17. Octal to Decimal with Decimal Point January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 17
  • 18. Octal to Decimal with Decimal Point January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 18
  • 19. Binary to Octal with floating point values January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 19
  • 20. Octal to Binary with floating point values January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 20
  • 21. Hexadecimal to Octal Conversion January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 21
  • 22. Octal to Hexadecimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 22
  • 23. Octal to Hexadecimal January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 23
  • 24. Octal(base=8) Decimal to Octal 8 98 2 8 12 4 8 1 1 0 Octal number system has the radix 8 and will have numbers starting from 0 → 7 (98)10→ (142)8 January 1, 2021 24 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 25. Octal(base=8) Octal to Decimal 1 4 2 2 1 0 (142)8 →(98)10 1×8^2+4×8^1+2×8^0 =1×64+32+2 =98 January 1, 2021 25 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 26. Binary to Octal • 0→000 • 1→001 • 2→010 • 3→011 • 4→100 • 5→101 • 6→110 • 7→111 2 7 1 2 3 1 2 1 1 0 January 1, 2021 26 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 27. Binary to Octal Conversion • (100001110011)2→(4163)8 1 0 0 0 0 1 1 1 0 0 1 1 4 1 6 3 January 1, 2021 27 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 28. Octal to Hexadecimal • (4163)8 → (?)16 1 0 0 0 0 1 1 1 0 0 1 1 4 1 6 3 1 0 0 0 0 1 1 1 0 0 1 1 8 7 3 January 1, 2021 28 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 29. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 29
  • 30. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 30
  • 31. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 31
  • 32. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 32
  • 33. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 33
  • 34. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 34
  • 35. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 35
  • 36. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 36
  • 37. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 37
  • 38. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 38
  • 39. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 39
  • 40. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 40
  • 41. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 41
  • 42. 42 1. Complement Theory 2. 1’s and, 2’s complement operation Number System Continued.... January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 43. 43 1.Complement Theory Example 1 Get 1’s complement of 50 Complement Digits 50 = 110010  001101 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 44. 44 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. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 45. 45 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 . January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 46. 46 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) January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 47. 47 2’s Complement Arithmetic 1. How to get 2’s complement form 2. Arithmetic operation using 2’s complement theory January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 48. 48 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 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 49. 49 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. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 50. POS + NEG → POS Answer 50 Take the 2’s complement of the negative number and use regular binary 8-bit addition. 00001001 9 + (-5) 4 ⎯→ ⎯ 11111011 + 00000101  11111010 +1 11111011 2’s Complement Process 100000100 Last Bit = 1: Answer is Positive Disregard 9th Bit January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 51. POS + NEG → NEG Answer 51 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 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 52. 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) January 1, 2021 52 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 53. 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) January 1, 2021 53 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 54. 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 January 1, 2021 54 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 55. 55 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 January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 56. 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 January 1, 2021 56 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 57. BCD Numbers January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 57
  • 58. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 58
  • 59. 1. Binary to decimal Conversion 2. BCD Addition January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 59
  • 60. 9’s & 10’s Complement January 1, 2021 60 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 61. Decimal(base 10) Addition January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 61
  • 62. Octal (base 8) Addition January 1, 2021 62 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 63. Hexadecimal(base 16) Addition January 1, 2021 63 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow
  • 64. Hexadecimal Number System January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 64
  • 65. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 65
  • 66. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 66
  • 67. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 67
  • 68. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 68
  • 69. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 69
  • 70. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 70
  • 71. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 71
  • 72. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 72
  • 73. Assignment Question Q. Solve for x: January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 73
  • 74. Assignment Solution January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 74
  • 75. January 1, 2021 Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow 75