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.

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
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Lucknow
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5. Binary Number System
Base Radix = 2
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8. Conversion Decimal to Binary
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9. Conversion Decimal to Binary with
decimal points
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10. Hexadecimal to Binary
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11. Binary to Hexadecimal
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12. Decimal to Octal
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13. Decimal to Hexadecimal
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14. Decimal to Hexadecimal with
Decimal Point
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Lucknow
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15. Hexadecimal to Decimal
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16. Hexadecimal to Decimal with
Decimal Point
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17. Octal to Decimal with Decimal
Point
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Lucknow
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18. Octal to Decimal with
Decimal Point
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19. Binary to Octal with
floating point values
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20. Octal to Binary with
floating point values
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Lucknow
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21. Hexadecimal to Octal
Conversion
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22. Octal to Hexadecimal
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23. Octal to Hexadecimal
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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
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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
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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
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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
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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
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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
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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
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Dr. Piyush Charan, Dept. of ECE, Integral University, Lucknow

57. BCD Numbers
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59. 1. Binary to decimal Conversion
2. BCD Addition
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60. 9’s & 10’s Complement
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61. Decimal(base 10) Addition
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62. Octal (base 8) Addition
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63. Hexadecimal(base 16) Addition
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64. Hexadecimal Number System
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73. Assignment Question
Q. Solve for x:
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74. Assignment Solution
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