The document provides an overview of fundamental digital number systems, including decimal, binary, octal, and hexadecimal systems. It explains the principles of conversion between these number systems and their positional value definitions, emphasizing the importance of each system in digital technology. Various methods for converting between bases, along with examples and exercises for practice, are also included.

1. Unit-1
Fundamentals of
Digital Systems
deepak.sigroha@recsonbhadra.ac.in
+91-9478856526
Electronics Engineering Department
Deepak Sigroha
Digital System Design
KEC302
Rajkiya Engineering College Sonbhadra

2. Digital Number Systems

3. Unit 1 – Fundamentals of Digital Systems 3
Digital Number System
 Many Number systems are used in digital technology. The most common are the
decimal, binary, octal and hexadecimal systems.
 A number is represented by as
 Here is number and is the base or radix.
 Positional value system in which value of a digit depends upon its position.
….…….…….

4. Unit 1 – Fundamentals of Digital Systems 4
Decimal Number System
 The decimal system contains ten unique symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
 Since counting in decimal involves ten symbols, we say that its base or radix is
ten. There is no symbol for its base, i.e. for ten.
 It is a positional weighted system. It means that the value attached to a symbol
depends on its location with respect to the decimal point. In this system, any
number (integer, fraction, or mixed) of any magnitude can be represented by the
use of these ten symbols only.
 Each symbol in the number is called a digit.

5. Unit 1 – Fundamentals of Digital Systems 5
Common Number System
System Base Symbols Used by Humans? Used in
Computers?
Decimal 10
0, 1, 2, 3, 4,
5, 6,7, 8, 9
Yes No
Binary 2 0,1 No Yes
Hexa-
decimal
16
0, 1, 2, 3, 4,
5, 6,7, 8, 9,
A, B, C, D,
E, F
No Yes
Octal 8
0, 1, 2, 3, 4,
5, 6, 7
No Yes

6. Unit 1 – Fundamentals of Digital Systems 6
Conversion Among Bases
 Possibilities
 Example
Hexadecimal
Decimal Octal
Binary
2510=110012 =318=1916
Base/Radix

7. Unit 1 – Fundamentals of Digital Systems 7
Quantities/Counting
Decimal Binary Octal
Hexa-
decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
Decimal Binary Octal
Hexa-
decimal
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

8. Unit 1 – Fundamentals of Digital Systems 8
Base to Decimal & Vice-versa
 Base to Decimal:
 Any positional value system number can be converted to its decimal equivalent simply by the
summation of multiplication of each digit/bit/symbol by its weights.
 Decimal to Base :
 The integral decimal number is repeatedly divided by and writing the remainders after each
division until a quotient is obtained. The last remainder is the MSB. The remainders read from
bottom to top give the equivalent integer number.
 For fraction number is multiplied by base . The given decimal fraction and the subsequent
decimal fractions are successively multiplied by , till the fraction part of the product is 0 or till
the desired accuracy is obtained. The first integer obtained is the MSB. Thus, the integers
read from top to bottom give the equivalent binary fraction.

9. Unit 1 – Fundamentals of Digital Systems 9
Binary Number System
 The binary number system is a positional weighted system. The base or radix of
this number system is 2. Hence, it has two independent symbols. The base itself
cannot be a symbol. The symbols used are 0 and 1.
 A binary digit is called a bit.
 A binary number consists of a sequence of bits, each of which is either a 0 or a 1.
 The binary point separates the integer and fraction parts.

10. Unit 1 – Fundamentals of Digital Systems 10
Decimal to Binary Conversion
 The integral decimal number is repeatedly divided by ‘2’ and writing the
remainders after each division until a quotient is obtained. The last
remainder is the MSB. The remainders read from bottom to top gives the
equivalent binary integer number.
 The fraction number is multiplied by base ‘2’. The given decimal fraction
and the subsequent decimal fractions are successively multiplied by , till
the fraction part of the product is or till the desired accuracy is obtained.
The first integer obtained is the MSB. Thus, the integers read from top to
bottom give the equivalent binary fraction.
This flowchart describes the process and can be used to convert from decimal to
binary number system.

11. Unit 1 – Fundamentals of Digital Systems 11
Decimal to Binary Conversion
 Example - 1
12510 =?2
2 125
1
2 62
0
2 31
1
2 15
1
2 7
1
2 3
1
2 1
1
0
12510 =11111012
 Example - 2
0.687510=?2
0.6875×2=¿
1.3750 1 0.3750
+¿
0.3750×2=¿
0.7500 0 0.7500
+¿
0.7500×2=¿
1.5000 1 0.5000
+¿
0.5000×2=¿
1.0000 1 0.0000
+¿
0.687510=0.10112
integer fraction

12. Unit 1 – Fundamentals of Digital Systems 12
Binary to Decimal Conversion
 Technique
 Multiply each bit by , where n is the “weight” of the bit
 The weight is the position of the bit w.r.t binary point, starting from on the left of binary point
and on the right of binary point. Finally, Add the results.
Binary Decimal

13. Unit 1 – Fundamentals of Digital Systems 13
Binary to Decimal Conversion
 Example - 1
1 0 1 0 1 1
1 20
1 21
0 22
123
0 24
1 25
+¿
+¿
+¿
+¿
+¿
1
2
0
0
32 +¿
+¿
+¿
+¿
+¿
1010112=4310
8
1010112=?10
 Example - 2
11.112=?10
1 1 . 1 1
1 20
1 21
+¿
1
2 +¿
1 2-2
1 2-1
+¿
0.25
0.5 +¿
+¿
+¿
11.112=3.7510

14. Unit 1 – Fundamentals of Digital Systems 14
Binary Counting Sequence
• The rightmost column in the binary
number begins with a and alternates
between and .
• The second column begins with (= )
and alternates between the groups
of and .
• The third column begins with (= )
and alternates between the groups
of and .
• The nth
column begins with and
alternates between the groups of
and .

15. Unit 1 – Fundamentals of Digital Systems 15
Decimal to Binary and Vice-versa Conversion Exercise
 (32)10 = ( )2
 (555)10 = ( )2
 (12999)10 = ( )2
 (157.63)10 = ( )2
 (64.125)10 = ( )2
 (11011)2 = ( )10
 (101101)2 = ( )10
 (11101111)2 = ( )10
 (110.011)2 = ( )10
 (1001.0010)2 = ( )10

16. Unit 1 – Fundamentals of Digital Systems 16
Octal Number System
 The octal number system was extensively used by early minicomputers.
 It is also a positional weighted system. Its base or radix is 8. It has 8 independent
symbols 0, 1, 2, 3, 4, 5, 6, and 7.
 Since its base , every 3-bit group of binary can be represented by an octal digit. An
octal number is, thus, the length of the corresponding binary number.

17. Unit 1 – Fundamentals of Digital Systems 17
Decimal To Octal Conversion
Technique
 The integral decimal number is repeatedly divided by ‘8’ and writing the
remainders after each division until a quotient ‘0’ is obtained. The last remainder
is the MSD. The remainders read from bottom to top give the equivalent binary
integer number.
 For fraction number is multiplied by base . The given decimal fraction and the
subsequent decimal fractions are successively multiplied by , till the fraction part
of the product is ‘0’ or till the desired accuracy is obtained. The first integer
obtained is the MSD. Thus, the integers read from top to bottom give the
equivalent octal fraction.

18. Unit 1 – Fundamentals of Digital Systems 18
Decimal To Octal Conversion
 Example - 1
12510 =?8
8 125
5
8 15
7
8 1
1
0
12510 =¿
0.687510=?8
0.6875×8=¿
5.5000 5 0.5000
+¿
0.5000×8=¿ 4 0.0000
+¿
0.687510=¿
0.54 8
integer fraction
 Example - 2

19. Unit 1 – Fundamentals of Digital Systems 19
Octal to Decimal Conversion
Technique
 Multiply each digit by , where n is the “weight” of the digit.
 The weight is the position of the bit w.r.t octal point, starting from on the left of
binary point and on the right of octal point. Finally, Add the results.

20. Unit 1 – Fundamentals of Digital Systems 20
Octal To Decimal Conversion
 Example - 1
7 2 4
4 80
2 81
7 82
+¿
+¿
7248=¿
46810
4
16
448 +¿
+¿
7248=?10
43.258 =?10
 Example - 2
4 3 . 2 5
3 80
4 81
+¿
3
32
43.258 =¿
+¿
35.328110
5 8-2
2 8-1
+¿
0.0781
0.25 +¿
+¿
+¿

21. Unit 1 – Fundamentals of Digital Systems 21
Octal Counting Sequence

22. Unit 1 – Fundamentals of Digital Systems 22
Decimal to Octal & Vice-versa Exercise
 (32)10 = ( )8
 (555)10 = ( )8
 (12999)10 = ( )8
 (157.63)10 = ( )8
 (64.125)10 = ( )8
 (32)8 = ( )10
 (555)8 = ( )10
 (12333)8 = ( )10
 (157.63)8 = ( )10
 (64.125)8 = ( )10

23. Unit 1 – Fundamentals of Digital Systems 23
Decimal to Hexadecimal Conversion
Technique
 The integral decimal number is repeatedly divided by ‘16’ and writing the
remainders after each division until a quotient ‘0’ is obtained. The last remainder
is the MSD/MSS. The remainders read from bottom to top give the equivalent
binary integer number.
 For fraction number is multiplied by base ‘16’. The given decimal fraction and the
subsequent decimal fractions are successively multiplied by , till the fraction part
of the product is or till the desired accuracy is obtained. The first integer obtained
is the MSD/MSS. Thus, the integers read from top to bottom give the equivalent
hexadecimal fraction.

24. Unit 1 – Fundamentals of Digital Systems 24
Decimal to Hexadecimal Conversion
 Example - 1
123410=?16
16 1234
2
123410=¿
16 77
16 4
4
0
0.0312510=?16
0.03125×16=¿
0.5000
0 0.5000
+¿
0.5000×16=¿
8.00008 0.0000
+¿
0.0312510=¿
0.08 16
integer fraction
 Example - 2

25. Unit 1 – Fundamentals of Digital Systems 25
Hexadecimal Number System
 Binary numbers are long. These numbers are fine for machines but are too lengthy
to be handled by human beings. So, there is a need to represent the binary numbers
concisely. One number system developed with this objective is the hexadecimal
number system (or Hex).
 The hexadecimal number system is a positional-weighted system. The base or radix
of this number system is 16, that means, it has 16 independent symbols. The symbols
used are and . Since its base is , every 4 binary digit combination can be represented
by one hexadecimal digit.
 So, a hexadecimal number is the length of the corresponding binary number, yet it
provides the same information as the binary number.

26. Unit 1 – Fundamentals of Digital Systems 26
Hexadecimal Number System
 A 4-bit group is called a nibble.
 Since computer words come in 8 bits, 16 bits, 32 bits and so on, that is, multiples
of 4 bits, they can be easily represented in hexadecimal.
 The hexadecimal system is particularly useful for human communications with
computers. By far, this is the most commonly used number system in computer
literature. It is used both in large and small computers.

27. Unit 1 – Fundamentals of Digital Systems 27
Hexadecimal To Decimal Conversion
Technique
 Multiply each digit by , where n is the “weight” of the digit.
 The weight is the position of the bit w.r.t octal point, starting from on the left of
binary point and on the right of hexadecimal point. Finally, Add the results.

28. Unit 1 – Fundamentals of Digital Systems 28
Hexadecimal to Decimal Conversion
 Example - 1
A B C
C 160
B 161
A 162
+¿
+¿
𝐴𝐵𝐶16=¿
274810
12 160
11 161
10 162
+¿
+¿
12
176
2560 +¿
+¿
ABC16=?10 43.2516=?10
 Example - 2
4 3 . 2 5
3 160
4 161
+¿
3
64
43.2516=¿
+¿
67.144510
5 16-2
2 16-1
+¿
0.0195
0.125+¿
+¿
+¿

29. Unit 1 – Fundamentals of Digital Systems 29
Hexadecimal Counting Sequence

30. Unit 1 – Fundamentals of Digital Systems 30
Decimal to Hexadecimal & Vice-versa Exercise
 (32)10 = ( )16
 (555)10 = ( )16
 (12999)10 = ( )16
 (157.63)10 = ( )16
 (64.125)10 = ( )16
 (FA8)16 = ( )10
 (9AC3)16 = ( )10
 (1A74D)16 = ( )10
 (1AC.9A)16 = ( )10
 (ABC.5AC)16 = ( )10

31. Unit 1 – Fundamentals of Digital Systems 31
Octal to Binary Conversion
Technique
 Convert each octal digit to a 3-bit equivalent binary representation
 Example Octal Binary
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111
7058 =?2
7 0 5
101
000
111
7058 =¿
1110001012
615.2348=110001101.0100111002

32. Unit 1 – Fundamentals of Digital Systems 32
Binary to Octal Conversion
Technique
 From given fractional point, group bits in threes to right and group bits in threes
to left
 If, left with less than 3 bits at the end then stuff 0s to make it group of three
 Convert to octal digits
 Example
1011010.1112=?8
1
011
001
1011010.1112=¿
132.7 8
111
010
3 2 7
.

33. Unit 1 – Fundamentals of Digital Systems 33
Octal to Binary and Vice-versa Conversion Exercise
 (463)8 = ( )2
 (2056)8 = ( )2
 (2057.64)8 = ( )2
 (6543.04)8 = ( )2
 (7476.47)8 = ( )2
 (11011)2 = ( )8
 (101101)2 = ( )8
 (11101111)2 = ( )8
 (110.011)2 = ( )8

34. Unit 1 – Fundamentals of Digital Systems 34
Hexadecimal to Binary Conversion
Technique
 Convert each hexadecimal digit to a 4-bit equivalent binary representation
 Example
Hexa-
Decimal
Binary Hexa-
Decimal
Binary
0 0000 8 1000
1 0001 9 1001
2 0010 1010
3 0011 1011
4 0100 1100
5 0101 1101
6 0110 1110
10 AF16 =?2
1 0 A F
1111
1010
0000
10 AF16=¿
10000101011112
0001
AB.9 F16=10101011.100111112

35. Unit 1 – Fundamentals of Digital Systems 35
Binary to Hexadecimal Conversion
Technique
 From given fractional point, group bits in fours to right and group bits in fours to
left
 If, left with less than 4 bits at the end then stuff 0s to make it group of four
 Convert to hexadecimal digits
 Example
101101.01112=?16
0010
101101.01112=¿
2 D .716
0111
1101
2 D 7
.

36. Unit 1 – Fundamentals of Digital Systems 36
Hexadecimal to Binary and Vice-versa Conversion Exercise
 (FA8)16 = ( )2
 (9AC3)16 = ( )2
 (1A74D)16 = ( )2
 (1AC.9A)16 = ( )2
 (ABC.5AC)16 = ( )2
 (11011)2 = ( )16
 (101101)2 = ( )16
 (11101111)2 = ( )16
 (110.011)2 = ( )16

37. Unit 1 – Fundamentals of Digital Systems 37
Octal to Hexadecimal Conversion
Technique
 Convert Octal to Binary
 From given fractional point, group bits in fours to right and group bits in fours
to left
 Convert Binary to Hexa-Decimal
 Example 10768=?16
1 0 7 6
110
111
000
10768=¿
23 E16
001
1110
0011
0010
E
3
2

38. Unit 1 – Fundamentals of Digital Systems 38
Hexadecimal to Octal Conversion
Technique
 Convert Hexa-Decimal to Binary
 From given fractional point, group bits in threes to right and group bits in threes
to left
 Convert Binary to Octal
 Example
1 F0C16=?8
1 F 0 C
1100
0000
1111
1 F0C16=¿
174148
0001
100
001
100
4
1
4
111
7
001
1
000
0

39. Unit 1 – Fundamentals of Digital Systems 39
Octal to Hexadecimal and Vice-versa Conversion Exercise
 (463)8 = ( )16
 (2056)8 = ( )16
 (2057.64)8 = ( )16
 (6543.04)8 = ( )16
 (7476.47)8 = ( )16
 (1020)16 = ( )8
 (1A0)16 = ( )8
 (ABCD)16 = ( )8
 (CD.F2)16 = ( )8

40. Unit 1 – Fundamentals of Digital Systems 40
Usefulness of the Octal and Hexadecimal Number System
 In computer work, binary numbers with up to 64-bits are not uncommon. These binary
numbers do not always represent a numerical quantity; they often represent some type
of code, which conveys non-numerical information.
 In computers, binary numbers might represent (a) the actual numerical data, (b) the
numbers corresponding to a location (address) in memory, (c) an instruction code, (d) a
code expressing alphabetic and other non-numerical characters, or (e) a group of bits
representing the status of devices internal or external to the computer.
 The ease with which conversions can be made between octal and binary, & hexadecimal
to binary makes the octal & hexadecimal system more attractive as a shorthand means
of expressing large binary numbers.
 When dealing with large binary numbers of many bits, it is convenient and more efficient
for us to write the numbers in octal or hexadecimal rather than binary.
 However, the digital circuits and systems work strictly in binary; we use octal or
hexadecimal; only for the convenience of the operators of the system.
The hexadecimal system is particularly useful for human communications with
computers. By far, this is the most commonly used number system in computer
literature.

41. Unit 1 – Fundamentals of Digital Systems 41
References
1. M. Morris Mano, M. D. Ciletti, “Digital Design” 6th Ed., USA : Prentice-
Hall.
2. A. Anand Kumar, “Fundamentals of Digital Circuits” 4th Ed., PHI.
3. T. L. Floyd, “Digital Fundamental”, 11th Ed., USA : Prentice-Hall.
4. R.J. Tocci, N. S. Widmer and G. L. Moss “Digital Systems: Principles and
Applications”, 12th Ed., USA : Prentice-Hall.

42. Thank You