This document discusses number systems and includes the following key points: 1. It introduces the four main number systems: decimal, binary, octal, and hexadecimal. Binary is widely used in digital systems. 2. It describes the different number bases and how they determine the digits used. Decimal uses base 10, binary uses base 2, octal uses base 8, and hexadecimal uses base 16. 3. It provides examples of converting between number systems, such as binary to decimal, octal to decimal, and hexadecimal to decimal. Addition and subtraction in binary systems is also demonstrated.

1. NUMBER SYSTEM
BY:DHARMENDRA CHAUHAN
CHAPTER-1
DIGITAL ELECTRONICS

2. Created By: Dharmendra A. Chauhan
Number System
 Contents
: Introduction of Number System
 Different Types Of RadixBase
 Base Conversion
 Addition, Subtraction

3. Created By: Dharmendra A. Chauhan
Number System
 Introductio
n:• The base of all system is a number system.
• There are four main number system in world.
1.Decimal
2.Binary
3.Octal
4.Hexadecimal
• The binary system are widely used in digital system.
• Everyday used in our life
• When binary system in no of digit is more then hexadecimal & octal system in digital
system.

4. Created By: Dharmendra A. Chauhan
Number System
 Different Types Of
RadixBase:
o Decimal Number System:
Radix: radix means base value.
• The decimal number system has 10 digits(0,1,2,3,4,5,6,7,8,9).
• The decimal number system has a base or radix of 10.
Decimal
Point
… … … · 10 𝑛 · 102 · 101· 100 · 10−1· 10−2 · 10−3 · 10−𝑛………..

5. Created By: Dharmendra A. Chauhan
Number System
o Binary Number System:
• The binary system used in digital system.
• i.e. computer, programming etc.
• The binary number system has 2 digits(0,1).
• The binary number system has a base or radix of 2.
• Weight:
MSB(Most Significant Bit)
LSB(Least Significant Bit)
… … … · 2 𝑛 · 22 · 21· 20 · 2−1· 2−2 · 2−3 · 2−𝑛………..
Binary
Point
MSB
LSB
0010100
(Least Significant Bit)(Most Significant Bit)

6. Created By: Dharmendra A. Chauhan
Number System
Page 5
o Octal Number System:
• The octal number system has 8 digits(0,1,2,3,4,5,6,7).
• The binary number system has a base or radix of 8.
… … … · 8 𝑛
· 82
· 81
· 80
· 8−1
· 8−2
· 8−3
· 8−𝑛
………..
Octal Point

7. Created By: Dharmendra A. Chauhan
Number System
o Hexadecimal Number
System:
• The hexadecimal number system has 16 digits(0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F).
• The hexadecimal number system has a base or radix of 16.
Hexadecimal
Point
… … … · 16 𝑛
· 162
· 161
· 160
· 16−1
· 16−2
· 16−3
· 16−𝑛
………..

8. Created By: Dharmendra A. Chauhan
Number System
 Base Conversion:
 Conversion from any Radix to
Decimal:
• Binary to Decimal
• Octal to Decimal
• Hexadecimal to Decimal

9. Created By: Dharmendra A. Chauhan
Number System
o Binary to Decimal:
 Integer Number Example:
Example 1: 0110 2 ? 10
Solution:
0 x 23 + 1 x 22 +1 x 21 +0 x 20
0 4 2 0
=
=
= 0+4+2+0
6=

10. Created By: Dharmendra A. Chauhan
Number System
o Binary to Decimal:
 Fraction Number Example:
Example 2: 010.101 2 ? 10
Solution:
0 x 22 + 1 x 21 + 1 x 20 · 1 x 2−1 +0 x 2−2 +1 x 2−3
0 2 0 0.5
=
=
= 0+2+0+0.5+0+0.1.25
2.625=
0 0.125

11. Created By: Dharmendra A. Chauhan
Number System
o Octal to Decimal:
 Example:
Example 3: 113 8 ? 10
Solution: 1 x 82 + 1 x 81 + 3x 80
64 8 3
=
=
= 64+8+3
75=

12. Created By: Dharmendra A. Chauhan
Number System
o Hexadecimal to Decimal:
 Example:
Example 4: 16F2 16 ? 10
Solution: 1 x 163 + 4 x 162 + 15x161 + 2x160
4096 1536
=
=
= 4096+1536+240+2
5874=
240
F=15

13. Created By: Dharmendra A. Chauhan
Number System
 Conversion from Decimal, Binary, Octal to
any Radix:
• Decimal to Binary
• Decimal to Octal
• Decimal to Hexadecimal
• Binary to Octal
• Binary to Hexadecimal
• Octal to Hexadecimal

14. Created By: Dharmendra A. Chauhan
Number System
o Decimal to Binary
2 105 1
2 52 0
2 26 0
2 13 1
2 6 0
2 3 1
2 1 1
0
Remainder
 Integer Number Example:
Example 5: 105 10 ? 2
Solution:
LSB
MSB
So, The Answer is
105 10 1101001 2

15. Created By: Dharmendra A. Chauhan
Number System
o Decimal to Binary
Remainder
Example 6: 0.17 10 ? 2
Solution:
MSB
LSB
So, The Answer is
0.17 10 0.0010101 2
0.17x2=0.34
0
0.34x2=0.68
0
0.68x2=1.36
1
0.36x2=0.72
0
 Fraction Numbers
Example

16. Created By: Dharmendra A. Chauhan
Number System
o Decimal to Octal
8 135 7
8 16 0
8 2 2
0
Remainder
Example:
Example 7: 105 10 ? 8
Solution:
LSB
MSB
So, The Answer is
105 10 207 2

17. Created By: Dharmendra A. Chauhan
Number System
o Decimal to Hexadecimal
16 3509 5
16 219 11=B
16 13 13=D
0
Remainder
Example 8: 3509 10 ? 16
Solution:
LSB
MSB
So, The Answer is
105 10 DB5 2

18. Created By: Dharmendra A. Chauhan
Number System
o Binary to Hexadecimal
Example 9: 001100 2 ? 8
Solution:
1 4 So, The Answer is
001100 2 14 8
001 100

19. Created By: Dharmendra A. Chauhan
Number System
o Binary to Octal
Example 10: 000111100000 2 ? 16
Solution:
1 14=E 0 So, The Answer is
000111100000 2 1E0 8
0001 1110 0000

20. Created By: Dharmendra A. Chauhan
Number System
o Octal to Hexadecimal
Example 11: 000111100000 2 ? 16
Solution:
1 14=E 0 So, The Answer is
000111100000 2 1E0 8
0001 1110 0000

21. Created By: Dharmendra A. Chauhan
Number System
o Addition
11
1101
1100
Example 12:Perform addition of
1101 2 𝑎𝑛𝑑 1100 2
+
11001
Carry
Answer
Answer:
Example 13:Perform addition of
101.111 2 𝑎𝑛𝑑 110.011 2
Addition Rule
Carry
0+0=0 -
0+1=1 -
1+0=1 -
1+1=10 10
1111.11
101.111
110.011
1100.01
0
Answer
Answer:
+
Carry

22. Created By: Dharmendra A. Chauhan
Number System
o Subtraction
1101
1100
Example 12:Perform addition of
1101 2 𝑎𝑛𝑑 1100 2
-
10001
Carry
Answer
Answer:
Example 13:Perform addition of
101.111 2 𝑎𝑛𝑑 110.011 2
Subtraction
Rule
1-1=0 0
1-0=0 -
0-1=1 1
0-0=0 0
1111.11
101.111
110.011
1100.01
0
Answer
Answer:
+
Carry

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