The document discusses number systems and conversions between different number systems. It introduces positional and symbolic number systems. The key number systems covered are binary, octal, decimal, and hexadecimal systems. The document explains how to count in each system and provides tables showing equivalent values. It then describes how to convert between different number systems by grouping bits or digits and using place value. Examples are provided for converting between binary, octal, decimal, and hexadecimal numbers.

1. Prof. D. D. Shrivastava
Assistant Professor
NUMBER SYSTEM

2. CONTENTS
1/15/20212
 Number System
 Need of Number System and Introduction
 Different Number Systems
 Number System Conversion
Number System

3. NUMBER SYSTEMS
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 Why Numbers?
 To measure
 To count
 To represent information
 To communicate
 Number system: Can be referred as analogue or digital?
 Positional Number System – Decimal number system
 Symbolic Number System – Roman number system

4. NUMBER SYSTEMS
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Positional Number System –
 Base/Radix of number system – number of symbols in a
number systems.
 Maximum count in a number system = Base – 1
 Different number systems –
 Binary number system
 Octal number system
 Decimal number system
 Hexadecimal number system

5. NUMBER SYSTEM
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 Number system, symbol and base.
Number
System
Base/
Radix
Symbols
Binary 2 0, 1
Octal 8 0, 1, 3, 4, 5, 6, 7
Decimal 10 0, 1, 3, 4, 5, 6, 7, 8, 9
Hexadecimal 16 0, 1, 3, 4, 5, 6, 7, 8, 9,A, B, C, D, E, F
Number
System
Base/
Radix
Symbols
Random 4 0, 1, 2, 3
Random 13 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,A, B, C

6. NUMBER SYSTEM
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Counting in various number systems.
 Decimal –
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
 0, 1, 2,…9,10, 11, 12,….19,20, 21, 22,…
 Octal –
 0, 1, 2, 3, 4, 5, 6, 7
 0, 1, 2,…7,10, 11, 12,….17,20, 21, 22,…
 Hexadecimal –
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,A, B, C, D, E, F
 0, 1, 2,…F, 10, 11, 12,….1F,20, 21, 22,…

7. NUMBER SYSTEM
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Counting in various number systems.
 Binary –
 0, 1
 0, 1, 10, 11
 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011,
1100,1101, 1110, 1111
 Random –
 0, 1, 2, 3, 4
 0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31,…

8. NUMBER SYSTEM
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Counting in various number systems and equivalent numbers.
Binary Octal Decimal Hexadecimal
0 0 0 0
1 1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
6 6 6
7 7 7
8 8
9 9
A
B
C
D
E
F
Binary Octal Decimal Hexadecimal
0 0 0 0
1 1 1 1
10 2 2 2
11 3 3 3
100 4 4 4
101 5 5 5
110 6 6 6
111 7 7 7
1000 10 8 8
1001 11 9 9
1010 12 10 A
1011 13 11 B
1100 14 12 C
1101 15 13 D
1110 16 14 E
1111 17 15 F

9. NUMBER SYSTEM
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Counting in various number systems and equivalent numbers.
Binary Octal Decimal Hexadecimal Base-5
0 0 0 0 0
1 1 1 1 1
10 2 2 2 2
11 3 3 3 3
100 4 4 4 4
101 5 5 5 10
110 6 6 6 11
111 7 7 7 12
1000 10 8 8 13
1001 11 9 9 14
1010 12 10 A 20
1011 13 11 B 21
1100 14 12 C 22
1101 15 13 D 23
1110 16 14 E 24
1111 17 15 F 30

10. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Binary Octal Decimal Hexadecimal
0 0 0 0
1 1 1 1
10 2 2 2
11 3 3 3
100 4 4 4
101 5 5 5
110 6 6 6
111 7 7 7
1000 10 8 8
1001 11 9 9
1010 12 10 A
1011 13 11 B
1100 14 12 C
1101 15 13 D
1110 16 14 E
1111 17 15 F

11. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Binary Octal
000 0
001 1
010 2
011 3
100 4
101 5
110 6
111 7
Binary Hexadecimal
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F

12. • Example – (101111110)B = (?)O
( 101 111 110 )B = (576)O
CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Conversion: Binary Octal.
The base of octal number system is third power of the base of
binary number system.
• Group three bits of binary number and write digit octal
equivalent number for each group.

13. (460)O = ( 100 110 000 )B
• Example – (460)O = (?)B
• Write three bit binary equivalent for every octal digit.
CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Conversion: Binary Octal.
The base of octal number system is third power of the base of
binary number system.

14. ( 1 0111 1110 )B = (17E)H
CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Conversion: Binary Hexadecimal.
The base of hexadecimal number system is fourth power of the
base of binary number system.
• Group four bits of binary number and write one digit
hexadecimal equivalent number for each group.
• Example – (101111110)B = (?)H

15. • Example – (460)H = (?)B
• Write four bit binary equivalent for every hexadecimal digit.
(460)H = ( 0100 0110 0000 )B
CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Conversion: Binary Hexadecimal.
The base of hexadecimal number system is fourth power of the
base of binary number system.

16. (615)O = ( 110 001 101 )B = (18D) H
CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Conversion: Octal Hexadecimal.
• Convert octal number to equivalent binary number.
• Convert this binary number to equivalent hexadecimal
number.
• Example – (615)O = (?)H
= (18D)H

17. (A6C)H = ( 1010 0110 1100 )B = (18D)O= (5154)O
CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Conversion: Octal Hexadecimal.
• Convert hexadecimal number to equivalent binary number.
• Convert this binary number to equivalent octal number.
• Example – (A6C)H = (?)O

18. Grouping . Grouping
CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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 What if the numbers have a radix point?
 Writing equivalent numbers
 Grouping and then writing equivalent numbers
1101.1101 A4D.907 52.345
Radix Point

19. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Conversion: Base-n Decimal.
• n can be 2, 8, 16 or any other value.
• Decimal number to an equivalent number in number system
with base-n.
• Integer part of the decimal number – division by n (base of
the number system in which equivalent number is to be
obtained) repeatedly till the quotient is 0.
• The remainder left from division is written as numeral answer
in reverse order.
• Fractional part of decimal number –multiplication by n till
result becomes 0 or is reoccurring.
• The integral part of the result of multiplication is written as
the fractional result.

20. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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• Example –
Decimal to Binary – divide by 2
(12)D = (?)B
(12)D = (1100)B
Division Quotient Remainder
12/2 6 0
6/2 3 0
3/2 1 1
1/2 0 1

21. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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• Example –
Decimal to Binary – divide by 2
(10.8125)D = (?)B
(10.56)D = (1010.1101)B
Division Quotient Remainder
10/2 5 0
5/2 2 1
2/2 1 0
1/2 0 1
Multiplica
tion
Result Integer
0.8125*2 1.625 1
0.625*2 1.25 1
0.25*2 0.5 0
0.5*2 1.0 1

22. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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• Example –
Decimal to Octal – divide by 8
(35.45)D = (?)O
(35.45)D = (43.34631)O
Division Quotient Remainder
35/8 4 3
4/8 0 4
Multiplica
tion
Result Integer
0.45*8 3.6 3
0.6*8 4.8 4
0.8*8 6.4 6
0.4*8 3.2 3
0.2*8 1.6 1

23. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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• Example –
Decimal to hexadecimal – divide by 16
(22)D = (?)H
(22)D = (16)H
Division Quotient Remainder
22/16 1 6
1/16 0 1

24. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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Conversion: Base-n Decimal.
• n can be 2, 8, 16 or any other value.
• Number in base-n number system to an equivalent
decimal number.
• Each digit of the number in base-n system is multiplied
with the positional weights.

25. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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• Example –
Binary to Decimal – multiply with positional weight
(1101.1)B = (?)D
Decimal No. = 1*23 + 1*22 + 0*21 + 1*20 + 1*2-1
= 1*8 + 1*4 + 0 + 1 + 0.5
= 8 + 4 +0 +1 + 0.5
= (13.5)D

26. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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• Example –
Octal to Decimal – multiply with positional weight
(540.25)O = (?)D
Decimal No. = 5*82 + 4*81 + 0*80 + 2*8-1 + 5*8-2
= 5*64 + 4*8 + 0 + 2*0.125
+5*0.0156
= 320 + 32 + 0 + 0.25 + 0.078125
= (352.32813)D

27. CONVERSION OF NUMBER (ONE NUMBER SYSTEM TO OTHER)
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• Example –
Hex to Decimal – multiply with positional weight
(5A.C)H = (?)D
Decimal No. = 5*161 + 10*160 + 12*16-1
= 5*16 + 10*1 + 12*0.0625
= 80 + 10 + 0.75
= (90.75)D

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