Tutorial-1
Dr. Manideepa Mukherjee
Department of CS & IS
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Number Systems
● Digits
● Positional Notation
● Conversion
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Digits
Digit is a single numerical symbol used to represent numbers. Digits vary
depending on the base of the number system.
○
○
○
○
Decimal (Base 10): 0,1, 2, 3, 4,…..9
Binary (Base 2): 0, 1
Octal (Base8): 0, 1, 2, 3….,7
Hexadecimal (Base 16): 0, 1, 2, 3….…, 9, A, B, C, …F
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Positional Notation
● Larger numbers can be constructed using positional notation.
● The position to the left of the radix (number base) point is always the units
position in any number system. For example the position left to the decimal
point is always 100, the position to the left of the binary point is 20, Octal point
is 80.
● Units position has the power 0 and weight 1. For Example. In decimal system
100, Binary system 20, Octal system 80, Hexadecimal System 160.
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Positional Notation
● The positions to the left of the units positions have positive powers. For
example 101, 21, 81, 161.
● The positions to the right of the radix point have negative powers. Example
10-1, 2-1, 8-1, 16-1.
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Binary Numbers
Base 2, Digits 0,1.
Conversion:
● Binary to decimal
● Decimal to binary
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Conversion from Binary to Decimal
Example : (101.1)2
Power
22
21
20
2-1
Weight
4
2
1
0.5
Number
1
0
1.
1
Numeric Value
4
0
1
0.5
Answer = (5.5)10
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Conversion from Decimal
To convert a decimal whole number to another number system divide the decimal
number by the radix and save the remainders as the significant digits of the result.
Example: Convert (68)10 to binary.
68/2 = 34 remainder is 0
34/2 = 17 remainder is 0
17/2 = 8 remainder is 1
8/2 = 4 remainder is 0
4/2 = 2 remainder is 0
2/2 = 1 remainder is 0
1/2 = 0 remainder is 1
Answer = (1000100)2
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Conversion from Decimal
To convert a decimal fraction to another number system fraction, multiplication by the
radix is carried out repetitively and the integer part of the result is saved and placed
after the radix point. This process can be stopped anytime when the desired accuracy
is achieved.
Example: Convert (0.68)10 to binary.
0.68 * 2 = 1.36 integer part is 1 Take the fractional part and continue the process
0.36 * 2 = 0.72 integer part is 0
0.72 * 2 = 1.44 integer part is 1
0.44 * 2 = 0.88 integer part is 0
Answer = (.1010)2
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Octal Number System
Digits – 0, 1, 2, …., 7
Base – 8
Conversion
● Octal to Decimal
● Decimal to Octal
● Binary to Octal and Octal to binary
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Octal to Decimal
Convert (3754)8 to decimal
=(3x83 ) + (7x82 ) + (5x81 ) + (4x80 )
= 1536 + 448 + 40 + 4
= (2028)10
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Decimal to Octal
(3479)10 to Octal
3479/8 = 434 rem = 7
434/8 = 54 rem = 2
54/8 = 6 rem = 6
6/8 = 0 rem = 6
Answer = (6627)8
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Binary to Octal and Octal to Binary
Group 3 bits starting from LSB and progressing to MSB
Example: Convert (11100111)2 to Octal
= 011 100 111
= (3 4 7)8
Example: Convert (537)8 to Binary
= (101011111)2
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Hexadecimal Number system
Base or radix 16 number system.
Numbers are 0,1,2…..8,9, A, B, C, D, E, F.
1 hex digit is equivalent to 4 bits.
Conversion
● Hexadecimal to Decimal
● Decimal to Hexadecimal
● Binary to Hexadecimal and Hexadecimal to Binary
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Conversion Hexadecimal to Decimal
(F4C)16 to decimal
(F x 162) + (4 x 161) + (C x 160)
= (15 x 256) + (4 x 16) + (12 x 1)
= (3916)10
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Conversion Decimal to Hexadecimal
(3479)10 to Hexadecimal
3479/16 = 217 rem = 7
217/16 = 13 rem = 9
13/16 = 0 rem = 13 (D)
Answer = (D97)16
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Binary to Hexadecimal and Hexadecimal to Binary
Grouping bits in the binary numbers into groups of four bits starting from LSB to
MSB.
Example: (1 1000 1010 1000 0111)2
= (0001 1000 1010 1000 0111)2
= (1 8 A 8 7)16
Example: (1BA2)16
= (0001101110100010)2
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ASCII
American Standard Code for Information Interchange (ASCII) data represent
alphanumeric characters in the memory of a computer system.
Developed by ANSI (American National Standards Institute)
Represents
○
○
○
Latin alphabet, Arabic numerals, standard punctuation characters
Plus small set of accents and other European special characters
7-bit code: 128 characters
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ASCII Reference Table
1110100
Copyright 2010 John Wiley & Sons, Inc.
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Conversion
Example2: Convert (23.5)6 to Decimal.
Power
61
60
6-1
Weight
6
1
0.167
Number
2
3.
5
Numeric
Value
12
3
0.835
(23.5)6 = 15.835
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Practice Problems
Convert to Decimal
1. (1011.11)2
2. (5AB.C)16
3. (753.23)8
Convert from Decimal to Binary, Octal and Hexadecimal
1. 788.38
2. 0.4325
3. 9365.23
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Conversion Problems
Convert each of the following binary numbers to decimal, octal, and hexadecimal
formats.
(111011101)2
(10101010111)2
(111100000)2
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THANK YOU
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