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Introduction to Number System
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Number System
When we type some letters or words, the computer translates them in binary numbers as computers can understand only binary numbers.
Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands and so on.
A value of each digit in a number can be determined using
The digit Symbol value (is the digit value 0 to 9) The position of the digit in the number Increasing Power of the base (i.e. 10) occupying
successive positions moving to the left
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Example
Decimal number (592):
Number Symbol Value
Position from the right end
Positional Value
Decimal Equivalent
5 9 22
9
5
0
1
2
100
101
102
2*100 = 2
9*101 = 90
5*102 = 500 592eITnotes.com
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Binary number system
Uses two digits, 0 and 1. Also called base 2 number system
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(110011)2 = (51)10
Number Symbol Value
Position from the right end
Positional Value
Decimal Equivalent
1 1 0 0 1 1
1
1
0
0
1
1
0
1
2
3
4
5
20
21
22
23
24
25
1*0 = 1
1*2 = 2
0*4 = 0
0*8 = 0
1*16= 16
1*32= 32 51
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Cont…
A Decimal number can converted into binary number by the following methods:
Double-Dabble Method Direct Method
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Double-Dabble Method
Divide the number by 2 Write the dividend under the number
. This become the new number Write the remainder at the right in
column Repeat these three steps until a ‘0’ is
produced as a new number Output (bottom to top).
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Convert decimal 17 into binary number
Step Remainder
1 Divide 17 by 2 2 17 8
1
2 Divide 8 by 2 2 8 4
0
3 Divide 4 by 2 2 4 2
0
4 Divide 2 by 2 2 2 1
0
5 Divide 1 by 2 2 1 0
1
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Direct Method
Write the positional values of the binary number
…. 26 25 24 23 22 21 20
…. 64 32 16 8 4 2 1 Now compare the decimal number with position
value listed above. The decimal number lies between 32 and 64. Now place 1 at position 32.
64 32 16 8 4 2 1
1 Subtract the positional value to the decimal
number i.e ( 45-32=13)
45
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Cont..
64 32 16 8 4 2 1
1 45-32 =13
1 1 13-8=5
1 1 1 5-4=1 1 1 1 1 1-1=0Place 0 at the rest of position value 0 1 0 1 1 0 1 (45)10=(101101)2
45
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Decimal number to fractional Binary number Multiply the decimal fraction by 2 Write the integer part in a column The fraction part become a new
fraction Repeat step 1 to 3 until the fractional
part become zero. Once the required number of digits
(say 4) have been obtained , we can stop.
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Example
Decimal number is (0.625)
Ans: (0.625)10= (0.101)2
Fractional decimal number
Operation
Product Fractional part of product
Integer part of product
0.625 Multiply by 2
1.250 .250 1
0.250 -do- 0.500 .500 0
0.500 -do- 1.000 0 1
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Questions
Convert decimal 89 into equivalent binary number by using Double-Dabble Method
(89)10= (1011001)2
Convert decimal 89 into equivalent binary number by using Direct Method
(89)10= (1011001)2
Convert decimal 0.8125 into fractional binary number
(0.8125)10 = (0.1101)2
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Convert Binary to Decimal Direct Method Double Dabble Method
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Direct Method
Binary Number
Positional value
operation
1 1 1 0 0 1 0 1 1
0
1
0
0
1
1
1
1*20
0*21
1*22
0*23
0*24
1*25
1*26
1*27
1
0
4
0
0
32
64
128 = 229eITnotes.com
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Double Dabble Method
Multiply left most digit by 2 add to the next digit and so on.
1 1 0 1
2+ 1 0 1 3 0 1
6+ 0 1
6 1
12+ 1 13
(1101)2= (13)10
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Example
Convert Binary number 10111011 to decimal
(10111011)2 = (187)10
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Convert fractional Binary number to Fractional Decimal number Write out the binary number as (-)ve
power of two. The various digits positions after binary points are 1,2,3,4…..and so on.
Convert each power of two into its decimal equivalent
Add these to give the decimal number
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Example
. 1 0 1 1
1*2-1 0*2-2 1*2-3 1*2-4
0.5 + 0 + 0.125 + 0.0625
= 0.6875
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Questions
Convert the fractional binary number to decimal number
(0.1101) ans= 0.8125 (0.1011) ans= 0.6875
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Octal number notation
Octal is base 8 counting system having digit values 0 through 7
The octal system groups three binary bits together into one digit symbol.
Octal Binary
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111eITnotes.com
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Convert binary number into octal Divide the given binary number into
group of three bits (from right to left) Replace each group by its octal
equivalent Examples: 11001 101010001110
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Convert decimal to octal
Divide the number by 8 Write the dividend under the
number. This become the new number
Write the remainder at the right in a column
Repeat steps 1 to 3 until a ‘0’ is produced as a new number
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Question
Convert decimal 17 to octal number Ans= (17)10 = (21)8
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Convert octal to decimal number
Write out the octal digits as power of 8
Convert each power of 8 into its decimal equivalent term
Add these terms to produce the required decimal number
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Example
(721)8= (465)10
Ques: Convert the octal 131 to its equivalent decimal number
ans: 89
7 2 1
=7*82
=448
465
2*81
16
1*80
1
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Hexadecimal
Hexadecimal number system is a base 16 counting system
It uses 16 Symbols: 0 to 9 and the capital letter A,B…F.
Each Hexadecimal is equivalent to a group of 4 binary bits.
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Hexadecimal
Binary Hexadecimal
Binary
0 0000 8 1000
1 0001 9 1001
2 0010 A 1010
3 0011 B 1011
4 0100 C 1100
5 0101 D 1101
6 0110 E 1110
7 0111 F 1111
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Convert binary to Hexadecimal
Divide the given binary number into groups of 4 bits each(from right to left).
Replace each group by its hexadecimal Equivalent.
Questions:1.Convert (101111100001)2 into its
hexadecimal. Ans: (BEI)16.
2. Convert (10101111.0010111)2 into its hexadecimal.
Ans: (AF.2E)16eITnotes.com
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Convert Decimal to Hexadecimal Divide the number by 16. Write the dividend under the number.
This become the new number. Write the remainder at the right in a
column. Repeat steps 1 to 3 until a ‘0’ is
produced as a new number.Question: Convert the Decimal 87 to
hexadecimal number. (87)10= (57)16
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Convert hexadecimal to Decimal Write out the Hexadecimal digits as
power of 16. Convert each power of 16 into its
decimal equivalent term. Add these terms to produce the
required decimal number.Question: (A2D)16=(2605)10
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Data Representation
We known that computer work with binary numbers and therefore the numbers, letters, and other symbols have to be converted into their binary equivalents.
However, this is not enough in the sense that still we do not know how to store this binary information so that it become suitable for computer processing.
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Cont..
The Representation of a positive integer number is quite straight forward but we are interested to represent positive as well as negative numbers.
For a Positive number , the sign bit set to 0 and for negative number the sign bit is set to 1.
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Integer Representation
An integer can be represented by fixed point representation
The left most bit is considered as sign bit.
The magnitude of the number can be represented in following three ways:
1. Signed magnitude representation.
2. Signed 1’s complement representation.
3. Signed 2’s complement representation.
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Signed Magnitude
In this representation , if n bit of storage is available then 1 bit is reserved for sign and n-1 bits for the magnitude.
The Disadvantage of this representation is that during addition and Subtraction, the sign bit has to be considered along with the magnitude.
Sign
bitmagnitude
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Signed 1’s Compliment
The 1’s Compliment of a binary integer can be obtained by simply replacing the digit 0 by 1 and digit 1 by 0
Example: 00001100 is 11100111
0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0
(+0)1
0
(-0)10
0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1
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Signed 2’s Compliment The 2’s Compliment of a binary number is
obtained by adding 1 to 1’s Compliment. Example: (+12)10= 1100
11110011 1’s Compliment 1 11110100 2,s Compliment
Therefore, Positive integer 2’s compliment is the negative integer
0 0 0 0 1 1 0 0
1 1 1 1 0 0 1 1 1’s
1 1 1 1 0 1 0 0(-
12)10
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Question
Express the following in signed magnitude form, 1’s Compliment, 2’s Compliment:
(35)10 = 100011
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Floating point representation We can represent a floating point binary number in
the following form: ±M * 2±e
Where M : is the mantissa or significant e : is the exponent Example: 101.11 10111 * 2-2
101.11 * 20
10.111 *21
1.0111 *22
.10111 * 23
.010111 * 24
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Cont.. .10111 * 23
M e The Mantissa part of the number is suitably
shifted (left or right) to obtain a non zero digit at a most significant position. The activity is known as normalization.
In a 16 bit representation, let us assume that 10 bits are reserved for mantissa and 6 for exponent.
Sign Sign
Mantissa exponent0 1 0 1 1 1 0 0 0 0 0 0 0 0 1 1
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Question
Represent floating point binary number in 16 bit representation (1110.001)
The normalization number is = .1110001 * 24
16 bit representation: Sign Sign
0 111000100 0 00100 M e
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