1. Introduction to Number System
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2. 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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3. Example Decimal number (592): Number Symbol Value
Position from the right end
Positional Value
Decimal Equivalent 2 9 5 1
100
101
102
2*100 = 2
9*101 = 90
5*102 = 500
592
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4. Binary number system Uses two digits, 0 and 1.
Also called base 2 number system
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5. (110011)2 = (51)10 Number Symbol Value Position from the right end
Positional Value
Decimal Equivalent 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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6. Cont…
A Decimal number can converted into binary number by the following methods:
Double-Dabble Method
Direct Method
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7. 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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8. Convert decimal 17 into binary number
Step
Remainder 1
Divide 17 by 2 17 8 2
Divide 8 by 2 4 3
Divide 4 by 2
Divide 2 by 2 5
Divide 1 by 2
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9. Direct Method Write the positional values of the binary number… …
Now compare the decimal number with position value listed above. The decimal number lies between 32 and 64. Now place 1 at position 32. 1
Subtract the positional value to the decimal number i.e ( 45-32=13) 45
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10. Cont.. 45
=13 =5 =1 =0
Place 0 at the rest of position value
(45)10=(101101)2
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11. 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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12. 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
1.000
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13. Questions
Convert decimal 89 into equivalent binary number by using Double-Dabble Method
(89)10= ( )2
Convert decimal 89 into equivalent binary number by using Direct Method
Convert decimal into fractional binary number
(0.8125)10 = (0.1101)2
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14. Convert Binary to Decimal
Direct Method
Double Dabble Method
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15. Direct Method Binary Number Positional value operation 1 1 1 0 0 1 0 1
1*20
0*21
1*22
0*23
0*24
1*25
1*26
1*27 4 32 64
128 = 229
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16. Double Dabble Method
Multiply left most digit by 2 add to the next digit and so on. 13
(1101)2= (13)10
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17. Example Convert Binary number 10111011 to decimal
( )2 = (187)10
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18. 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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19. Example
1* * * *2-4 =
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20. Questions Convert the fractional binary number to decimal number
(0.1101)
ans=
(0.1011)
ans=
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21. 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
000 1
001 2
010 3
011 4
100 5
101 6
110 7
111
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22. 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
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23. 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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24. Question Convert decimal 17 to octal number Ans= (17)10 = (21)8
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25. 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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26. Example 7 2 1
=7*82
=448
465
2*81 16
1*80
(721)8= (465)10 Ques: Convert the octal 131 to its equivalent decimal number ans: 89
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27. 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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28. Hexadecimal
Binary
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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29. 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:
Convert ( )2 into its hexadecimal.
Ans: (BEI)16.
2. Convert ( )2 into its hexadecimal.
Ans: (AF.2E)16
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30. 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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31. 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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32. 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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33. 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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34. 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:
Signed magnitude representation.
Signed 1’s complement representation.
Signed 2’s complement representation.
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35. 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 bit
magnitude
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36. (+0)10 1
(-0)10
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: is 1
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37. 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
’s Compliment 1
,s Compliment
Therefore, Positive integer 2’s compliment is the negative integer 1
1’s 1
(-12)10 1
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38. Question
Express the following in signed magnitude form, 1’s Compliment, 2’s Compliment:
(35)10 =
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39. 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:
10111 * 2-2
* 20
*21
*22
* 23
* 24
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40. Cont..
* 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 exponent 1
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41. Question
Represent floating point binary number in 16 bit representation ( )
The normalization number is = * 24
16 bit representation:
Sign Sign
M e
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