Data Presentation
The set of symbols to express
quantities as the basis of counting is referred to be a number system. There
has been various number systems under human use for the purpose of counting and
measurement. For instance decimal number system, octal number system, binary
number system and hexadecimal number system have been most popular number
systems which human beings have been utilizing for the purpose of counting and
measurement.
Though a computer can process only binary numbers i.e. only 0 and 1, we give a brief account of each of the number system mentioned above. Inter conversion of these number systems would also be discussed in this chapter.
Though a computer can process only binary numbers i.e. only 0 and 1, we give a brief account of each of the number system mentioned above. Inter conversion of these number systems would also be discussed in this chapter.
Decimal Number
System
This is the most commonly used number
system and which is very natural to man, hence very easy to use and remember.
It consists of ten digits, i.e. from 0 to 9. As the total number of digits used
in a number system is called its base or radix, hence base of radix of decimal
number system is said to be ten.
Binary Number
System
The base or radix of binary number is
two, implying that there are only two digits 0 and 1 used in binary number
system. This number system has become most popular system because of its use by
the computers for processing data.
Octal Number
System
This number system is consist of
eight i.e. 0,1,2,3,4,5,6 and 7. Hence the base or the radix of octal number
system is eight. Actually this number is a short hand version of binary
numbers, Hence one octal represents three binary digits.
Hexadecimal
Number System
Hexadecimal numbers are extensively
used in microcomputers. As they are much shorter than binary numbers, hence
easy to write and remember. They are also a short hand version of binary
numbers. The hexadecimal also called Hex is equivalent to four binary digits.
This number system consists of 0,1,2,3,4,5,6,7,8,9, A,B,C,D,E and F. The base
of this number system is 16. The symbol A……..F stand respectively for 10…..15.
Binary to
Decimal Conversion
All the number system may be easily
converted into other number systems. In order to convert binary numbers into
decimal numbers we multiply each bit of a particular binary number with (n-1)
power of 2.
Add all the resultant multiplied
bits.
Example No. 1
To convert the binary number 10010 to
a decimal number we proceed as follows:
100102 = (1 x 24) + (1 x 23) + (1 x
22) + (1 x 21) + (1 x 20)
= (1 x 16) + (0 x
+ (0 x 4) + (1
x 2) + (0 x 1)
= 16 + 0 + 2 + 0
= 18
It should be kept in the mind that
any number raised to the power 0 equals 1.
Example No. 2
Similarly in order to convert the
binary number 110111 to a decimal number we may proceed as follows:
1101112 = (1 x 25) + (1 x 24) + (1 x
23) + (1 x 22) + (1 x 21) + (1 x 20)
= (1 x 32) + (1 x 16) + (1 x
+ (1 x 4) + (1
x 2) + (1 x 1)
= 32 + 16 + 0 + 0 + 4 + 2 + 1
= 56
Example No. 3
If there is a period (.) in the
binary number e.g. (1011.10). Then the conversion of such binary number can be
illustrated as follows:
(10112) = (1 x 23) + (1 x 22) + (1 x
21) + (1 x 20)
= (1 x
+ (0 x 4) + (1
x 2) + (1 x 1)
= (1 x
= 8 + 0 + 2 + 1
= 11
AND
AND
(.102) = (1 x 2-4) + (1 x 2-2)
= (1 x 1/2) + (1 x 1/4)
= 1/2 + 0 = 0.5
Hence (1011.102) = 11.5
Decimal to Binary
Conversion
A decimal number can easily be
converted to a binary number. For this purpose we take following steps.
1. Divide the decimal number by 2 and
note down the remainder.
2. Take the quotient and again divide by 2, then note down the remainder again.
3. Repeat the 2nd step unit you get 1 as the quotient.
2. Take the quotient and again divide by 2, then note down the remainder again.
3. Repeat the 2nd step unit you get 1 as the quotient.
4. The remainders in the reverse
order would be our required binary
number.
Example No. 4
For example in order to convert
decimal number 15 to binary number. we may proceed as follows.
15/2 = 7 and the remainder is 1
7/2 = 3 and the remainder is 1
3/2 = 1 and the remainder is 1
1/2 = 0 and the remainder is 1
Hence the required binary number is
1111
Converting
Decimal Fractions to Binary Numbers
The most simple method of converting
fractional decimal numbers to binary number is repeated multiplication by two.
In this method numbers to the right of the decimal point are repeatedly
multiplied by 2 till we get a whole number. The product in fractions is
assigned as 0 bit while in whole number it is assigned as 1 bit.
The process of converting a
fractional decimal number to binary number would be explained by the following
example.
Example No. 5
Let us suppose the we want to convert
0.125 in a binary numbers then proceed as follow.
0.125 x 2 = 0.250 it carries 0
0.250 x 2 = 0.500 it carries 0
0.500 x 2 = 1.00 it carries 1
Hence we can conclude that (0.125)10
= (0.001)2
Addition to
Binary Numbers
In order to add binary number below
given table should be kept in mind
0 + 0 = 0
0 + 0 = 0
0 + 1 + 1
1 = 0 = 1
1 + 1 = 0 with a carry of 1
The following example will illustrate
the addition of binary numbers.
Example No.6
Add 1001 and 1010
1001 + 1010 = 11011
Subtraction of
Binary Numbers
In order to subtract a binary number
from another binary number you must
keep the following table in your
mind.
0 – 0 = 0
1 – 0 = 1
1 – 1 = 0
0 – 1 = 1 with a barrow of 1
The process of binary subtraction an
be illustrated with the help of example given below
Example No. 7
For subtracting binary number 101
from 1001 we proceed as follows
1001 – 100.1 = 1.11
1001 – 100.1 = 1.11
Octal to
Decimal Conversion
For converting an octal number to a
decimal number we use the same sort of polynomial as was used in the binary
case. However in octal to decimal conversion the base or radix is 8 instead of
2.
The process of converting an octal
number into a decimal number would be clarified with the help of following
example.
Example No. 9
The Octal number 2134 can be
converted into decimal number by proceeding as follows
(2134)8 = (2 x 83) + (2 x 82) + (2 x
81) + (2 x 80)
= (2 x 512) + (1 x 64) + (3 x
+ (4 x 1)
= 1024 + 64 + 24 + 4
= 1116
Hence (2134)8 = (1116)10
Example No.
10
The octal number 1.123 can be
converted to decimal number as follows
(1.123)8 = (1 x 80) + (1 x 8-1) + (1 x 8-2) + (1 x 8-3)
(1.123)8 = (1 x 80) + (1 x 8-1) + (1 x 8-2) + (1 x 8-3)
= (1 x 1) + (1 x 1/8) + (1 x 1/64) +
(3 x 1/512)
= 1 + 1/8 + 2/64 + 3/512
= 1 83/512
Decimal to
Octal Conversion
For converting decimal number to
octal number. we apply a method of repeatedly dividing decimal by 8 and use each
reminder as a digit in the octal number. The process of converting decimal to
octal number may be illustrated by the following example.
Example No.
11
To convert the decimal number 429 to
octal number we process as follows
429/8 = 53 Reminder is 5
429/8 = 53 Reminder is 5
53/8 = 6 Reminder is 5
6/8 = 0 Reminder is 6
Hence (429)8 = (655)10
Octal to
Binary Conversion
As we have mentioned earlier that
octal number system is a short hand version of binary numbers and one octal
represents three binary digits as shown by the following table
Equivalent Octal and Binary Number
Octal 0
1
2
3
4
5
6
7
2
3
4
5
6
7
Binary
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
In order to convert an octal number
to a binary number we just replace each octal digit with the appropriate three
bits. This produce an be explained with the help of following example
Example No.
12
(25)8 = (010101)2
Because (2)8 = (010)2 and (5)8 =
(101)2 as shown by the above table.
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