data representation class11 notes

 Chapter-13
data representation class11 notes

data representation class11 notes

Introduction :

Digital techniques have found their way into innumerable areas of technology, but the area of automatic digital computers is by far the most notable and most extensive. As you know, a computer is a system of hardware that perform arithmetic operations, manipulates data, and make decisions.

DIGITAL NUMBER SYSTEMS

In digital representation, various number systems are used. The most common number used are decimal, binary, octal, and hexadecimal systems.

Decimal Number System

Base – 10

Minimum – 0      { digits }

Maximum – 9      { digits }

Ex —–> (125)10

Binary Number System

Base – 2

Minimum – 0        { digits }

Maximum – 1        { digits }

Ex —–> (100)2

Octal Number System

Base – 8

Minimum – 0     { digits }

Maximum – 7     { digits }

Ex —–> (200)8

Hexadecimal Number System

Base – 16

Minimum – 0                   { digits }

Maximum – 15(F)            { digits }

data representation class11 notes

Number Conversions

The binary number system is the most important one in digital systems as it is very easy to implement in circuitry. The decimal system is important because it is universally used to represent quantities outside a digital system.

In a digital system, tree or four of these number system may be in use at the same time, so that understanding of the system operation requires the ability to convert from one number system to another. This section discusses how to perform these conversions. So , let us discuss them one by one.

Decimal-to-Binary  Conversion

Q1. Convert (43)10 to binary using repeated division method.

solution.

data representation class11 notes

Reading the remainders from bottom to the top,

(43)10 = (101011)

Q2. Convert (200)10 to binary using repeated division method.

solution.

data representation class11 notes

Reading the remainders from the bottom to top, the result is : (200)10 = (11001000)2

Binary-to-Decimal Conversion

data representation class11 notes

 

Decimal-to-Octal Conversion

Adecimal integer can be converted to octal by using the same repeated-division method that we used in the decimal-to-binary conversion, but with a division factor of 8 instead of 2. An example is shown below :

data representation class11 notes

Reading up, (266)10 = (412)8

Note that the remainder becomes the least significant digit (LSD) of the octal number, and the last remainder becomes the most significant digit (MSD).

Octal-to-decimal conversion

example:

(372)8 = 3 X (8²) + 7 X (8¹) + 2 X (8 power 0)

= 3 X 64 + 7 X 8 + 2 X 1 = (250)10

Another example :

24.68 = 2 X (8¹) + 4 X (8power 0) + 6 X (8౼¹) = (20.75)10

Octal -to-Binary Conversion

The primary advantage of the octal number system is the ease with which which conversion can be made between binary and octal numbers. The conversion from octal to binary is performed by converting each octal digit to its 3-bit binary equivalent. The eight possible digits are converted as indicated in table 13.3.

data representation class11 notes

data representation class11 notes

Binary -to-octal conversion

data representation class11 notes

 

Decimal-to Hex conversion

data representation class11 notes

 
 

Hex-to-Decimal Conversion

A Hex number can be converted to its decimal equivalent by using the fact that each hex digit position has weight that is a power of 16. The LSD has a weight of 160 = 1, the next higher digit has a weight of 161 = 16, the next higher digit has a weight of 162 = 256, and so on. The conversion process is demonstrated in example below :

(356)16 = 3 X 16² + 5 X 16¹ + 16 X 16 power 0

= 768 + 80 + 6

= (854)10

(56.08)16 = 5 X 16¹ + 6 X 16power 0 + 0 X 16-¹ + 8 X 16-²

= 80 + 6 + 0 + 8/256 = 86 + 0.03125

= (86.03125)10

Binary -to-Hex Conversion

Binary numbers can be easily converted to hexadecimal by grouping in groups of four starting at the binary point.

Q1. convert (1010111010)2 to hexadecimal.

solution          group in fours                     10,1011,1010

convert each numbers         2     B       A

thus, the solution is 2 BA

Q2. Convert (10101110.010111)2 to hexadecimal.

solution.     groups in four (inserting zeros before MSB or in the end, wherever required)

1010, 1110 .  0101, 1100

A         E    .      5        C

10101110.0101112 = (AE.5C)16

Hex to Binary Conversion

 
data representation class11 notes
data representation class11 notes
 
 
 
 
data representation class11 notes
 
 
 
 
Hey guys are you looking for best Data representation notes in easy language
 

data representation class11 notes