Data Representation : Number System and Encoding

 Data Representation in Computers

Binary Representation of Data:

In order to work with data, the data must be represented inside the computer. Digital computers represent data by means of an easily identified symbol called a digit. 

Numbering Systems:

Each number system has a base also called a Radix. A decimal number system is a system of base 10; binary is a system of base 2; octal is a system of base 8; and hexadecimal is a system of base 16.

Number System              Base                                    Symbols used

Binary                             2                                             0,1

Octal                               8                                     0,1,2,3,4,5,6,7

Decimal                           10                                     0,1,2,3,4,5,6,7,8,9

Hexadecimal                    16                        0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F                                                           where A = 10; B = 11; C = 12; D = 13;                                                                                    E = 14; F = 15

Converting a number from one Base to another

Decimal to Binary

Method to convert a Decimal number into its Binary equivalent

1. Divide the decimal number by 2.

2. Take the remainder and record it on the side.

3. Divide the quotient by 2.

4. REPEAT UNTIL the decimal number cannot be divided further.

5. Record the remainders in reverse order and you get the resultant binary number.

Example

Convert the Decimal number 125 into its Binary equivalent.

125 / 2 = 62     1

62 / 2 = 31       0

31 / 2 = 15       1

15 / 2 = 7         1

7 / 2 = 3           1

3 / 2 = 1           1

1 / 2 = 0           1

Answer: (1111101)2

2. Converting Decimal fraction to Binary

Example:

Convert (0.75)10 to binary

• Multiply the given fraction by 2. 
• Keep the integer in the product as it is and multiply the new fraction in the product by 2. 
• Continue the process till the required number of decimal places or till you get zero in the fraction part. 
• Record the integers in the products from top to bottom.

Given fraction 0.75

Multiply 0.75 by 2  =1.50

Multiply 0.50 by 2 = 1.00

Reading the integers from top to bottom 0.75 in decimal number system is 0.11 in binary number system.

Binary to Decimal

Method to convert Binary to Decimal:

1. Start at the rightmost bit.

2. Take that bit and multiply by 2n where n is the current position beginning at 0       and increasing by 1 each time. This represents a power of two.

3. Sum each terms of product until all bits have been used.

Example

Convert the Binary number 101011 to its Decimal equivalent.

1*2⁵ + 0*2⁴ + 1*2³ + 0*2² + 1*2¹ + 1*2⁰

32 + 0 + 8 + 0 +2 + 1 = (43)10

Example

Convert the Binary number 1001 to its Decimal equivalent.

1*2³ + 0*2² + 0*2¹ + 1*2⁰

8 + 0 + 0 + 1 = (9)10

Binary fraction to decimal

Example

Convert (11011.101)2 to decimal

1 1 0 1 1 .1 0 1

= (1 x 2⁴) + (1 x 2³) + (0 x 2²) + (1 x 2¹) + (1 x 2⁰) + (1 x 2^-1) + (0 x 2^-2) + (1 x 2^-3)

= 16+8+0+2+1+0.5+0+0.125

= (27.625)10

Decimal to Octal

The method to convert a decimal number into its octal equivalent:

1. Divide the decimal number by 8.

2. Take the remainder and record it on the side.

3. Divide the quotient by 8.

4. REPEAT UNTIL the decimal number cannot be divided further.

5. Record the remainders in reverse order and you get the resultant binary

Example

Convert the Decimal number 125 into its Octal equivalent.

125 / 8 = 15              5

15/ 8 = 1                   7

1/8 =0                      1

Answer: (175)8

Converting Decimal fraction to Octal

Example

Convert (0.75)10 to Octal

• Multiply the given fraction by 8. Keep the integer in the product as it is and multiply the new fraction in the product by 8. 
• Continue the process and read the integers in the products from top to bottom.

Given fraction              0.75

Multiply 0.75 by 8        6.00

Reading the integers from top to bottom 0.75 in decimal number system is 0.6 in octal number system.

Octal to Decimal

Method to convert Octal to Decimal:

1. Start at the rightmost bit.

2 . Take that bit and multiply by 8n where n is the current position beginning at 0       and increasing by 1 each time. This represents the power of 8.

3. Sum each of the product terms until all bits have been used.

Example

Convert the Octal number 321 to its Decimal equivalent.

3*8² + 2*8¹ + 1*8⁰

192+16+ 1 = (209)10

Octal fraction to decimal

Example

Convert (23.25)8 to decimal

               8¹ 8⁰ . 8^-1 8^-2

= (2 x 8¹) + (3 x 8⁰) + (2 x 8^-1) + (5 x 8^-2)

= 16 + 3  +  0.25  +   0.07812

= (19.32812)10

Decimal to Hexadecimal

Method to convert a Decimal number into its Hexadecimal equivalent:

1. Divide the decimal number by 16.

2. Take the remainder and record it on the side.

3. REPEAT UNTIL the decimal number cannot be divided further.

4.Record the remainders in reverse order and you get the equivalent             hexadecimal number.

Example

Convert the Decimal number 300 into its hexadecimal equivalent.

300 / 16 = 18      12-(C)

18 / 16 = 1          2

1 / 16 = 0            1

Answer: (12C)16

Converting Decimal fraction to Hexadecimal

Example

Convert (0.75)10 to hexadecimal

• Multiply the given fraction by 16. Keep the integer in the product as it is and multiply the new fraction in the product by 16. 
• Continue the process and read the integers in the products from top to bottom.

Given fraction                 0.75

Multiply 0.75 by 16         12.00 - C

Reading the integers from top to bottom 0.75 in decimal number system is 0C in Hexadecimal number system.

Hexadecimal to Decimal

Method to convert Hexadecimal to Decimal:

1. Start at the rightmost bit.

2. Take that bit and multiply by 16n where n is the current position beginning at        0 and increasing by 1 each time. This represents a power of 16.

3. Sum each terms of product until all bits have been used.

Example

Convert the Hexadecimal number AB to its Decimal equivalent.

=A * 16¹    +  B * 16⁰

=10 * 16¹ + 11 * 16⁰

=160+11 = (171)10

Hexadecimal fraction to decimal

Example

Convert (1E.8C)16 to decimal

           16¹     16⁰ .     16^-1     16^-2

          1               E             8              C

= (1 x 16¹)+ (14 x 16⁰)+ (8 x 16^-1)+ (12 x 16^-2)

= 16        +    14        +    0.5    +    0.04688

= (30.54688)10

Binary to Hexadecimal

The hexadecimal number system uses the digits 0 to 9 and A, B, C, D, E, F.

Method to convert a Binary number to its Hexadecimal equivalent is:

We take a binary number in groups of 4 and use the appropriate hexadecimal digit in it’s place. We begin at the rightmost 4 bits. If we are not able to form a group of four, insert 0s to the left until we get all groups of 4 bits each. Write the hexadecimal equivalent of each group. Repeat the steps until all groups have been converted.

Example

Convert the binary number 1000101 to its Hexadecimal equivalent.

0100 0101       (Note that we needed to insert a 0 to the left of 100.)

4         5

Answer: (45)16

In case of a fractional binary number form groups of four bits on each side of decimal point. Then replace each group by its corresponding hexadecimal number.

Example

Convert (11100.1010)2 to hexadecimal equivalent.

0001     1100     .     1010

1          C           .     A

Answer : (1C.A)16

Hexadecimal to Binary

Method to convert a Hexadecimal number to its Binary equivalent is:

Convert each digit of Hexadecimal Number to it’s binary equivalent and write them in 4 bits. Then, combine each 4 bit binary number and that is the resulting answer.

Example

Convert the Hexadecimal number (10AF)16 to its Binary equivalent.

1           0         A         F

0001 | 0000 | 1010 | 1111

Answer: (0001000010101111)2

Example

Convert the Hexadecimal number (A2F)16 to its Binary equivalent.

    A       2         F

1010 | 0010 | 1111

Answer: (1010 0010 1111)2

Binary to Octal and Octal to Binary

To convert Binary to Octal, as the octal system is a power of two (23), we can take the bits into groups of 3 and represent each group as an octal digit. The steps are the same for the binary to hexadecimal conversions except we are dealing with the octal base now.

To convert from octal to binary, we simply represent each octal digit in it’s three bit binary form.

Example

Convert the Octal number (742)8 to its Binary equivalent.

    7 |    4 |   2

111 | 100 | 010

Answer: (111100010)2

Hexadecimal to Octal and Octal to Hexadecimal

To convert Hexadecimal to Octal, Convert each digit of Hexadecimal Number to it’s binary equivalent and write them in 4 bits. Then, combine each 3 bit binary number and that is converted into octal.

Example

Convert the Hexadecimal number (A42)16 to its Octal equivalent.

    A  |     4   |     2

1010 | 0100 | 0010

101 | 001 | 000 | 010

Answer: (5102)8

To convert Octal to hexadecimal, convert each digit of Octal Number to it’s binary equivalent and write them in 3 bits. Then, combine each 4 bit binary number and that is converted into hexadecimal.

Example

Convert the Octal number (562)8 to its hexadecimal equivalent.

5     | 6     | 2

101 | 110 | 010

0001 | 0111 | 0010

Answer: (172)16

The following table summarizes the number representation in decimal, binary, octal and hexadecimal number system:

Decimal         Binary         Octal         Hexadecimal

0                     0000            0                  0

1                     0001             1                 1

2                     0010             2                 2

3                     0011             3                 3

4                     0100            4                  4

5                     0101            5                  5

6                     0110            6                  6

7                     0111             7                 7

8                     1000           10                 8

9                     1001           11                 9

10                   1010           12                 A

11                   1011           13                 B

12                   1100           14                 C

13                   1101           15                 D

14                   1110           16                 E

15                   1111           17                 F

BINARY ADDITION

Consider following during Binary Addition-

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10 , important point to notice here is that 1 goes for carry which is to be added with next place bit.

or

See the figure given below:

Character Representation

Computers are designed to work internally with numbers. In order to handle characters, we need to choose a number for each character.  

The complete set of characters / symbols are called alphanumeric codes. The complete alphanumeric code typically includes −

• 26 upper case letters (A-Z)
• 26 lower case letters (a-z)
• 10 digits (0-9)
• 7 punctuation marks
• 20 to 40 special characters

Following are some forms of character set

• ASCII
• ISCII
• UNICODE

ASCII Code: American Standard Code for Information Interchange

• The ability of a computer system to understand signals or letters depends on its character set.
• Character set has its standards known as character set code like - ASCII, ISCII, UNICODE etc.
• ASCII ( American Standard Code for Information Interchange) most of the micro computers, mini computers and some mainframe computers uses this code.
• ASCII code has two versions - ASCII – 7 and ASCII – 8.
• ASCII – 7 code use 7 bits for one signal or character. By this, 27 = 128 , different characters can be used.
• ASCII – 8 code use 8 bits for one signal or character. By this, 28 = 256 , different characters can be used.

 

ISCII: Indian Script Code for Information Interchange

ISCII stands for Indian Script Code for Information Interchange for Indian languages. 

In order to facilitate the use of Indian languages on Computers, a common  standard for coding Indian scripts called ISCII was developed in India during mid 1980s.

It is an 8-bit code representation for Indian languages which means it can      represent 28=256 characters. It retains all 128 ASCII codes and uses rest of  the codes 128 for additional Indian language character set. Additional codes have been assigned in the upper region (160– 255) for the ‘aksharas’ of the  language. These codes are used for ten Indian scripts - Devnagari, Punjabi, Gujarati, Oriya, Bengali, Assami, Telugu, Kannada, Malayalam and Tamil. 

Unicode: (Universal Code)

• This is Universal Character Set which represents a signal or a character in a group of 32 bit.
• It has the capability to include signals and characters from all scripts of all languages of world.
• Before the development of Unicode, various encoding system were in use.
• Problems related to language on internet has been resolved by the use of Unicode.
• 
  
• The Unicode Standard is the universal character encoding standard for written characters and text. Each character or symbol is assigned a unique numeric value,  largely within the framework of ASCII. Earlier, each script had its own encoding system, which could conflict with each other.
• The purpose of Unicode is to − 

            provide a unique number for every character, 

            no matter what the platform, 

            no matter what the program, 

            no matter what the language.

• Unicode 13.0 adds 5930 characters, for a total of 143,859 characters. These additions include 4 new scripts, for a total of 154 scripts, as well as 55 new emoji characters.

Unicode ( Devnagari- Hindi Script)

Unicode characters are represented in one of three encoding forms: 

a 32-bit form (UTF32), 

a 16-bit form (UTF-16), and 

an 8-bit form (UTF-8). 

The 8-bit, byte-oriented form, UTF-8, has been designed for ease of use with existing ASCII-based systems. 

Unicode Transformation Format 8-bit

• It was originally designed by Ken Thompson and Rob Pike in 1992.  
• UTF-8 encodes each Unicode character as a variable number of 1 to 4 octets,   where the number of octets depend on the integer value assigned to the Unicode character. 
• At the time of its introduction, ASCII was the most popular character encoding. In ASCII all     letters, digits and symbols were assigned a number. And this number, being fixed to 8 bits, it could only represent a maximum of 255 characters, and it wasn’t enough.
• UTF-8 was designed to be backward compatible with ASCII. Because ASCII was much older (1963) and widespread, and moving to UTF-8 was challenging.
• The first 128 characters of UTF-8 map exactly to ASCII. 
• Why 128? Because ASCII uses 7-bit encoding, which allows up to 128 combinations. Why 7 bits? Later when ASCII was conceived, 7 bit systems were popular as well. Being 100% compatible with ASCII makes UTF-8 also very efficient. 

Unicode Transformation Format 32-bit

• UTF-32 always uses 4 bytes, without optimizing for space usage, and  as such it wastes a lot of bandwidth.
• This constrain makes it faster to operate on because you have less to  check, as you can assume 4 bytes for all characters.
• It’s not as common as UTF-8 and UTF-16, but it has its applications.     The key use of UTF-32 is in internal APIs where the data is single code points / glyphs, rather than strings of characters. Used on Unix systems sometimes for storage of information.
• It is a fixed length encoding scheme that uses exactly 4 bytes to represent all Unicode code points. 

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BOOLEAN ALGEBRA

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BOOLEAN ALGEBRA

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