Encoding Schemes and Number System: Class 11 NCERT Computer Science Guide

Introduction to Encoding Schemes and Number Systems

In Class 11 NCERT Computer Science, the chapter on Encoding Schemes and Number System introduces how computers represent and process data using different number systems. Humans typically use the decimal system (base 10), but computers operate using the binary system (base 2). To simplify binary data, octal (base 8) and hexadecimal (base 16) systems are often used. Encoding schemes define how characters and data are represented in binary form, essential for computer communication and storage.

Converting Decimal Numbers to Binary, Octal, and Hexadecimal

Converting decimal numbers to other number systems is crucial for understanding computer data encoding.

Method:

• Divide the decimal number by the base of the target system (2 for binary, 8 for octal, 16 for hexadecimal).
• Record the remainder each time.
• Write the remainders in reverse order to get the converted number.

Example: Convert decimal 122 to binary:

1. 122 ÷ 2 = 61 remainder 0 2. 61 ÷ 2 = 30 remainder 1 3. 30 ÷ 2 = 15 remainder 0 4. 15 ÷ 2 = 7 remainder 1 5. 7 ÷ 2 = 3 remainder 1 6. 3 ÷ 2 = 1 remainder 1 7. 1 ÷ 2 = 0 remainder 1

Reading remainders backwards: 1111010

So, $122_{10} = 1111010_2$.

Converting Binary, Octal, and Hexadecimal Numbers to Decimal

To convert numbers from binary, octal, or hexadecimal to decimal, use the positional notation method:

• Write the position of each digit from right to left, starting at 0.
• Calculate the positional value by raising the base to the position number.
• Multiply each digit by its positional value.
• Sum all products to get the decimal equivalent.

Example: Convert binary (1101)₂ to decimal:

Positions: 3 2 1 0

Digits: 1 1 0 1

Calculations:

$1 \times 2^3 = 8$

$1 \times 2^2 = 4$

$0 \times 2^1 = 0$

$1 \times 2^0 = 1$

Sum = 8 + 4 + 0 + 1 = 13

Thus, $(1101)_2 = 13_{10}$.

Conversions Between Binary, Octal, and Hexadecimal Systems

Binary numbers can be easily converted to octal or hexadecimal by grouping bits:

• Binary to Octal: Group bits in sets of 3 from right to left.
• Binary to Hexadecimal: Group bits in sets of 4 from right to left.

Each group is replaced by the corresponding octal or hexadecimal digit.

Example: Convert binary 1101011 to octal:

Group bits in 3s: 1 101 011 (add leading zero to make full groups: 001 101 011)

Groups: 001 = 1, 101 = 5, 011 = 3

Octal equivalent: 153

Converting Numbers with Fractional Parts Between Systems

Fractional parts require a different approach:

• Multiply the fractional part by the base of the target system.
• Extract the integer part of the result as the next digit.
• Repeat with the new fractional part.

Example: Convert decimal 0.25 to binary:

1. $0.25 \times 2 = 0.5$ → integer part 0 2. $0.5 \times 2 = 1.0$ → integer part 1

Result: 0.01 in binary

For binary fractions to octal or hexadecimal, group bits in 3s or 4s for integer and fractional parts separately.

Example: Convert binary 100101.101 to decimal:

Integer part 100101 = 37 decimal

Fractional part .101 = $1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} = 0.5 + 0 + 0.125 = 0.625$

Total = 37.625 decimal.

Understanding Encoding Schemes in Computer Science

Encoding schemes define how characters and symbols are represented in binary for computers. Common schemes include:

• ASCII: Uses 7 or 8 bits to represent English characters, digits, and symbols.
• ISCII: Indian Script Code for Information Interchange, used for Indian languages.

Encoding schemes ensure data consistency and interoperability between devices and software. For example, ASCII-8 can represent up to 256 characters.

Knowing encoding schemes helps understand how text and data are stored and transmitted digitally.