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

Understanding Number Systems in Computer Science

In Class 11 NCERT Computer Science, number systems are crucial for representing data. The main systems are:

• Binary (Base-2): Uses digits 0 and 1. Fundamental to computers.
• Octal (Base-8): Uses digits 0 to 7. Groups binary bits in 3s.
• Decimal (Base-10): Uses digits 0 to 9. Common in daily life.
• Hexadecimal (Base-16): Uses digits 0-9 and letters A-F. Groups binary bits in 4s.

Each system has a positional value for digits. For example, in decimal, the number 345 means $3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0$. Similarly, in binary, $1011_2$ equals $1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 11_{10}$. Understanding these systems helps in converting numbers efficiently.

Hexadecimal Number System Explained

Hexadecimal is a base-16 number system using sixteen symbols: 0-9 and A-F. Here, A to F represent decimal values 10 to 15.

Why use hexadecimal?

• Compact representation of binary numbers.
• Each hex digit corresponds to exactly 4 binary digits (bits).
• Simplifies reading and writing long binary sequences.

Hexadecimal to Binary Conversion: Group binary digits in 4s, then convert each group to its hex equivalent.

Example: Convert binary $1100000011110001_2$ to hexadecimal.

Group bits: 1100 0000 1111 0001

Convert each group:

• 1100 = C
• 0000 = 0
• 1111 = F
• 0001 = 1

So, $1100000011110001_2 = C0F1_{16}$.

Applications of Hexadecimal in Computing

Hexadecimal is widely used in computer science for:

• Memory Addressing: Memory locations are easier to read and write in hex than binary.
• Color Coding in Web Design: Colors in RGB format use 24-bit color codes represented in hex.

Example: The color red in RGB is (255, 0, 0) in decimal.

• Decimal to Hex:
• 255 = FF
• 0 = 00

So, red is represented as #FF0000 in hexadecimal.

This notation simplifies coding and digital graphics.

Encoding Schemes for Data Representation

Encoding schemes convert data into binary for computer processing. Important encoding schemes include:

• ASCII (American Standard Code for Information Interchange): Uses 7 or 8 bits to represent characters. ASCII-8 can represent 256 characters.
• ISCII (Indian Script Code for Information Interchange): Designed for Indian scripts.

Each character is assigned a unique binary code. For example, the letter 'A' in ASCII is $01000001_2$ (decimal 65).

Example: To encode the word "CAT" in ASCII:

This binary data is used internally by computers for text processing.

Conversions Between Number Systems

Class 11 NCERT emphasizes converting numbers between binary, octal, decimal, and hexadecimal.

Key Methods:

• Binary to Octal: Group bits in 3s from right to left.
• Binary to Hexadecimal: Group bits in 4s from right to left.
• Hexadecimal to Decimal: Use positional notation.

Positional Notation Formula: For a hexadecimal number $H_nH_{n-1}...H_1H_0$, decimal value is:

$$\sum_{i=0}^{n} H_i \times 16^i$$

where $H_i$ is the decimal value of the hex digit.

Example: Convert $1C3_{16}$ to decimal.

• $1 = 1 \times 16^2 = 1 \times 256 = 256$
• $C = 12 \times 16^1 = 12 \times 16 = 192$
• $3 = 3 \times 16^0 = 3 \times 1 = 3$

Sum = 256 + 192 + 3 = 451

So, $1C3_{16} = 451_{10}$.