Number System Class 11 CBSE Computer Science: Decimal, Binary, Octal, Hexadecimal and Conversion to Decimal
Class 11 · Computer Science
Number System (CBSE Class 11 Computer Science)
A computer understands and processes data in the form of numbers. Different number systems are used in computer science to represent and process information efficiently. Humans commonly use the Decimal Number System, whereas computers internally use the Binary Number System. Other important number systems include Octal and Hexadecimal.
Learning Objectives
- Understand the concept of a Number System.
- Learn about Decimal, Binary, Octal and Hexadecimal Number Systems.
- Understand Base (Radix) and Positional Notation.
- Convert Binary, Octal and Hexadecimal numbers into Decimal.
What is a Number System?
A Number System is a method of representing numbers using a set of symbols or digits according to specific rules.
Base (Radix)
The Base or Radix of a number system is the total number of unique digits used to represent numbers.
Example
- Binary → Base 2
- Octal → Base 8
- Decimal → Base 10
- Hexadecimal → Base 16
Types of Number Systems
| Number System | Base | Digits Used |
|---|---|---|
| Binary | 2 | 0, 1 |
| Octal | 8 | 0–7 |
| Decimal | 10 | 0–9 |
| Hexadecimal | 16 | 0–9, A–F |
Decimal Number System
The Decimal Number System is the most commonly used number system in daily life. It has a base of 10 and uses ten digits from 0 to 9.
Example
458₁₀
Here, the subscript 10 indicates that the number belongs to the Decimal Number System.
Binary Number System
The Binary Number System has a base of 2 and uses only two digits: 0 and 1.
Since electronic circuits have two stable states (ON and OFF), computers internally use binary numbers.
Example
101101₂
Octal Number System
The Octal Number System has a base of 8 and uses digits from 0 to 7.
Example
725₈
Hexadecimal Number System
The Hexadecimal Number System has a base of 16 and uses sixteen symbols.
| Decimal | Hexadecimal |
|---|---|
| 10 | A |
| 11 | B |
| 12 | C |
| 13 | D |
| 14 | E |
| 15 | F |
Example
2AF₁₆
Positional Number System
In a positional number system, the value of each digit depends on:
- The digit itself.
- Its position.
- The base of the number system.
Example
453₁₀
= (4 × 10²) + (5 × 10¹) + (3 × 10⁰)
= 400 + 50 + 3
= 453
Place Value in Different Number Systems
| Position | Binary | Octal | Decimal | Hexadecimal |
|---|---|---|---|---|
| 0 | 2⁰ | 8⁰ | 10⁰ | 16⁰ |
| 1 | 2¹ | 8¹ | 10¹ | 16¹ |
| 2 | 2² | 8² | 10² | 16² |
| 3 | 2³ | 8³ | 10³ | 16³ |
Conversion of Binary to Decimal
Multiply each binary digit by its corresponding power of 2 and add the results.
Example
101101₂
= (1 × 2⁵)
+ (0 × 2⁴)
+ (1 × 2³)
+ (1 × 2²)
+ (0 × 2¹)
+ (1 × 2⁰)
= 32 + 0 + 8 + 4 + 0 + 1
= 45₁₀
Conversion of Octal to Decimal
Multiply each octal digit by its corresponding power of 8 and add the results.
Example
725₈
= (7 × 8²)
+ (2 × 8¹)
+ (5 × 8⁰)
= 448 + 16 + 5
= 469₁₀
Conversion of Hexadecimal to Decimal
Multiply each hexadecimal digit by its corresponding power of 16 and add the results.
Example
2AF₁₆
= (2 × 16²)
+ (10 × 16¹)
+ (15 × 16⁰)
= 512 + 160 + 15
= 687₁₀
Comparison of Number Systems
| Number System | Base | Largest Digit | Common Use |
|---|---|---|---|
| Binary | 2 | 1 | Computer Processing |
| Octal | 8 | 7 | Short Representation of Binary |
| Decimal | 10 | 9 | Daily Calculations |
| Hexadecimal | 16 | F | Memory Addressing and Programming |
Real-Life Example
Think of different clocks:
- A normal counting system uses ten digits (Decimal).
- A light switch has only two states (ON/OFF), similar to Binary.
- Hexadecimal is commonly used by programmers to represent colours (for example, #FF0000 for red) and memory addresses.
Common Errors
- Using incorrect powers while converting numbers.
- Confusing the base of Binary, Octal and Hexadecimal.
- Forgetting that A–F represent decimal values 10–15.
- Writing hexadecimal digits beyond F.
- Ignoring the positional value of digits.
Exam Tips
- Remember the bases: Binary (2), Octal (8), Decimal (10), Hexadecimal (16).
- Always multiply digits by the powers of the base while converting to Decimal.
- Memorize hexadecimal values A to F.
- Write the base as a subscript whenever possible.
- Practice conversion problems regularly.
Frequently Asked Questions (FAQs)
1. What is a Number System?
A Number System is a method of representing numbers using a fixed set of digits and a base.
2. Why do computers use the Binary Number System?
Because electronic circuits have two stable states: ON and OFF, represented by 1 and 0.
3. What is the base of the Hexadecimal Number System?
The base of the Hexadecimal Number System is 16.
4. What does the hexadecimal digit 'A' represent?
The hexadecimal digit A represents the decimal value 10.
5. What is positional notation?
It is a method in which the value of a digit depends on its position and the base of the number system.
Summary
- A Number System represents numbers using a fixed set of digits and a base.
- The four commonly used number systems are Binary, Octal, Decimal and Hexadecimal.
- Binary uses base 2, Octal uses base 8, Decimal uses base 10 and Hexadecimal uses base 16.
- In positional notation, the value of a digit depends on its position and the base.
- Conversion to Decimal is performed by multiplying each digit with the corresponding power of its base.
- Understanding number systems is essential for computer science and digital electronics.