Binary, Octal and Hexadecimal Conversions Class 11 CBSE Computer Science: Binary ↔ Octal, Binary ↔ Hexadecimal and Octal ↔ Hexadecimal
Class 11 · Computer Science
Conversions Between Binary, Octal and Hexadecimal (CBSE Class 11 Computer Science)
Since Binary, Octal, and Hexadecimal number systems are closely related, conversions between them are much easier than converting through Decimal. These conversions use simple grouping techniques based on powers of 2.
Learning Objectives
- Convert Binary to Octal.
- Convert Octal to Binary.
- Convert Binary to Hexadecimal.
- Convert Hexadecimal to Binary.
- Convert Octal to Hexadecimal.
- Convert Hexadecimal to Octal.
Relationship Between Number Systems
| Number System | Base | Binary Group Size |
|---|---|---|
| Binary | 2 | 1 bit |
| Octal | 8 | 3 bits |
| Hexadecimal | 16 | 4 bits |
Binary to Octal Conversion
To convert Binary into Octal, group the binary digits into sets of 3 bits starting from the right. Add leading zeros if required.
Example
Convert 1101011₂ into Octal.
1101011₂
= 001 101 011
001 = 1
101 = 5
011 = 3
1101011₂ = 153₈
Octal to Binary Conversion
Replace each octal digit with its equivalent 3-bit binary number.
Binary Equivalents
| Octal | Binary |
|---|---|
| 0 | 000 |
| 1 | 001 |
| 2 | 010 |
| 3 | 011 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
Example
153₈
1 → 001
5 → 101
3 → 011
153₈ = 001101011₂
Binary to Hexadecimal Conversion
To convert Binary into Hexadecimal, group the binary digits into sets of 4 bits starting from the right. Add leading zeros if necessary.
Example
101101111₂
= 0001 0110 1111
0001 = 1
0110 = 6
1111 = F
101101111₂ = 16F₁₆
Hexadecimal to Binary Conversion
Replace each hexadecimal digit with its 4-bit binary equivalent.
Example
2AF₁₆
2 → 0010
A → 1010
F → 1111
2AF₁₆ = 001010101111₂
Hexadecimal to Binary Reference Table
| Hex | Binary | Hex | Binary |
|---|---|---|---|
| 0 | 0000 | 8 | 1000 |
| 1 | 0001 | 9 | 1001 |
| 2 | 0010 | A | 1010 |
| 3 | 0011 | B | 1011 |
| 4 | 0100 | C | 1100 |
| 5 | 0101 | D | 1101 |
| 6 | 0110 | E | 1110 |
| 7 | 0111 | F | 1111 |
Octal to Hexadecimal Conversion
There is no direct conversion rule. Convert the Octal number to Binary first, then convert the Binary number to Hexadecimal.
Example
725₈
↓
111010101₂
↓
1 1101 0101
↓
1D5₁₆
Hexadecimal to Octal Conversion
Convert the Hexadecimal number to Binary first, then convert the Binary number to Octal.
Example
3A₁₆
↓
00111010₂
↓
000 111 010
↓
072₈
= 72₈
Shortcut Rules
| Conversion | Shortcut |
|---|---|
| Binary → Octal | Group into 3 bits |
| Octal → Binary | Replace each digit with 3 bits |
| Binary → Hexadecimal | Group into 4 bits |
| Hexadecimal → Binary | Replace each digit with 4 bits |
| Octal ↔ Hexadecimal | Convert through Binary |
Comparison of Grouping Methods
| Target Number System | Group Size |
|---|---|
| Octal | 3 Bits |
| Hexadecimal | 4 Bits |
Real-Life Example
Imagine packing chocolates into boxes:
- For Octal, pack chocolates in boxes of 3.
- For Hexadecimal, pack chocolates in boxes of 4.
- If the first box is not full, add empty spaces (leading zeros) to complete the box.
Common Errors
- Grouping binary digits from the left instead of the right.
- Forgetting to add leading zeros.
- Using 4-bit groups for Octal conversion.
- Using 3-bit groups for Hexadecimal conversion.
- Incorrect hexadecimal values for A–F.
Exam Tips
- Remember: Octal uses 3-bit groups.
- Remember: Hexadecimal uses 4-bit groups.
- Always group binary digits from the right.
- Add leading zeros if the leftmost group is incomplete.
- Practice direct Binary ↔ Octal and Binary ↔ Hexadecimal conversions.
Frequently Asked Questions (FAQs)
1. Why are binary digits grouped in sets of 3 for Octal conversion?
Because 2³ = 8, which is the base of the Octal Number System.
2. Why are binary digits grouped in sets of 4 for Hexadecimal conversion?
Because 2⁴ = 16, which is the base of the Hexadecimal Number System.
3. Is there a direct method to convert Octal to Hexadecimal?
No. Convert Octal to Binary first, then Binary to Hexadecimal.
4. Why are leading zeros added during grouping?
Leading zeros complete the leftmost group without changing the value of the number.
5. Which direction should binary digits be grouped?
Binary digits should always be grouped from right to left.
Summary
- Binary to Octal conversion uses groups of 3 bits.
- Binary to Hexadecimal conversion uses groups of 4 bits.
- Octal and Hexadecimal conversions are performed through Binary.
- Leading zeros are added when necessary to complete the leftmost group.
- Shortcut grouping methods make conversions faster and easier.
- Regular practice helps improve speed and accuracy in number system conversions.