How to Convert Decimal to Octal: Step-by-Step Guide
The concept of Octal Number System plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Octal Number System?
The Octal Number System is a base-8 number system. It uses only eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. Each digit in an octal number represents a power of 8, making this system especially important in computer science and digital electronics where binary values are converted for easier reading and grouping. Students often encounter octal numbers when learning about number systems, fundamental in mathematics and programming.
Key Formula for Octal Number System
Here’s the standard formula for converting an octal number to decimal:
\( (d_nd_{n-1}...d_0)_8 = d_n \times 8^n + d_{n-1} \times 8^{n-1} + \cdots + d_0 \times 8^0 \)
Cross-Disciplinary Usage
Octal Number System is not only useful in Maths but also plays an important role in Computer Science, Digital Logic, and daily logical reasoning. Students preparing for exams like JEE or those studying coding will see its relevance in various topics, especially while converting between binary, octal, decimal, and hexadecimal systems.
Step-by-Step Illustration
Let’s see how to convert a decimal number to octal and vice versa.
- Convert Decimal 125 to Octal:
1. Divide 125 by 8. Quotient: 15, Remainder: 5
2. Divide 15 by 8. Quotient: 1, Remainder: 7
3. Divide 1 by 8. Quotient: 0, Remainder: 1
4. Read the remainders from last to first: (175)8
- Convert Octal (145)8 to Decimal:
1. Write as: (1 × 8²) + (4 × 8¹) + (5 × 8⁰)
2. Calculate: (1 × 64) + (4 × 8) + (5 × 1) = 64 + 32 + 5 = 101
| Octal Digit | Binary Equivalent |
|---|---|
| 0 | 000 |
| 1 | 001 |
| 2 | 010 |
| 3 | 011 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for converting Binary to Octal:
- Group the binary digits in sets of three, starting from the right. If needed, add zeros to make a complete group.
- Convert each group to its octal equivalent using the Octal-Binary table above.
- Combine results to form the octal number.
Example: Convert (101101)2 to Octal
Groups: 101, 101 → 5, 5 → (55)8
Tricks like this help students during timed exams and are used in classes at Vedantu to build speed and accuracy.
Try These Yourself
- Write the octal equivalents of decimal numbers 8, 16, 32, and 64.
- Is 9 a valid octal digit? Why or why not?
- Convert (101110)2 to octal.
- Expand (231)8 in decimal form.
Frequent Errors and Misunderstandings
- Including digits 8 or 9 in octal numbers (which is not allowed).
- Forgetting to read remainders in reverse order during conversion.
- Incorrectly grouping binary digits for octal conversion.
Relation to Other Concepts
The idea of Octal Number System connects closely with topics such as Decimal Number System, Hexadecimal Number System, and Number System Conversion. Understanding octal makes learning digital electronics and computer programming easier.
Classroom Tip
A quick way to know an octal digit is valid: If every digit is between 0 and 7, it’s octal. Teachers at Vedantu often use color-coded charts and finger counting (up to 7) as memory aids in live sessions.
We explored Octal Number System — from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept.
Useful Internal Links
- Number System for the basics and comparison among all systems.
- Binary Number System to learn binary-octal relationships.
- Hexadecimal Number System for conversion and digital applications.
- Number System Conversion for stepwise guides and shortcuts.
FAQs on Octal Number System: Definition, Conversion & Examples
1. What is an octal number system?
The octal number system is a base-8 numeral system that uses eight distinct digits: 0, 1, 2, 3, 4, 5, 6, 7. Each position in an octal number represents a power of 8. For example, in the octal number 237, each digit has the following value:
- $2 \times 8^2$
- $3 \times 8^1$
- $7 \times 8^0$
2. Is 786 an octal number?
No, 786 is not an octal number because the octal system only includes digits from 0 to 7. The digit 8 or 9 cannot appear in an octal number. To be a valid octal number, all digits must range from 0–7. Vedantu’s expert teachers can help clarify such number system concepts with interactive examples in their online classes.
3. What is an example of octal?
An example of an octal number is 345 (octal). To understand its decimal equivalent:
- $3 \times 8^2 = 192$
- $4 \times 8^1 = 32$
- $5 \times 8^0 = 5$
4. How do you find the octal number?
To find the octal representation of a decimal number, repeatedly divide the number by 8 and record the remainders. Arrange the remainders in reverse order to get the octal number. For example, to convert decimal 65 to octal:
- 65 ÷ 8 = 8, remainder 1
- 8 ÷ 8 = 1, remainder 0
- 1 ÷ 8 = 0, remainder 1
5. How does the octal number system differ from decimal and binary systems?
Octal uses base 8 (digits 0–7), decimal uses base 10 (digits 0–9), and binary uses base 2 (digits 0 and 1). Each system has a unique way of representing values, and conversion between these systems is a common exercise in mathematics and computer science. Vedantu’s curriculum covers number system differences and conversions, preparing students for competitive exams and coding basics.
6. What are the practical uses of the octal number system?
Octal numbers are primarily used in computer systems, particularly in programming and digital circuits, because three binary digits correspond to one octal digit. This makes reading and writing large binary numbers simpler. Vedantu teaches students how octal is used in computer science, including memory addressing and coding.
7. How do you convert a binary number to octal?
To convert binary to octal, group the binary digits into sets of three (starting from the right) and replace each group with their octal equivalent. For example, binary 1101011:
- Group: 1 101 011
- Octal: 1 5 3
8. Why is octal used in digital electronics?
Octal notation simplifies binary representations in digital electronics since each octal digit translates directly to three binary bits. This can make error-checking and circuit design more manageable. Vedantu’s learning resources explain the application of octal in logic circuits and microprocessors.
9. Can octal numbers have fractional parts?
Yes, octal numbers can have fractional parts. For example, $12.5_8$ represents $1 \times 8^1 + 2 \times 8^0 + 5 \times 8^{-1} = 8 + 2 + 0.625 = 10.625$ in decimal. Vedantu’s math experts teach both whole numbers and fractional representations in octal to strengthen foundational knowledge.
