Octal Number System

The Octal Number System is a base-8 numeral system that uses only 8 digits: 0 to 7. It is widely used in digital electronics and computing as a simpler way to represent long binary numbers. By understanding how to convert octal to decimal, binary, or hexadecimal, and where it’s used in real-life systems, you can gain a strong grasp of number systems in mathematics and computer science.

 

Table of Contents

• Definition
• Octal Number System Chart
• Decimal to Octal Conversion
• Octal to Decimal Conversion
• Binary to Octal Conversion
• Octal to Hexadecimal Conversion
• Octal Multiplication Table
• Uses of Octal Number System
• Problems and Solutions
• Practice Questions
• Conclusion
• FAQs on Octal Number System

 

Definition  

The octal number system is one of the positional numeral systems used in mathematics and digital electronics. It is a base-8 system, meaning it uses only eight digits: 0, 1, 2, 3, 4, 5, 6, and 7.  

 

Define Octal Number System:  

To define the octal number system clearly, it represents numbers using base 8, where the positional values are powers of 8. The rightmost digit represents 8^0, the next represents 8^1, then 8^2, and so on.  

 

What is Octal Number System?  

When someone asks, "What is the octal number system?" the simple answer is that it is a numerical system based on 8 digits (0 to 7) used in computing to simplify binary coding.  

 

Explain Octal Number System:  

To explain the octal number system in detail, it is important to note that each digit has a positional value like in the decimal system, but with powers of 8 instead of 10. 

For example, the octal number (325)₈ represents:  

3×8² + 2×8¹ + 5×8⁰ = 192 + 16 + 5 = 213  

 

Octal Number System Chart  

This chart helps relate octal values with decimal and binary numbers, which is essential in conversions.

 

 

This chart is frequently used in binary-to-octal and octal-to-binary conversions.  

 

Decimal to Octal Conversion  

To convert a number from decimal (base 10) to octal (base 8), use repeated division by 8 and collect the remainders.  

Steps:  

1. Divide the decimal number by 8.  

2. Write down the remainder.  

3. Divide the quotient by 8 again.  

4. Repeat until the quotient is zero.  

5. Read the remainders in reverse order.  

 

Octal Number System Example:  

Example: Convert 156 to octal  

156 ÷ 8 = 19 remainder 4  

19 ÷ 8 = 2 remainder 3  

2 ÷ 8 = 0 remainder 2  

Now reverse the remainders: 2 3 4  

So, (156)₁₀ = (234)₈  

This is a typical example that shows decimal to octal conversion.  

 

Octal to Decimal Conversion  

To convert a number from the octal number system to decimal, multiply each digit by 8 raised to its positional power and add the results.  

Formula:  

If the octal number is abc, then:  

(abc)₈ = a×8² + b×8¹ + c×8⁰  

 

Octal Number System Example:  

Convert 745 to decimal.  

7 × 8² = 448  

4 × 8¹ = 32  

5 × 8⁰ = 5  

Total = 448 + 32 + 5 = 485  

So, (745)₈ = (485)₁₀  

This helps explain how to convert from octal to decimal.  

 

Binary to Octal Conversion  

Since 8 is a power of 2 (8 = 2³), converting from binary to octal is straightforward.  

Steps:  

1. Group binary digits in sets of 3 from right to left.  

2. Convert each group of 3 into an octal digit.  

 

Octal Number System Example:  

Convert binary number 101101 to octal.  

Group: 101 101  

Convert: 101 (binary) = 5, 101 (binary) = 5  

Answer: (101101)₂ = (55)₈  

This is another commonly used example in the octal number system.  

 

Octal to Hexadecimal Conversion  

To convert a number from the octal number system (base-8) to the hexadecimal number system (base-16), follow these simple steps:

Step

1. Convert the octal number to a binary number.
Each octal digit is converted to a 3-bit binary equivalent.

2. Group the binary digits into 4-bit groups starting from the right (add leading 0s if necessary).

3. Convert each 4-bit group to its hexadecimal equivalent.

 

Example

Convert 123₈ to hexadecimal

Step 1: Octal to Binary

• 1 → 001

• 2 → 010

• 3 → 011

123₈ = 001010011₂

Group: 0001 0100 11 → add one zero to the left: 0001 0100 1100

• 0001 → 1

• 0100 → 4

• 1100 → C
  
  

Answer: 123₈ = 14C₁₆

 

Octal Multiplication Table

This table is useful in learning multiplication in the octal number system.

All values are written in octal.

 

Uses of Octal Number System  

Common uses of the octal number system include:  

• Simplifying binary representation: Every 3 bits of binary equals one digit in octal, which helps reduce long binary sequences.  

• File permissions in Unix/Linux: Permissions like 755 or 644 are written in octal format.  

• Microprocessors: Instructions in some old microprocessors were easier to write in octal.  

• Digital clocks and timers: Octal values simplify segment displays.  

• Data encoding and bit representation: It is efficient in systems that use data sizes of 3-bit groupings.  

These practical uses show the relevance of the octal number system today.  

 

Problems

Problem 1: Convert 63 in octal to decimal
6×8 = 48, 3×1 = 3 → Total = 51

 

Problem 2: Convert 110 in decimal to octal
110 ÷ 8 = 13 remainder 6
13 ÷ 8 = 1 remainder 5
1 ÷ 8 = 0 remainder 1
Answer: 156 in octal

 

Problem 3: Convert 100110 in binary to octal

Group: 100 110 → 4 6 → 

Answer: 46 in octal

 

Problem 4: Convert 110101 (binary) to octal.
Group: 110 101 → 6 5
Answer: 65 (octal)

 

Problem 5: Convert 56 (octal) to hexadecimal.
Octal → Binary: 5 = 101, 6 = 110 → 101110
Binary → Hex: 0010 1110 → 2E
Answer: 2E (hexadecimal)

 

Practice Questions

1. Convert 41 (octal) to decimal

2. Convert 98 (decimal) to octal

3. Convert 111000 (binary) to octal

4. Convert 21 (octal) to hexadecimal

5. What is the octal of decimal 64

6. Define octal number system

7. Explain octal number system with example

8. State 3 uses of octal number system
   
   

Conclusion  

The octal number system is significant in computing and electronics. To understand what the octal number system is, one must know its digits, base, and how to convert it to and from other number systems. By defining the octal number system and practicing with octal number system examples, you develop a solid understanding. Its uses appear in programming, binary compression, and digital systems.

 

Related Links

Convert Octal to Binary - Learn how to convert octal numbers into binary format with clear steps and examples for better understanding.

Binary to Octal Conversion - Understand the process of converting binary numbers to octal numbers, with rules, shortcuts, and examples.

Binary number system -  Learn how the binary number system works using only 0s and 1s. Explore its role in computers with simple explanations and fun examples!

 

Frequently Asked Questions on Octal Number System

1. How do you write 8 in octal?

Ans: The number 8 is written as 10 in the octal number system, because octal uses base-8 and does not include the digit 8.

2. Is the octal number system consists of 8 digit 0 1 2 3 4 5 6 and 7 True or false?

Ans: True. The octal number system consists of only 8 digits: 0 to 7.

3. Why is 8 in octal 10?

Ans: In the octal number system, the number after 7 is written as 10 (just like in decimal after 9, we write 10). So, 8 in decimal becomes 10 in octal.

4. What are the 4 types of number system?

Ans: The 4 types of number systems are:

• Binary (Base 2)

• Octal (Base 8)

• Decimal (Base 10)

• Hexadecimal (Base 16)
  
  

5. What is an octal number system?

Ans: The octal number system is a number system that uses base-8 and includes digits from 0 to 7. It is often used in computing and digital systems as a shorthand for binary numbers.

Practice these conversions and examples to understand the topic better. For more learning support, visit Orchids The International School.