Octal Number System

The Octal Number System is a base-8 number system that uses only the digits 0 to 7. It is used in computing and digital electronics to make long binary numbers shorter and easier to work with. Learning the octal system helps us convert numbers between decimal, binary, octal, and hexadecimal systems. Octal is also used in real life, such as computer file permissions, microprocessors, and digital clocks. This guide will explain the octal number system, show how to convert numbers, and give examples and practice questions to help you understand it better.

 

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 a positional numeral system based on base 8, which means it uses only eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. Each digit in an octal number has a positional value, determined by powers of 8, similar to how the decimal system uses powers of 10. The rightmost digit represents 8⁰, the next digit to the left represents 8¹, then 8², 8³, and so on.

 

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  

The Binary Number System uses base 2 (only digits 0 and 1), while the Octal Number System uses base 8 (digits 0 to 7). Since 8 = 2³, every three binary digits (bits) can be directly grouped into one octal digit.

This makes conversion between binary and octal very simple compared to converting through decimal.

Steps to Convert Binary to Octal

1. Write down the binary number.

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

   • If the last group has less than 3 digits, add leading zeros.

3. Convert each group of 3 into its octal equivalent.

4. Combine all octal digits to get the final number.

Octal Number System Example:  

Example 1: Convert 101101₂ to octal

Solution:
 Write down the binary number.
101101₂

Group the binary digits into sets of 3, starting from the right side.
101101 → 101 101

Convert each group of 3 bits into decimal (0–7).
101₂ = 5₈
101₂ = 5₈

Combine the digits.
(101101)₂ = (55)₈

Final Answer:
101101₂ = 55₈

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 1: Convert 123₈ to hexadecimal

Solution :

Octal number: 123₈

Convert each octal digit to binary:

1 → 001

2 → 010

3 → 011

Combined binary: 001 010 011 → 001010011₂

Group into 4 bits: 0001 0100 11 → add 0 → 0001 0100 1100

Convert each group:

0001 = 1

0100 = 4

1100 = C

Final 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 the 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

To convert an octal number to decimal, multiply each digit by 8 raised to the power of its position, starting from 0 on the right.

Solutions:

Octal number: 63₈

Identify the positions:

3 is at the 8⁰ place

6 is at the 8¹ place

Multiply each digit by its positional value:

6 × 8¹ = 6 × 8 = 48

3 × 8⁰ = 3 × 1 = 3

Add the results: 48 + 3 = 51

Answer: 63₈ = 51₁₀

 

Problem 2: Convert 110 in decimal to octal

Use repeated division by 8 and write down the remainders.

Solutions:

Decimal number: 110

Divide by 8: 110 ÷ 8 = 13 remainder 6

Divide the quotient by 8: 13 ÷ 8 = 1 remainder 5

Divide the new quotient by 8: 1 ÷ 8 = 0 remainder 1

Write the remainders in reverse order: 1 5 6

Answer: 110₁₀ = 156₈

 

Problem 3: Convert 100110 in binary to octal

Solutions:

Since 8 = 2³, group binary digits in sets of 3 from right to left.

Binary number: 100110₂

Group digits in 3s: 100 110

Convert each group to octal:

100₂ = 4₈

110₂ = 6₈

Combine the octal digits: 4 6

Answer: 100110₂ = 46₈

 

Problem 4: Convert 110101 (binary) to octal

Binary number: 110101₂

Group digits in 3s from right: 110 101

Solutions:

Convert each group to octal:

110₂ = 6₈

101₂ = 5₈

Combine the octal digits: 6 5

Answer: 110101₂ = 65₈

 

Problem 5: Convert 56 (octal) to hexadecimal

Convert octal → binary → hexadecimal.

Solutions:

Convert each octal digit to 3-bit binary:

5₈ = 101₂

6₈ = 110₂

Combine binary digits: 101110₂

Group binary digits into 4-bit sets for hexadecimal (add leading zeros if needed): 0010 1110

Convert each 4-bit group to hexadecimal:

0010₂ = 2₁₆

1110₂ = E₁₆

Answer: 56₈ = 2E₁₆

 

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.

 

Frequently Asked Questions on Octal Number System

1. How do you write 8 in octal?

Answer: 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?

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

3. Why is 8 in octal 10?

Answer: 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?

Answer: 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?

Answer: 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.