Octal to Binary Conversion Examples

Octal to Binary Conversion is an important topic in number system conversion used in computer science, programming, and digital electronics. It helps students understand how base-8 numbers can be easily converted into base-2 using a direct 3-bit mapping technique. This concept is widely used in memory addressing, digital systems, and low-level computing operations.

In this article, we will discuss octal to binary conversion examples and some examples of the fractional part as well. Let’s start

Understanding Octal to Binary Conversion

Octal to binary conversion is a quick and efficient method because each octal digit directly corresponds to a 3-bit binary number. This makes it easier to convert large numbers without complex calculations. The list of important concepts and steps is given below:

1. What is Octal Number System?

The octal number system is a base-8 number system that uses digits from 0 to 7. It is widely used in computing as a compact representation of binary numbers.

• Base: 8
• Digits: 0, 1, 2, 3, 4, 5, 6, 7
• Each position represents powers of 8

2. What is Binary Number System?

The binary number system is a base-2 number system used internally by computers. It uses only two digits, making it ideal for digital systems.

• Base: 2
• Digits: 0 and 1
• Each position represents powers of 2

3. Relationship Between Octal and Binary

Octal and binary are closely related because 8 is equal to 2³. This means each octal digit can be directly converted into a 3-bit binary equivalent.

• 1 octal digit = 3 binary bits
• No division or multiplication required
• The direct substitution method is used

Octal to Binary Conversion Methods

There are two main methods for converting octal to binary:

1. Octal to Binary Conversion using Standard Table

In this method, each octal digit is directly converted into its equivalent 3-bit binary form using a standard conversion table.

2. Octal to Binary Conversion using the Decimal Number System

In this method, the octal number is first converted into a decimal number, and then the decimal number is converted into binary.

 

Let’s explain both methods with examples

1. Octal to Binary Conversion Using Standard Table

The octal to binary conversion table helps students quickly map each octal digit to its binary equivalent. This table is essential for solving problems efficiently.
The list of octal digits and their binary equivalents is given below:

Steps for Octal to Binary Conversion

Octal to binary conversion is a simple process based on direct substitution, where each octal digit is converted into a 3-bit binary value. This method is widely used in computer science for fast and accurate conversion between number systems. The step-by-step process with a simple example is given below.

Step 1: Write the Given Octal Number

Start by writing the octal number clearly, ensuring each digit is separated if needed.

• Example: (25)₈

Step 2: Convert Each Octal Digit into 3-bit Binary

Replace each octal digit with its corresponding 3-bit binary equivalent using the standard table.

• 2₈ = 010₂
• 5₈ = 101₂

Step 3: Combine All Binary Groups

Join all the binary groups together to form the final binary number.

• 010₂ + 101₂ = 010101₂

Step 4: Remove Leading Zeros (If Required)

Remove unnecessary leading zeros to simplify the final binary result.

• 010101₂ = 10101₂

Examples are the best way to understand octal to binary conversion in a practical and exam-oriented manner. Below are solved examples with step-by-step explanations.
The list of solved examples is given below:

Example 1: Convert (25)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (25)₈ into its equivalent binary

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (25)₈

• The binary of 2₈ is 010₂
• The binary of 5₈ is 101₂

By joining all the binary groups together to form the final binary number. So, the final result is (25)₈ = 010101₂

Example 2: Convert (64)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (64)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (64)₈.

• The binary of 6₈ is 110₂
• The binary of 4₈ is 100₂

By joining all the binary groups together to form the final binary number.  So, the final result is (64)₈ = 110100₂

Example 3: Convert (67)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (67)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (67)₈.

• The binary of 6₈ is 110₂
• The binary of 7₈ is 111₂

By joining all the binary groups together to form the final binary number.  So, the final result is (67)₈ = 110111₂

Example 4: Convert (77)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (77)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (77)₈.

• The binary of 7₈ is 111₂
• The binary of 7₈ is 111₂

By joining all the binary groups together to form the final binary number.  So, the final result is (77)₈ = 111111₂

Example 5: Convert (127)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (127)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (127)₈.

• The binary of 1₈ is 001₂
• The binary of 2₈ is 010₂
• The binary of 7₈ is 111₂

By joining all the binary groups together to form the final binary number.  So, the final result is (127)₈ = 001010111₂

Example 6: Convert (305)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (305)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (305)₈.

• The binary of 3₈ is 011₂
• The binary of 0₈ is 000₂
• The binary of 5₈ is 101₂

By joining all the binary groups together to form the final binary number.  So, the final result is (305)₈ = 011000101₂

Example 7: Convert (777)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (777)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (777)₈.

• The binary of 7₈ is 111₂
• The binary of 7₈ is 111₂
• The binary of 7₈ is 111₂

By joining all the binary groups together to form the final binary number.  So, the final result is (777)₈ = 111111111₂

Example 8: Convert (452)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (452)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (452)₈.

• The binary of 4₈ is 100₂
• The binary of 5₈ is 101₂
• The binary of 2₈ is 010₂

By joining all the binary groups together to form the final binary number.  So, the final result is (452)₈ = 100101010₂

Example 9: Convert (100)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (100)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (100)₈.

• The binary of 1₈ is 001₂
• The binary of 0₈ is 000₂
• The binary of 0₈ is 000₂

By joining all the binary groups together to form the final binary number.  So, the final result is (100)₈ = 001000000₂

Example 10: Convert (735)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (735)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (735)₈.

• The binary of 7₈ is 111₂
• The binary of 3₈ is 011₂
• The binary of 5₈ is 101₂

By joining all the binary groups together to form the final binary number.  So, the final result is (735)₈ = 111011101₂

Example 11: Convert (1234)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (1234)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (1234)₈.

• The binary of 1₈ is 001₂
• The binary of 2₈ is 010₂
• The binary of 3₈ is 011₂
• The binary of 4₈ is 100₂

By joining all the binary groups together to form the final binary number.  So, the final result is (1234)₈ = 001010011100₂

Example 12: Convert (5670)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (5670)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (5670)₈.

• The binary of 5₈ is 101₂
• The binary of 6₈ is 110₂
• The binary of 7₈ is 111₂
• The binary of 0₈ is 000₂

By joining all the binary groups together to form the final binary number.  So, the final result is (5670)₈ = 101110111000₂

Example 13: Convert (70125)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (70125)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (70125)₈.

• The binary of 7₈ is 111₂
• The binary of 0₈ is 000₂
• The binary of 1₈ is 001₂
• The binary of 2₈ is 010₂
• The binary of 5₈ is 101₂

By joining all the binary groups together to form the final binary number.  So, the final result is (70125)₈ = 111000001010101₂

Example 14: Convert (34567)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (34567)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (34567)₈.

• The binary of 3₈ is 011₂
• The binary of 4₈ is 100₂
• The binary of 5₈ is 101₂
• The binary of 6₈ is 110₂
• The binary of 7₈ is 111₂

By joining all the binary groups together to form the final binary number.  So, the final result is (34567)₈ = 011100101110111₂

Example 15: Convert (120345)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (120345)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (120345)₈.

• The binary of 1₈ is 001₂
• The binary of 2₈ is 010₂
• The binary of 0₈ is 000₂
• The binary of 3₈ is 011₂
• The binary of 4₈ is 100₂
• The binary of 5₈ is 101₂

By joining all the binary groups together to form the final binary number.  So, the final result is (120345)₈ = 001010000011100101₂

Example 16: Convert (7654321)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (7654321)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (7654321)₈.

• The binary of 7₈ is 111₂
• The binary of 6₈ is 110₂
• The binary of 5₈ is 101₂
• The binary of 4₈ is 100₂
• The binary of 3₈ is 011₂
• The binary of 2₈ is 010₂
• The binary of 1₈ is 001₂

By joining all the binary groups together to form the final binary number.  So, the final result is (7654321)₈ = 111110101100011010001₂

Example 17: Convert (12345670)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (12345670)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (12345670)₈.

• The binary of 1₈ is 001₂
• The binary of 2₈ is 010₂
• The binary of 3₈ is 011₂
• The binary of 4₈ is 100₂
• The binary of 5₈ is 101₂
• The binary of 6₈ is 110₂
• The binary of 7₈ is 111₂
• The binary of 0₈ is 000₂

By joining all the binary groups together to form the final binary number.  So, the final result is (12345670)₈ = 001010011100101110111000₂

Example 18: Convert (70123456)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (70123456)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (70123456)₈.

• The binary of 7₈ is 111₂
• The binary of 0₈ is 000₂
• The binary of 1₈ is 001₂
• The binary of 2₈ is 010₂
• The binary of 3₈ is 011₂
• The binary of 4₈ is 100₂
• The binary of 5₈ is 101₂
• The binary of 6₈ is 110₂

By joining all the binary groups together to form the final binary number.  So, the final result is (70123456)₈ = 111000001010011100101110₂

Example 19: Convert (765401234)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (765401234)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (765401234)₈.

• The binary of 7₈ is 111₂
• The binary of 6₈ is 110₂
• The binary of 5₈ is 101₂
• The binary of 4₈ is 100₂
• The binary of 0₈ is 000₂
• The binary of 1₈ is 001₂
• The binary of 2₈ is 010₂
• The binary of 3₈ is 011₂
• The binary of 4₈ is 100₂

By joining all the binary groups together to form the final binary number.  So, the final result is (765401234)₈ = 111110101100000001010011100₂

Example 20: Convert (1234567012)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (1234567012)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (1234567012)₈.

• The binary of 1₈ is 001₂
• The binary of 2₈ is 010₂
• The binary of 3₈ is 011₂
• The binary of 4₈ is 100₂
• The binary of 5₈ is 101₂
• The binary of 6₈ is 110₂
• The binary of 7₈ is 111₂
• The binary of 0₈ is 000₂
• The binary of 1₈ is 001₂
• The binary of 2₈ is 010₂

By joining all the binary groups together to form the final binary number.  So, the final result is (1234567012)₈ = 001010011100101110111000001010₂

Fractional Octal to Binary Conversion Examples

Here are 10 examples of Fractional Octal to Binary conversions

Example 21: Convert (17.35)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (17.35)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (17.35)₈.

The binary of 1₈ is 001₂
The binary of 7₈ is 111₂
The binary of 3₈ is 011₂
The binary of 5₈ is 101₂

By joining all the binary groups together to form the final binary number. So, the final result is (17.35)₈ = 001111.011101₂

Example 22: Convert (245.7)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (245.7)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (245.7)₈.

The binary of 2₈ is 010₂
The binary of 4₈ is 100₂
The binary of 5₈ is 101₂
The binary of 7₈ is 111₂

By joining all the binary groups together to form the final binary number. So, the final result is (245.7)₈ = 010100101.111₂

Example 23: Convert (603.14)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (603.14)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (603.14)₈.

The binary of 6₈ is 110₂
The binary of 0₈ is 000₂
The binary of 3₈ is 011₂
The binary of 1₈ is 001₂
The binary of 4₈ is 100₂

By joining all the binary groups together to form the final binary number. So, the final result is (603.14)₈ = 110000011.001100₂

Example 24: Convert (7521.63)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (7521.63)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (7521.63)₈.

The binary of 7₈ is 111₂
The binary of 5₈ is 101₂
The binary of 2₈ is 010₂
The binary of 1₈ is 001₂
The binary of 6₈ is 110₂
The binary of 3₈ is 011₂

By joining all the binary groups together to form the final binary number. So, the final result is (7521.63)₈ = 111101010001.110011₂

Example 25: Convert (13467.25)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (13467.25)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (13467.25)₈.

The binary of 1₈ is 001₂
The binary of 3₈ is 011₂
The binary of 4₈ is 100₂
The binary of 6₈ is 110₂
The binary of 7₈ is 111₂
The binary of 2₈ is 010₂
The binary of 5₈ is 101₂

By joining all the binary groups together to form the final binary number. So, the final result is (13467.25)₈ = 001011100110111.010101₂

Example 26: Convert (567012.4)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (567012.4)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (567012.4)₈.

The binary of 5₈ is 101₂
The binary of 6₈ is 110₂
The binary of 7₈ is 111₂
The binary of 0₈ is 000₂
The binary of 1₈ is 001₂
The binary of 2₈ is 010₂
The binary of 4₈ is 100₂

By joining all the binary groups together to form the final binary number. So, the final result is (567012.4)₈ = 101110111000001010.100₂

Example 27: Convert (701234.56)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (701234.56)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (701234.56)₈.

The binary of 7₈ is 111₂
The binary of 0₈ is 000₂
The binary of 1₈ is 001₂
The binary of 2₈ is 010₂
The binary of 3₈ is 011₂
The binary of 4₈ is 100₂
The binary of 5₈ is 101₂
The binary of 6₈ is 110₂

By joining all the binary groups together to form the final binary number. So, the final result is (701234.56)₈ = 111000001010011100.101110₂

Example 28: Convert (123456.07)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (123456.07)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (123456.07)₈.

The binary of 1₈ is 001₂
The binary of 2₈ is 010₂
The binary of 3₈ is 011₂
The binary of 4₈ is 100₂
The binary of 5₈ is 101₂
The binary of 6₈ is 110₂
The binary of 0₈ is 000₂
The binary of 7₈ is 111₂

By joining all the binary groups together to form the final binary number. So, the final result is (123456.07)₈ = 001010011100101110.000111₂

Example 29: Convert (7654321.3)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (7654321.3)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (7654321.3)₈.

The binary of 7₈ is 111₂
The binary of 6₈ is 110₂
The binary of 5₈ is 101₂
The binary of 4₈ is 100₂
The binary of 3₈ is 011₂
The binary of 2₈ is 010₂
The binary of 1₈ is 001₂
The binary of 3₈ is 011₂

By joining all the binary groups together to form the final binary number. So, the final result is (7654321.3)₈ = 111110101100011010001.011₂

Example 30: Convert (45670123.76)₈ to Binary

Solution:

The following diagram shows the conversion of the octal number (45670123.76)₈ into its equivalent binary.

Description:

By replacing each octal digit with its corresponding 3-bit binary equivalent using the standard table, we get the binary of (45670123.76)₈.

The binary of 4₈ is 100₂
The binary of 5₈ is 101₂
The binary of 6₈ is 110₂
The binary of 7₈ is 111₂
The binary of 0₈ is 000₂
The binary of 1₈ is 001₂
The binary of 2₈ is 010₂
The binary of 3₈ is 011₂
The binary of 7₈ is 111₂
The binary of 6₈ is 110₂

By joining all the binary groups together to form the final binary number. So, the final result is (45670123.76)₈ = 100101110111000001010011.111110₂

2. Octal to Binary Conversion (Using Decimal Number System)

This method has two main Steps:

Step 1: Convert Octal to Decimal

1. Write the given octal number
2. Start from the rightmost digit
3. Multiply each digit by powers of 8:
   • Rightmost digit → × 8⁰
   • Next digit → × 8¹
   • Next → × 8² and so on
4. Add all the results

The sum is your decimal number

For Fractional Part (if present) Digits after the decimal use negative powers of 8: First digit → × 8⁻¹ Next → × 8⁻²

Step 2: Convert Decimal to Binary

1. Take the decimal number
2. Divide it by 2
3. Write down the remainder
4. Repeat division until the quotient becomes 0
5. Write remainders in reverse order

 This gives the binary result

Fractional Part (if present) For decimal fractions, use the multiplication by 2 method to convert it into decimal Take the fractional part Multiply it by 2 Write down the integer part (0 or 1) Keep the fractional part only Repeat until: Fraction becomes 0 or Required accuracy is reached Write the collected digits in the same order

Example 01: Convert (12.3)₈ to Binary

Step 1: Convert Octal to Decimal

Integer Part (12)₈ Conversion

(12)₈ =1 × 8¹ + 2 × 8⁰
(12)₈  = 8 + 2
(12)₈  = (10)₁₀

Fractional Part (.3)₈ Conversion

(0.3)₈  = 3 × 8⁻¹ = 3/8 = (0.375)₁₀

Combine the integer and the fractional Part

(12)₈ + (0.3)₈  = (10)₁₀ + (0.375)₁₀

(12.3)₈ = (10.375)₁₀

Step 2: Convert Decimal to Binary

Integer Part (10)₁₀ Conversion

10 ÷ 2 = 5 remainder 0
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1

(10)₁₀ = (1010)₂

Fractional Part (0.375)₁₀ Conversion

0.375 × 2 = 0.75 → 0
0.75 × 2 = 1.5 → 1
0.5 × 2 = 1.0 → 1

(0.375)₁₀ = (.011)₂

Combine the integer and the fractional Part

(10.375)₁₀ = (1010.011)₂

Hence, the given octal (12.3)₈ is converted to binary (1010.011). so we can write

(12.3)₈ = (10.375)₁₀ = (1010.011)₂

Example 02: Convert (25.4)₈ to Binary

Step 1: Convert Octal to Decimal

Integer Part (25)₈ Conversion

(25)₈ = 2 × 8¹ + 5 × 8⁰
(25)₈ = 16 + 5
(25)₈ = (21)₁₀

Fractional Part (.4)₈ Conversion

(0.4)₈ = 4 × 8⁻¹ = 4/8 = (0.5)₁₀

Combine the integer and the fractional Part

(25)₈ + (0.4)₈ = (21)₁₀ + (0.5)₁₀

(25.4)₈ = (21.5)₁₀

Step 2: Convert Decimal to Binary

Integer Part (21)₁₀ Conversion

21 ÷ 2 = 10 remainder 1
10 ÷ 2 = 5 remainder 0
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1

(21)₁₀ = (10101)₂

Fractional Part (0.5)₁₀ Conversion

0.5 × 2 = 1.0 → 1

(0.5)₁₀ = (.1)₂

Combine the integer and the fractional Part

(21.5)₁₀ = (10101.1)₂

Hence, the given octal (25.4)₈ is converted to binary (10101.1). so we can write

(25.4)₈ = (21.5)₁₀ = (10101.1)₂

Example 03: Convert (7.2)₈ to Binary

Step 1: Convert Octal to Decimal

Integer Part (7)₈ Conversion

(7)₈ = 7 × 8⁰
(7)₈ = (7)₁₀

Fractional Part (.2)₈ Conversion

(0.2)₈ = 2 × 8⁻¹ = 2/8 = (0.25)₁₀

Combine the integer and the fractional Part

(7)₈ + (0.2)₈ = (7)₁₀ + (0.25)₁₀

(7.2)₈ = (7.25)₁₀

Step 2: Convert Decimal to Binary

Integer Part (7)₁₀ Conversion

7 ÷ 2 = 3 remainder 1
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1

(7)₁₀ = (111)₂

Fractional Part (0.25)₁₀ Conversion

0.25 × 2 = 0.5 → 0
0.5 × 2 = 1.0 → 1

(0.25)₁₀ = (.01)₂

Combine the integer and the fractional Part

(7.25)₁₀ = (111.01)₂

Hence, the given octal (7.2)₈ is converted to binary (111.01). so we can write

(7.2)₈ = (7.25)₁₀ = (111.01)₂

Example 04: Convert (34.1)₈ to Binary

Step 1: Convert Octal to Decimal

Integer Part (34)₈ Conversion

(34)₈ = 3 × 8¹ + 4 × 8⁰
(34)₈ = 24 + 4
(34)₈ = (28)₁₀

Fractional Part (.1)₈ Conversion

(0.1)₈ = 1 × 8⁻¹ = 1/8 = (0.125)₁₀

Combine the integer and the fractional Part

(34)₈ + (0.1)₈ = (28)₁₀ + (0.125)₁₀

(34.1)₈ = (28.125)₁₀

Step 2: Convert Decimal to Binary

Integer Part (28)₁₀ Conversion

28 ÷ 2 = 14 remainder 0
14 ÷ 2 = 7 remainder 0
7 ÷ 2 = 3 remainder 1
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1

(28)₁₀ = (11100)₂

Fractional Part (0.125)₁₀ Conversion

0.125 × 2 = 0.25 → 0
0.25 × 2 = 0.5 → 0
0.5 × 2 = 1.0 → 1

(0.125)₁₀ = (.001)₂

Combine the integer and the fractional Part

(28.125)₁₀ = (11100.001)₂

Hence, the given octal (34.1)₈ is converted to binary (11100.001). so we can write

(34.1)₈ = (28.125)₁₀ = (11100.001)₂

Example 05: Convert (51.6)₈ to Binary

Step 1: Convert Octal to Decimal

Integer Part (51)₈ Conversion

(51)₈ = 5 × 8¹ + 1 × 8⁰
(51)₈ = 40 + 1
(51)₈ = (41)₁₀

Fractional Part (.6)₈ Conversion

(0.6)₈ = 6 × 8⁻¹ = 6/8 = (0.75)₁₀

Combine the integer and the fractional Part

(51)₈ + (0.6)₈ = (41)₁₀ + (0.75)₁₀

(51.6)₈ = (41.75)₁₀

Step 2: Convert Decimal to Binary

Integer Part (41)₁₀ Conversion

41 ÷ 2 = 20 remainder 1
20 ÷ 2 = 10 remainder 0
10 ÷ 2 = 5 remainder 0
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1

(41)₁₀ = (101001)₂

Fractional Part (0.75)₁₀ Conversion

0.75 × 2 = 1.5 → 1
0.5 × 2 = 1.0 → 1

(0.75)₁₀ = (.11)₂

Combine the integer and the fractional Part

(41.75)₁₀ = (101001.11)₂

Hence, the given octal (51.6)₈ is converted to binary (101001.11). so we can write

(51.6)₈ = (41.75)₁₀ = (101001.11)₂

Example 06: Convert (123.45)₈ to Binary

Step 1: Convert Octal to Decimal

Integer Part (123)₈ Conversion

(123)₈ = 1 × 8² + 2 × 8¹ + 3 × 8⁰
(123)₈ = 64 + 16 + 3
(123)₈ = (83)₁₀

Fractional Part (.45)₈ Conversion

(0.45)₈ = 4 × 8⁻¹ + 5 × 8⁻²
= 4/8 + 5/64
= 0.5 + 0.078125
= (0.578125)₁₀

Combine the integer and the fractional Part

(123)₈ + (0.45)₈ = (83)₁₀ + (0.578125)₁₀

(123.45)₈ = (83.578125)₁₀

Step 2: Convert Decimal to Binary

Integer Part (83)₁₀ Conversion

83 ÷ 2 = 41 remainder 1
41 ÷ 2 = 20 remainder 1
20 ÷ 2 = 10 remainder 0
10 ÷ 2 = 5 remainder 0
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1

(83)₁₀ = (1010011)₂

Fractional Part (0.578125)₁₀ Conversion

0.578125 × 2 = 1.15625 → 1
0.15625 × 2 = 0.3125 → 0
0.3125 × 2 = 0.625 → 0
0.625 × 2 = 1.25 → 1
0.25 × 2 = 0.5 → 0
0.5 × 2 = 1.0 → 1

(0.578125)₁₀ = (.100101)₂

Combine the integer and the fractional Part

(83.578125)₁₀ = (1010011.100101)₂

Hence, the given octal (123.45)₈ is converted to binary (1010011.100101). so we can write

(123.45)₈ = (83.578125)₁₀ = (1010011.100101)₂

Example 07: Convert (456.12)₈ to Binary

Step 1: Convert Octal to Decimal

Integer Part (456)₈ Conversion

(456)₈ = 4 × 8² + 5 × 8¹ + 6 × 8⁰
= 256 + 40 + 6
= (302)₁₀

Fractional Part (.12)₈ Conversion

(0.12)₈ = 1/8 + 2/64
= 0.125 + 0.03125
= (0.15625)₁₀

Combine the integer and the fractional Part

(456)₈ + (0.12)₈ = (302)₁₀ + (0.15625)₁₀

(456.12)₈ = (302.15625)₁₀

Step 2: Convert Decimal to Binary

Integer Part (302)₁₀ Conversion

302 ÷ 2 = 151 r0
151 ÷ 2 = 75 r1
75 ÷ 2 = 37 r1
37 ÷ 2 = 18 r1
18 ÷ 2 = 9 r0
9 ÷ 2 = 4 r1
4 ÷ 2 = 2 r0
2 ÷ 2 = 1 r0
1 ÷ 2 = 0 r1

(302)₁₀ = (100101110)₂

Fractional Part (0.15625)₁₀ Conversion

0.15625 × 2 = 0.3125 → 0
0.3125 × 2 = 0.625 → 0
0.625 × 2 = 1.25 → 1
0.25 × 2 = 0.5 → 0
0.5 × 2 = 1.0 → 1

(0.15625)₁₀ = (.00101)₂

Combine the integer and the fractional Part

(302.15625)₁₀ = (100101110.00101)₂

Hence, the given octal (456.12)₈ is converted to binary (100101110.00101). so we can write

(456.12)₈ = (302.15625)₁₀ = (100101110.00101)₂

Example 08: Convert (701.7)₈ to Binary

Step 1: Convert Octal to Decimal

Integer Part (701)₈ Conversion

(701)₈ = 7×8² + 0×8¹ + 1×8⁰
= 448 + 0 + 1
= (449)₁₀

Fractional Part (.7)₈ Conversion

(0.7)₈ = 7/8 = (0.875)₁₀

Combine

(701.7)₈ = (449.875)₁₀

Step 2: Convert Decimal to Binary

Integer Part (449)₁₀ Conversion

449 → (111000001)₂

Fractional Part (0.875)₁₀ Conversion

0.875 ×2 =1.75 →1
0.75 ×2 =1.5 →1
0.5 ×2 =1.0 →1

(.111)₂

Combine

(449.875)₁₀ = (111000001.111)₂

Hence

(701.7)₈ = (449.875)₁₀ = (111000001.111)₂

Example 09: Convert (2345.67)₈ to Binary

(2345)₈ = 2×512 +3×64 +4×8 +5
=1024+192+32+5 = (1253)₁₀

(0.67)₈ = 6/8 +7/64 =0.75+0.109375 = (0.859375)₁₀

(2345.67)₈ = (1253.859375)₁₀

Binary: (10011100101.110111)₂

Example 10: Convert (777.77)₈ to Binary

(777)₈ = 511₁₀

(0.77)₈ = 7/8 +7/64 = 0.984375

Binary: (111111111.111111)₂

Example 11: Convert (3456.21)₈ to Binary

(3456)₈ = (1838)₁₀

(0.21)₈ = 2/8 +1/64 = 0.265625

Binary: (11100101110.010001)₂

Example 12: Convert (6021.34)₈ to Binary

(6021)₈ = (3097)₁₀

(0.34)₈ = 3/8 +4/64 = 0.4375

Binary: (110000011001.0111)₂

Example 13: Convert (1573.5)₈ to Binary

(1573)₈ = (891)₁₀

(0.5)₈ = 5/8 = 0.625

Binary: (1101111011.101)₂

Example 14: Convert (4210.16)₈ to Binary

(4210)₈ = (2184)₁₀

(0.16)₈ = 1/8 +6/64 = 0.21875

Binary: (100010001000.00111)₂

Example 15: Convert (7654.32)₈ to Binary

(7654)₈ = (4012)₁₀

(0.32)₈ = 3/8 +2/64 = 0.40625

Binary: (111110101100.01101)₂

Common Mistakes in Octal to Binary Conversion

Understanding common mistakes helps students avoid errors and improve accuracy in exams. These mistakes often occur due to a misunderstanding of the conversion method.
The list of common mistakes is given below:

Here are the same points with clear examples for each mistake in octal → binary conversion:

1. Ignoring 3-bit Representation

Each octal digit must always be written as exactly 3 binary bits.

Example: Correct:

• 7₈ = 111₂

Incorrect (wrong method):

• 7₈ = 11 (missing one bit)

This gives an incomplete and incorrect binary value.

2. Removing Necessary Zeros

 Only leading zeros can be removed. Zeros inside the number must stay.

Example: Correct:

• 3₈ = 011₂

Incorrect:

• 3₈ = 11  (zero removed, value becomes unclear in grouped conversion)

When combining digits like 7.3₈ → 111.011₂, removing the middle zero changes structure.

3. Incorrect Mapping

Using wrong binary equivalents for octal digits leads to wrong results.

Example: Correct mapping:

• 6₈ = 110₂

Incorrect mapping:

• 6₈ = 101  (wrong value)

This completely changes the final binary number.

4. Combining Digits Incorrectly

Binary groups must be written in the same order as octal digits.

Example: Convert 7.3₈:

Correct:

• 7 → 111
• 3 → 011
  → 111.011₂

Incorrect combination:

• 3.7₈ → 011.111  (order reversed)

This changes the actual value completely.

Advantages of Octal to Binary Conversion

Octal to binary conversion provides multiple benefits, especially in simplifying digital computations. It is widely used in programming and electronics.
The list of advantages is given below:

1. Fast Conversion

Direct mapping eliminates the need for complex calculations.

2. Error Reduction

Using a fixed table reduces chances of mistakes.

3. Useful in Computer Systems

Helps in understanding memory representation and machine-level data.

4. Efficient for Large Numbers

Large octal numbers can be quickly converted into binary.

Practice Questions for Students

Practicing different problems helps students master octal to binary conversion effectively. Try solving the following questions for better understanding.
The list of practice questions is given below:

• Convert (17)₈ to binary
• Convert (234)₈ to binary
• Convert (506)₈ to binary
• Convert (701)₈ to binary
• Convert (1234)₈ to binary

Conclusion

Octal to binary conversion is one of the easiest and most important number system conversions in computer science. By understanding the 3-bit relationship and practicing examples, students can quickly master this topic and perform conversions accurately in exams and real-world applications.