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
- Write the given octal number
- Start from the rightmost digit
- Multiply each digit by powers of 8:
- Rightmost digit → × 8⁰
- Next digit → × 8¹
- Next → × 8² and so on
- Add all the results
The sum is your decimal number
For Fractional Part (if present)
|
Step 2: Convert Decimal to Binary
- Take the decimal number
- Divide it by 2
- Write down the remainder
- Repeat division until the quotient becomes 0
- 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
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.