Converting Hexadecimal to Octal

In the number system, we come across four different types of the number system, i.e. hexadecimal, octal, decimal and binary. The conversion of these numbers from one form to another is possible. To convert hexadecimal to octal numbers, we need to convert hexadecimal to its equivalent decimal number first and then decimal to octal. Before this conversion, first, go through the basic definition of hexadecimal and octal numbers.

What Are Hexadecimal Numbers?

Hexadecimal numbers are the numbers which have base 16. It uses 16 different digits to represent the numbers. It is denoted as h16, where h is a hexadecimal number. It may be a combination of alphabets and numbers. Thus, it includes numbers from 0 to 9 and alphabets A to F.

Example: (AB2)₁₆ , (98D1)₁₆, (AFD)₁₆

What are Octal Numbers?

Octal numbers have base 8. These numbers use digits from 0 to 7, total 8 digits and hence, they are called octal number system. Octal numbers have base 8. It is denoted as o8 and o is an octal number. It does not use digits 8 and 9 to represent a number.

Example: (112)₈, (275)₈,(45)₈

Hexadecimal to Octal Conversion

Conversion of hexadecimal to octal cannot be done directly. Firstly we need to convert hexadecimal into its equivalent decimal number then decimal to octal. Follow the steps below to understand the process.

Consider the given hexadecimal number.
First, count the number of digits in the number.
If n is the position of the digit from the right end, then multiply each digit with 16^(n-1).
Add the terms after multiplication. The resultant is the equivalent decimal form.
Divide the decimal number by 8.
Note down the remainder.
Repeat the previous two steps with the quotient until the quotient is zero.
Write the remainders in reverse order.
The obtained number is the required result.

Another Method to Convert Hex to Octal

There is another method to convert any hexadecimal to its equivalent octal. As we know, hexadecimal numbers include binary digits; therefore, we can club these binary numbers into a pair so that we can relate it with the octal numbers. Let us check the method with steps and example:

For each given hexadecimal number digit, write the equivalent binary number. If any of the binary equivalents are less than 4 digits, add 0’s to the left side.
Combine and make the groups of binary digits from right to left, each containing 3 digits. Add 0’s to the left if there are less than 3 digits in the last group.
Find the octal equivalent of each binary group.

Example

Understanding Hexadecimal-to-Octal Conversion

Hexadecimal-to-octal conversion involves converting numbers in hexadecimal (base 16) into their equivalent octal (base 8) values. Hexadecimal uses digits 0-9 and letters A-F to represent values from 0 to 15, while octal uses digits 0-7.

The general approach to converting hexadecimal numbers to octal includes:

Convert the hexadecimal number to its binary equivalent (base 2).
Group the binary digits into sets of three, starting from the rightmost digit.
Convert each group of three binary digits into its octal equivalent.

Steps for Hexadecimal-to-Octal Conversion

Step 1: Write down the hexadecimal number.

Step 2: Convert the hexadecimal number to binary (each hexadecimal digit corresponds to 4 binary digits).

Step 3: Group the binary digits into sets of three, starting from the rightmost digit. Add leading zeros if necessary to complete a group.

Step 4: Convert each group of three binary digits into its octal equivalent.

Example: Converting Hexadecimal to Octal

Convert \( 2F \) to octal:

Step 1: Write the hexadecimal number \( 2F \).
Step 2: Convert to binary:
- \( 2 = 0010 \)
- \( F = 1111 \)
- Binary: \( 0010 \, 1111 \)
Step 3: Group into sets of three:
- Binary: \( 00\,101\,111 \) (adding leading zeros to complete groups).
Step 4: Convert each group to octal:
- \( 00 = 0 \)
- \( 101 = 5 \)
- \( 111 = 7 \)
Final octal result: \( 57 \).

Conversion Table for Quick Reference

Here is a quick reference for hexadecimal to binary (used in conversion):

0 → 0000
1 → 0001
2 → 0010
3 → 0011
4 → 0100
5 → 0101
6 → 0110
7 → 0111
8 → 1000
9 → 1001
A → 1010
B → 1011
C → 1100
D → 1101
E → 1110
F → 1111

Applications of Hexadecimal-to-Octal Conversion

Hexadecimal-to-octal conversion is commonly used in:

Digital electronics and microcontroller programming.
Understanding data representations in lower-level computing.
Simplifying large binary representations in debugging.

Practice Problem

Convert \( 3A \) to octal:

Solution:
- Hexadecimal \( 3A \) → Binary \( 0011 \, 1010 \).
- Group binary digits: \( 000 \, 111 \, 010 \) (adding leading zeros).
- Convert to octal:
  - \( 000 = 0 \)
  - \( 111 = 7 \)
  - \( 010 = 2 \)
- Final octal result: \( 72 \).

Hexadecimal-to-Octal Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a single hexadecimal digit to its octal equivalent.	Convert the hexadecimal digit to binary (4 bits). Group the binary digits into 3-bit groups, starting from the right. Convert each group to its octal equivalent.	For `B`: ‘B’ in binary: `1011`. Group into 3 bits: `001 011`. Convert to octal: `13`. Octal: `13`.
Handling Larger Numbers	Converting a multi-digit hexadecimal number to octal.	Convert each hexadecimal digit to its 4-bit binary equivalent. Combine all binary groups into one binary string. Group the binary string into 3-bit groups from the right. Convert each 3-bit group to its octal equivalent.	For `3A7`: ‘3’ in binary: `0011`. ‘A’ in binary: `1010`. ‘7’ in binary: `0111`. Combine: `0011 1010 0111`. Group into 3 bits: `000 111 010 011 1`. Convert to octal: `1673`. Octal: `1673`.
Verifying Conversion	Checking the accuracy of hexadecimal-to-octal conversion.	Convert the octal number back to binary (3 bits per digit). Group the binary digits into 4-bit groups starting from the right. Convert each 4-bit group to its hexadecimal equivalent. Ensure the hexadecimal result matches the original input.	For `1673` (octal for `3A7`): ‘1’ in binary: `001`. ‘6’ in binary: `110`. ‘7’ in binary: `111`. ‘3’ in binary: `011`. Combine: `0011 1010 0111`. Group into 4 bits: `0011 1010 0111`. Hexadecimal: `3A7`, matches the original input.
Handling Special Cases	Converting hexadecimal numbers with letters (A-F) to octal.	Convert the letters to their binary equivalents (4 bits). Group the binary into 3 bits for octal conversion.	For `FE`: ‘F’ in binary: `1111`. ‘E’ in binary: `1110`. Combine: `1111 1110`. Group into 3 bits: `001 111 111 110`. Convert to octal: `776`. Octal: `776`.
Applications	Understanding where hexadecimal-to-octal conversion is used.	Hexadecimal and octal are both used in computing to simplify binary representation. Commonly applied in memory addressing, permissions, and compact data representation.	Example: Converting hexadecimal memory address `1A3F` to octal for compactness: ‘1’ in binary: `0001`. ‘A’ in binary: `1010`. ‘3’ in binary: `0011`. ‘F’ in binary: `1111`. Combine: `0001 1010 0011 1111`. Group into 3 bits: `001 101 000 111 111`. Convert to octal: `15077`. Octal: `15077`.