What Are Hexadecimal Numbers?
The numbers whose base value is equivalent to 16 are known as hexadecimal numbers. A single number is usually represented by 16 other numbers. It is denoted as n16, where n is any random hexadecimal number. The 16 digit number can be a combination of numbers as well as alphabets. The numbers used to represent ranges from 0 - 9 and the alphabets range from A - F.
What Are Octal Numbers?
The numbers whose base value is equivalent to 8 are known as hexadecimal numbers. A single number is usually represented by 8 other numbers. It is denoted as n8, where n is any random octal number. The 8 digit number can be a combination of numbers. The numbers used to represent ranges from 0 - 7. Octal numbers cannot be represented by 8 and 9.
Hexadecimal to Octal Conversion
There are two hex to octal conversion methods. They are:
Method 1
You cannot convert a hexadecimal number to octal decimal number directly. First, you will have to convert the hexadecimal number to its decimal equivalent and then convert the decimal equivalent to octal decimal number. To understand better, follow the steps given below.
Note down the given hexadecimal number
Count the number of digits and note down the number of digits in the number
Multiply each digit with 16x-1, if x is the position of the digit from the right end.
Find the sum of the terms after multiplication
The result what is obtained is in the equivalent decimal form
Divide the obtained decimal number with 8
Write down the value of the remainder
Repeat step 6 and step 7 with the quotient, until the value of the quotient is zero
Note the remainders in reverse order
The number that is obtained is the required result.
Method 2
That was method 1. In case you find that difficult, you can also follow the other method to find the octal decimal value of a hexadecimal number. We know that the hexadecimal number comprises of binary digits. In this method, you can combine these binary digits in the pairs of three and you can relate them to the octal numbers. The steps involved in this method are given below.
For every given hexadecimal number’s digit, note down the number equivalent to the binary number. Add 0’s to the left side, if any of the binary equivalents are less than 4 digits.
Next, combine the numbers and make the groups of binary digits from right to left, each containing 3 digits. In case there are less than 3 digits in the last group, add 0’s to the group.
Find the value equivalent to the octal numbers of each binary group.
Below are the Hexadecimal to Octal Conversion Examples
Question 1: Convert the following hexadecimal number to octal number 2CD16.
Answer: Given,
2CD16 is a hexadecimal number.
2 → 0010, C →1100, D → 1101,
Now you will be grouping them from right to left, each having 3 digits.
001, 011, 001, 101
001→1, 011 →3, 001→1, 101→5
Hence, 2CD16 = 13158
Question 2: Convert the following hexadecimal number to octal number 3EC16.
Answer: Given,
3EC16 is a hexadecimal number.
3 → 0010, E → 1110, C →1100,
Now you will be grouping them from right to left, each having 3 digits.
001, 011, 101, 100
001→1, 011 →3, 101→5, 100→4
Hence, 3EC16 = 13548
Now that you’ve understood the hexadecimal to octal example problems, look at the table given below for easier conversions.
Hex to Octal Conversion Table
Let’s deep-dive into hexadecimal to octal questions.
Solved Examples
Question 1: What is the equivalent form of F216?
Solution
F216 = ( F × 161 ) + ( 2 × 160 )
= F × 16 + 2 × 1
= 17 × 16 + 2
= 272 + 2
F216 = 272 (Decimal form)
The octal number is 4208.
Therefore, F216 = 4208.
FAQs on Hexadecimal to Octal Conversion
1. What exactly is the hexadecimal number system?
The hexadecimal number system is a base-16 system. This means it uses 16 unique symbols to represent numbers. It uses the digits 0 through 9 for the first ten values, and the letters A through F for the values 10 through 15. It's widely used in computer science for representing data in a more human-readable format than binary.
2. And what about the octal number system?
The octal number system is a base-8 system, meaning it uses only eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. Numbers in this system never include the digits 8 or 9. Like hexadecimal, it's an important system in computing, especially in older systems and for file permissions.
3. What is the most direct way to convert a hexadecimal number to an octal number?
The most straightforward method is to use binary as an intermediate step, since both hexadecimal (base-16) and octal (base-8) are powers of two. The steps are:
- Step 1: Convert each hexadecimal digit into its 4-bit binary equivalent.
- Step 2: Combine all the binary bits into a single long string.
- Step 3: Regroup this binary string into sets of 3 bits, starting from the right.
- Step 4: Convert each 3-bit group into its corresponding octal digit.
4. Could you provide an example of a hexadecimal to octal conversion?
Certainly. Let's convert the hexadecimal number (1B3)₁₆ to octal.
- First, convert each digit to 4-bit binary: 1 → 0001, B (11) → 1011, 3 → 0011.
- Combine them: 000110110011.
- Now, group into 3-bit sets from the right: (000) (110) (110) (011).
- Convert each 3-bit group to octal: 0 → 0, 110 → 6, 110 → 6, 011 → 3.
- So, (1B3)₁₆ = (663)₈.
5. Why is converting through binary the recommended method for hex to octal conversions?
This method is recommended because of the direct mathematical relationship between the number bases. Since 16 = 2⁴ and 8 = 2³, both systems are directly compatible with binary (base-2). Converting each hex digit to a 4-bit binary number and then regrouping into 3-bit octal numbers is much faster and less prone to calculation errors than the alternative method of converting to decimal first and then to octal.
6. How is converting from hexadecimal to octal different from converting octal back to hexadecimal?
The process is simply the reverse, with one key difference: the size of the binary groups. For octal to hexadecimal, you convert each octal digit to its 3-bit binary equivalent first. Then, you regroup the resulting binary string into sets of 4 bits (from right to left) and convert each 4-bit group to its hexadecimal digit. The core difference is the binary grouping: 3-bits for octal, 4-bits for hex.
7. What is a common mistake to avoid when converting hexadecimal numbers to octal?
The most common mistake is incorrect binary grouping. After converting the hexadecimal number to its full binary string, you must group the bits into sets of three starting from the right-hand side. If you group from the left, you will get an incorrect answer. Also, if the leftmost group has fewer than three bits, you must add leading zeros to complete the group before converting it to an octal digit.
