Converting Decimal to Octal means changing a number from the decimal system (base 10) to the octal system (base 8). To do this by hand, you keep dividing the decimal number by 8 until the result is 0, and then write down the remainders in reverse order.

A number system is a way of writing numbers using a specific set of digits. Each system has a base, which tells you how many digits it uses. For example, the decimal system has 10 digits (0 to 9), while the octal system has 8 digits (0 to 7).

What is the Octal Number System?

“OCT” stands for “eight.” The octal number system is a base-8 number system, which means that any number in the octal system requires 8 separate symbols to represent. 0, 1, 2, 3, 4, 5, 6, and 7 are the symbols. In this system, the smallest two-digit number is \((10)_8\), which is identical to decimal 8. The octal number system was widely utilized in early minicomputers. The number \((352)_8\) is written in this system as an example.

What is the Decimal Number System?

The decimal number system is also known as the base 10 numbers system because it uses ten digits from 0 to 9. The Chinese invented the decimal numbering system about 1350 B.C. Base-10 and denary are two other names for it. A decimal place value chart can be used to depict a value using decimal numbers.

How to Convert Decimal to Octal Number?

To convert a decimal number to an octal number, there are several direct and indirect methods. They are as given below:

• Decimal to Octal Conversion Using by Direct Method: As the name suggests decimal numbers are directly converted to octal numbers.
• Decimal to Octal Conversion Using by Indirect Method: This method converts the decimal number into a binary number or hexadecimal first and then converts that to an octal number.

Decimal to Octal Conversion Using Direct Method

To convert a Decimal number to an Octal number directly, you must start dividing the number by 8 until you get 0 as the quotient. This is a straightforward method that involves dividing the number to be converted.

Step 1: If the decimal number is N, divide it by 8 because the octal number system’s base is 8.

Step 2: Note the value of the residual, which will be one of the following: 0, 1, 2, 3, 4, 5, 6, or 7. Divide the remaining decimal number until it equals 0 and record the remainder of each step.

Step 3: Then, from bottom to top (or in reverse order), write the remainders, which will be the equivalent octal number of the provided decimal number.

Let’s see this with the help of an example:

Note: The dividend (here given decimal number) is the number to be divided, the divisor (here base of octal, i.e., 8) is the number to be divided by, and the quotient (remaining divided decimal number) is the outcome of the division.

Decimal to Octal Conversion Using Indirect Method

As mentioned above this method converts the decimal number into a binary number or hexadecimal first and then converts that binary or hexadecimal number to an octal number.

Convert Decimal to Binary to Octal

Let’s see how to convert decimal numbers to binary to octal conversion. By repeatedly dividing a number by two and recording the result, decimal values can be transformed into binary.

Conversion of Integral Decimal Numbers 

Step 1: Divide the number by 2.

Step 2: Get the integer quotient for the next iteration.

Step 3: Get the remainder for the binary digit.

Step 4: Repeat the above steps until the quotient is equal to 0.

Take a look at an example to see how this works.

The remainders are to be read from bottom to top to obtain the binary equivalent.

\(43_{10} = 101011_{2}\)

A binary number can be converted to an octal number in a variety of ways. Direct and indirect approaches can both be used to convert. To begin, you must convert a binary into a different base system (e.g., into decimal, or into hexadecimal). After that, you must convert it to an octal number.

Because the octal number system has only eight digits (from 0 to 7), we may express each octal digit using only three bits, as seen below.

The process to convert a binary number to an octal number is as follows:

Step 1: Consider the binary number. For the integer component, divide the binary digits into three groups (beginning from the right), and for the fraction part, start from the left.

Step 2: Each set of three binary digits should be converted to one octal digit.

Let’s see with the help of an example.

Convert binary number 1010111100 into an octal number.

Therefore, Binary to octal is

= (1010111100)

= (001 010 111 100)

= (1 2 7 4)

= (1274)

Convert Decimal to Hexadecimal to Octal

Let’s see how to convert decimal numbers to hexadecimal to octal.

Converting with Remainders (for the integer part)

This is a simple procedure that involves dividing the number to be transformed by two.

Step 1: If the decimal number is N, divide it by 16 because the hexadecimal number system’s base is 16. Make a note of the value of the remainder, which will range from 0 to 15 (replace 10, 11, 12, 13, 14, 15 with A, B, C, D, E, and F respectively). Divide the remaining decimal number until it equals 0 and record the remainder of each step. Then, from bottom to top (or in reverse order), write the remainders, which will be the equivalent hexadecimal number of the supplied decimal number.

Example:

Converting with Division

This approach works by estimating a decimal number’s hexadecimal equivalent. Any decimal number can be used as a starting point. Make a list of 16’s abilities. Multiply the decimal number by the 16th power. Find the rest of the items. Multiply the residual by the 16th power. Repeat until you’ve figured out the whole solution.

Example:

Now we will convert the hexadecimal number to octal number. Here are the steps:

Step 1: To begin, count the digits in the number.

Step 2: If n is the digit’s position from the right end, multiply each digit by \(16^{n-1}\).

Step 3: After you’ve multiplied the terms, add them together. The comparable decimal form is the resultant.

Step 4: Write down the rest of the information.

Step 5: With the quotient, repeat the previous two steps until the quotient is zero. Reverse the order of the remainder. The obtained number corresponds to the desired outcome.

Example:

Convert Decimal to Octal including Decimal Point

To convert Decimal to Octal including decimal points, we first convert it to hexadecimal and then convert it to octal.

Here’s how we convert to hexadecimal.

Converting with Remainders (for fractional part)

If the decimal fractional portion is M, multiply it by 16 because the hexadecimal number system’s base is 16. Take note of the integer part’s value, which will range from 0 to 15. (replace 10, 11, 12, 13, 14, and 15 by A, B, C, D, E, and F respectively). Multiply the remaining decimal fractional number until it equals 0 and record each integer part of the result. Then write the integer part’s results, which will be a fraction hexadecimal number comparable to the specified decimal value.

Once we get the hexadecimal number we convert it to octal using the steps we discussed above.

This table shows how regular (decimal) numbers are written in octal (base-8) form.

Solved Examples of Decimal to Octal Conversion

Let’s see some more solved examples from decimal to Octal.

Example 1: Convert 440 (base 10) to octal

Solution: To convert 440 from decimal to octal, divide the number repeatedly by 8 and record the remainders.

440 ÷ 8 = 55, remainder = 0

55 ÷ 8 = 6, remainder = 7

6 ÷ 8 = 0, remainder = 6

Now, read the remainders in reverse order: 6 7 0

So,

440₁₀ = 670₈

Example 2: Convert 4321.356 (base 10) to octal (up to 6 decimal places)

Step 1: Convert the whole number part (4321)

4321 ÷ 8 = 540, remainder = 1

540 ÷ 8 = 67, remainder = 4

67 ÷ 8 = 8, remainder = 3

8 ÷ 8 = 1, remainder = 0

1 ÷ 8 = 0, remainder = 1

Reading remainders in reverse: 1 0 3 4 1

So,

4321₁₀ = 10341₈

Step 2: Convert the fractional part (0.356)

Multiply the fractional part by 8 and take the integer part each time:

0.356 × 8 = 2.848 → 2

0.848 × 8 = 6.784 → 6

0.784 × 8 = 6.272 → 6

0.272 × 8 = 2.176 → 2

0.176 × 8 = 1.408 → 1

0.408 × 8 = 3.264 → 3

So,

0.356₁₀ ≈ 0.266213₈

Final Answer: 4321.356₁₀ = 10341.266213₈.

Example 3: Convert the decimal number 75 to an octal number.

Solution:
To change a decimal number to octal, divide the number by 8 until the quotient becomes 0. Write down the remainders and read them from bottom to top.

Now read the remainders from bottom to top: 1 1 3

So, (75)₁₀ = (113)₈

I hope this article on Decimal to Octal Conversion was informative. Get some practice of the same on our free Testbook App. Download Now!