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# Binary to Octal conversion

## Octal number system

The octal number system consists of eight digits from 0 to 7. Numbers 8 and 9 are excluded from the octal number system. It is denoted by A8, where A is a number from 0 to 7. For example, 2348, 7648, 32678, etc., are some numbers represented in the octal number system.

## Binary number system

In a binary number system, two numbers are used to represent everything. 0 and 1 are the primary bases of the binary number system. It is denoted by A2, where A is 0 or 1. For example, 0101012, 1110002, etc., are numbers represented in a binary number system. These are commonly used in computers and electronic devices.

## Conversion of binary to octal number system

Method 1

We cannot convert binary to octal directly. We have to convert the binary number to decimal and then that decimal to the octal number system. We need to perform the following steps to convert binary to octal number system-

1. We need to take the binary digit and multiply each digit by 2n-1, where n is the digit's position from the decimal.
2. The number that we get is the result of the binary number that we took at the start.
3. Divide the whole number 8, and note the remainder on the right side as we do in the prime factorization method.
4. Continue the steps till we get the remainder 0.
5. Now, write the remainder in the reverse order to get the required octal number. The final result is the answer for binary to octal number.

## Example

Convert 1010101₂ to octal number system

## Solution

We have the binary number as 10101012. We need to convert it to a decimal number first.

We need to raise it to the power of 2n-1.

10101012 = (1 * 26) + (0 * 25 ) + (1 * 24) + (0 * 23) + (1 * 22) + (0 * 21) + (1 * 20)

= 64 + 0 + 16 + 0 + 4 + 0 + 1

= 64 + 21 = 85

This 85 is the required decimal form.

Now, we need to divide 85 by 8 and write the corresponding number on the right hand side. We get,

We need to write the remainder from bottom to top to get the required octal number. Therefore, the required octal number is 1258.

Method 2

We can convert binary to octal by grouping the numbers. Perform the following steps to convert the number system by another method-

• Recognize whether the number is of binary number system or not.
• If yes, then group them in triplets starting from the right-hand side.
• If the triplet remains incomplete, add 0s to complete the set.
• Now, from the table below, write the corresponding octal number.
• The final result is the required octal number system from the binary number system.
 Octal Number Binary Number 0 0 1 01 2 010 3 011 4 100 5 101 6 110 7 111

## Example

Convert 1010111100₂ to octal number system

## Solution

We need to make pairs of these binary numbers in triplets, starting from the right hand side.

We get, 1 010 111 100. We need to add 0s to the left to complete the left over triplet.

We get, 001 010 111 100

From the table above, we will get the corresponding octal number for these binary numbers.

Therefore, the required octal number is 1274₈.

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