Binary to octal conversion is a topic commonly found in number system questions, especially in exams. To convert binary numbers into octal, we use a simple method that involves grouping binary digits, not multiplying by 8. Each group of three binary digits (from right to left) is replaced with its equivalent octal digit.

Binary Numbers
Binary numbers use only two digits: 0 and 1. This system is called base-2. Computers use binary to store and process all types of information like text, images, audio, and video. Since there is no “2” in this system, adding 1 + 1 gives 10 in binary.

Octal Numbers
Octal numbers use base-8 and include digits from 0 to 7. In octal, each digit represents a value from 0 to 7. It is often used in computing as a shorter way to write long binary numbers.

Binary to Octal Conversion

The Octal number system helps make large binary numbers easier to read and work with. Since binary numbers can be very long, converting them into octal form makes them shorter and simpler. This is because one octal digit can represent three binary digits. So instead of writing many 0s and 1s, we can group them in sets of three and replace each group with one octal digit. There are a few easy methods to do this, and it’s a quick way to handle binary numbers in a more compact form.

Binary to Octal Conversion using Direct and Indirect Method

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.

Numbers are a form of the positional numbering system. That is, for the integer part, the weight of the positions from right to left is \(8^{0}\), \(8^{1}\), \(8^{2}\), \(8^{3}\) and so on, and for the fractional part, the weight of the positions from left to right is \(8^{-1}\), \(8^{-2}\), \(8^{-3}\) and so on.

There is a direct method to transform a binary number into an octal number grouping.

Binary to Octal Conversion using Grouping Method

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 in the table below.

Steps to Convert Binary to Octal Number

The step-by-step procedure to convert binary to octal numbers is as follows:

Step 1: Consider the binary number.

Step 2: For the integer component, divide the binary digits into three groups (beginning from the right)

Step 3: For the fraction part, start from the left.

Step 4: Each set of three binary digits should be converted to one octal digit.

Properties of Binary to Octal Conversion

1. Base Relationship:
   Binary numbers have base 2 (only digits 0 and 1), and octal numbers have base 8 (digits 0 to 7). Since 8 is 23, each octal digit represents exactly 3 binary digits.
2. Grouping of Digits:
   To convert binary to octal, digits are grouped in sets of three from right to left (for whole numbers). If necessary, zeros are added to the left to make complete groups.
3. Compact Representation:
   Octal numbers are shorter and easier to read than binary numbers. This makes them useful when dealing with long binary sequences, especially in computing.
4. No Decimal Point Loss:
   Conversion between binary and octal is exact — there is no rounding or approximation.
5. Faster Calculation:
   Since each octal digit stands for 3 binary digits, the conversion process is quick and reliable for manual calculations.

Widely Used in Programming:
Octal numbers are often used in computer programming, especially in file permissions (e.g., Unix systems).

Example 1: Convert binary number 1010111100 into an octal number.

Therefore, Binary to octal is.

= (1010111100)

= (001 010 111 100)

= (1 2 7 4)

= (1274)

Example 2: Convert binary number \(0110 011.1011\) into octal number.

Solution: Since there is a binary point here and fractional part. So,

Now, using the grouping method, set the binary number into three numbers in each group

Binary to octal conversion of \((0110011.1011)_{2}\): 

\((0110011.1011)_{2}\)

\(=(\underbrace{0}\underbrace{110}\underbrace{011}.\underbrace{101}\underbrace{1})_{2}\) 

\(=(\underbrace{110}\underbrace{011}.\underbrace{101}\underbrace{100})_{2}\)

\(=(\underbrace{6}\underbrace{3}.\underbrace{5}\underbrace{4})_{8}\)

\(=(63.54)_{8}\)

Therefore, the binary to octal conversion of \((0110011.1011)_{2}\) is \((63.54)_{8}\).

Example 3: Convert binary number \(01110101\) into octal number.

Solution: Using the grouping method, set the binary number into three numbers in each group. 

Binary to octal conversion of \((01110101)_{2}\):

\((01110101)_{2}\)

\(=(\underbrace{001}\underbrace{110}\underbrace{101})_{2}\)

\(=(\underbrace{1}\underbrace{6}\underbrace{5})_{8}\)

\(=(165)_{8}\)

Therefore, the binary to octal conversion of \((01110101)_{2}\) is \((165)_{8}\).

Example 4: Convert binary number \(001101\) into octal number.

Solution: Using the grouping method, set the binary number into three numbers in each group. 

Binary to octal conversion of \((001101)_{2}\):

\((001101)_{2}\)

\(=(\underbrace{001}\underbrace{101})_{2}\)

\(=(\underbrace{1}\underbrace{5})_{8}\)

\(=(15)_{8}\)

Therefore, the binary to octal conversion of \((001101)_{2}\) is \((15)_{8}\).

Example 5: Convert binary number \(101010011\) into octal number.

Solution: Using the grouping method, set the binary number into three numbers in each group. 

Binary to octal conversion of \((101010011)_{2}\):

\((101010011)_{2}\)

\(=(\underbrace{101}\underbrace{010}\underbrace{011})_{2}\)

\(=(\underbrace{5}\underbrace{2}\underbrace{3})_{8}\)

\(=(523)_{8}\)

Therefore, the binary to octal conversion of \((101010011)_{2}\) is \((523)_{8}\).

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