[Maths Class Notes] on Binary to Octal Conversion Pdf for Exam

A number system is a way of representing the numbers. We have studied different types of numbers, like real numbers, whole numbers, rational numbers, etc. But in digital electronics and computer systems, we used other systems of numbers. There are four different types of number systems. They are given below:

• Binary number system which has the base 2.

• Octal number system which has the base 8.

• Decimal number system which has the base 10.

• hexadecimal number system which has the base 16.

Computers understand only binary numbers in the form of bits and bytes. whereas the octal number system is used in electronics. To convert the binary to octal numbers, first, we have to convert binary numbers to decimal numbers, then decimal numbers to octal numbers. Before studying how to convert binary to octal, let us understand what are binary numbers and what are octal numbers.

What are Binary Numbers?

Computers can understand only two conditions “On” and “Off” i.e. 1 and 0. The binary number system deals with the study of 0s and 1s.

A binary number system is a method of representing the number with the base 2, it uses the digits 1 and 0. As it uses only two digits 0 and 1 and has a base of 2, it is called binary.

All digital devices use a binary number system in their electronic circuit. The input 0 indicates the “OFF” state and the input 1 indicates the “ON” state. Because of these implementations, binary number systems are most widely used in modern computer technology.

Each digit is referred to as a bit.

There are no 2, 3, 4, 5, 6, 7, 8, 9 in the binary number system.

Examples of the binary number system are given below:

What are Octal Numbers?

The octal number system is the numbers with the base 8 and uses digits from 0 to 7, the digits 8 and 9 are not used in the octal number system. The octal number system is also called the base 8 number system.

The octal number system is generally used in a minicomputer.

Examples of octal number system are given below:

As we are quite familiar with the binary number system and octal number system,  now let us study how to convert binary to octal? and binary to octal conversion examples.

How to Convert Binary to Octal?

The computer doesn’t understand the octal number system. So, the additional circuit is required for the digital systems which convert the octal to binary number. So now, let us study conversion from binary to octal steps and solve binary to octal conversion examples. Binary to octal numbers can be converted into two methods.

Method 1: Converting binary to decimal then to decimal to octal.

Conversion from binary to octal steps are as follows:

• First, recognize if the given is a binary number.

• Then binary number to the decimal number system.

• Multiply each digit by 2^n-1, where n is the total number of digits.

• We will get the decimal number for the given binary number.

• Divide the decimal number by 8.

• Write down the remainder.

• Continue the above two steps with the quotient till the quotient is zero.

• Write the remainder in the reverse order.

• The result is the required octal number for the given binary number.

Method 2: Converting binary to octal by grouping.

Conversion from binary to octal steps are as follows:

• First, recognize if the number is binary or not. The numbers with 0 and 1 with the base 2 are binary numbers.

• Group all the 0 and 1 in the binary numbers in a set of three starting from the right side (Least Significant Bit: LSB).

• Add 0’s to the left (Most Significant Bit: MSB) if it doesn’t form a group of three. Make sure each group must have three numbers.

• Now write the octal equivalent number for each group and the result is the number in the octal number system.

Below image shows how to convert binary to octal examples.

Following this conversion from binary to octal steps, you can easily convert any binary number to an octal number. Now let us solve these binary to octal conversion examples, which will make the concept more clear.

Binary to Octal Conversion Examples

Example 1: Convert the binary number 1010101₂ to the octal number system.

Solution:

We will be using the first method of conversion of binary to octal.

The binary number is 10101012.

First, we will convert given binary number to decimal number.

(1010101)2

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

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

= 85

(010101)₂= 85₁₀

• Now we will convert this decimal number to an octal number.

• Let us divid
  e 85 by 8, we will get the remainder 5 and quotient 10.

• Again divide 10 by 8, we will get remainder 2 and quotient 1.

• Again divide 1 by 8, we will get remainder 1 and quotient 0.

• Now collecting the remainders in reverse order, we get 1, 2, and 5.

Therefore, the equivalent octal number is 125₈.

Example 2:  Convert binary number 1010111100₂ into an octal number.

Solution:

We will be using method 2 for the conversion of binary to octal.

Let us group the binary numbers into a set of three.

Therefore, binary to octal is.

(1010111100)₂

Making a group of three

=  1 010 111 100

Adding two 0 to complete the set

= 001 010 111 100

Now write the octal equivalent number

=   1    2     7     4

So, (1010111100)₂= (1274)₈.

Practise some more binary to octal conversion examples which are given below.

Convert Binary to Octal Examples

Convert the following binary numbers to octal number system.

• 1010102

• 1100112

• 0001112

• 111001112