Octal Number System

Octal Number System is a type of number system that has a base of eight and uses digits from 0 to 7. A number system is a system of naming, representing, or expressing numbers in different types of forms. The basic ways of representing numbers are done in four ways i.e. Octal Number System, Binary Number System, Decimal Number System, and Hexadecimal Number System.

Types of Number System

Definition of Octal Number System

A number system with its base as eight and uses digits from 0 to 7 is called Octal Number System. The word octal is used to represent the numbers that have eight as the base. The octal numbers have many applications and importance such as it is used in computers and digital numbering systems. In the number system, octal numbers can be converted to binary numbers, binary numbers to octal numbers by first converting a binary number to decimal number then decimal number to octal number.

Similar to the octal number system, the binary number system is represented by the base 2, the decimal number system is represented by the base 10 and the hexadecimal number system is represented by the base 16. A few of the examples of these number systems are:

$(10)_{2}$ is a binary number
$(119)_{10}$ is a decimal number
$(51)_{6}$ is a hexadecimal number

While solving an octal number, each place is a power of eight. For example: $(347)_{8}$ = 3 x 8² + 4 x 8¹ + 7 x 8⁰

Conversion from Octal to Binary Numbers

For the process of conversion, we need to convert each number from the octal number to the binary number. Every digit has to be converted to a 3-bit binary number and hence arriving at the binary equivalent of the octal number. Below is a table representation of the binary numbers to the octal numbers and vice versa.

Conversion table for Octal and Binary Number System

Example 1 - Convert $(14)_{8}$ into a binary number

Solution - Given $(14)_{8}$ is an octal number, with the help of the above table we can write $(14)_{8}$ = $(001100)_{2}$ . Zeros on the left do not have any significance. Hence, $(14)_{8}$ = $(001100)_{2}$ .

Example 2 - Convert $(11100101)_{2}$ into an octal number.

Solution - Given $(11100101)_{2}$ is a binary number, with the help of the above table we first write the number into its 3-bit binary number as a zero needs to be added before digits to form the 3-bit binary number. So, the number can be written as $(011100101)_{2}$ . Hence, the 3-bit binary number is 011, 100, 101. Looking at the same table above we can convert these binary numbers to their octal numbers to derive the final number. Hence, the numbers are 3, 4, 5

Therefore, $(11100101)_{2}$ = $(345)_{8}$

Conversion from Octal to Decimal number

The conversion of octal numbers to decimals numbers is done in a simple way. The number is expanded with the base of eight where each number is multiplied with the reducing power of 8. The decimal number system has a base of 10 after the conversion.

For example - Convert octal number $(121)_{8}$ to its decimal form.

Solution - $(121)_{8}$ = 1 x 8² + 2 x 8¹ + 1 x 8⁰

= 1 x 64 + 2 x 8 + 1 x 1

= 64 + 16 + 1

Therefore, $(121)_{8}$ = $(81)_{10}$

Conversion from Decimal to Octal Number

To convert decimal to octal number, a different method is used. In this method, the decimal number is divided by 8 each time a reminder is obtained from the previous digit. The first remainder obtained is the least significant digit(LSD) and the last remainder is the most significant digit(MSD). Let us understand the conversion with the help of an example.

For example - Convert the decimal number 321 to its octal form.

Solution - We need to start dividing the number 321 by 8

321/8 gives quotient 40 and the remainder is 1

40/8 gives quotient 5 and the remainder is 0

So, here quotient is 5 and the remainder is 0. The octal number starts from MSD to LSD, i.e 501

Therefore, $(321)_{10}$ = $(501)_{8}$

Conversion from Octal to Hexadecimal Numbers

Hexadecimal is represented with base 16 and consists of both numbers and alphabets. The numbers from 0-9 are represented in the usual form, but from 10 to 15, it is denoted as A, B, C, D, E, F. Conversion of Octal to Hexadecimal is done in two steps i.e. first convert the octal number to decimal number and then convert it to a hexadecimal number. Let us look at an example to understand this method better.

For example - $(121)_{8}$ = $(81)_{10}$

Solution - We already have the decimal number 8110, so we only need to convert this to a hexadecimal number. To determine the hexadecimal number we need to divide the number 81 by 16 until the remainder is less than 16. It is completely divisible with the answer as 5 and the remainder as 1.

Therefore, $(121)_{8}$ = $(51)_{16}$