Octal Number System

By the name “Octal number system” we can understand that it is a type of a number system. So before knowing the octal number system let us first know what is a number system? A number system, also known as a numeral system is the system of naming or representing or expressing numbers. In mathematics, we can name or represent numbers in various different forms. Four of the basic ways to represent numbers are the binary number system, decimal number system, hexadecimal number system, and octal number system. Now, let us know the entire concepts of the numeral system along with their types, conversions, and examples.

What Is A Number System?

A number system is a way or a system of writing that we use to express numbers. It is a mathematical notation used for the representation of numbers of a given set by using digits or other symbols in a logical manner. The number system allows us to represent every number in a unique way. It represents the arithmetic and algebraic structure of the figures. Not only that but it also lets us perform arithmetic operations like addition, subtraction, and division. 


The value of a digit in a number is usually determined by:

  • The digit

  • The position of it in the number

  • The base of the number system

Types of Number System

In mathematics, we can represent numbers in various types but the four most basic number systems are:

  1. The decimal number system (Base- 10)

  2. The binary number system (Base- 2)

  3. The octal number system (Base-8)

  4. The hexadecimal number system (Base- 16)

Octal Number System

Octal Number System is a system that has a base of eight and uses the number from 0 to 7. It is one of the classifications of number systems apart from the Binary Numbers, the Decimal Numbers, and the Hexadecimal Numbers. The symbol of the octal is used to represent the numbers that have base 8. There are various applications and importance of octal numbers. One of the most common uses of it is in computer basics. We can convert the octal numbers to decimal numbers, Binary numbers to octal numbers where we first have to convert a binary number to a decimal number and a decimal number to octal number. So, let us first discuss the octal number with its definition, table, example, and application.

What Is An Octal Number System?

A number system with its base as ‘eight’ is known as an Octal number system and uses numbers from 0 to 7 i.e., 0, 1, 2, 3, 4, 5, 6, and 7. We can take an example, to understand the concept better. Like we already know, any number that has a base 8 is called an octal number like \[24_{8}\], \[109_{8}\], \[55_{8}\], etc.


Just like Octal numbers are represented with a base of 8, in the same way, a binary number is represented with a base of 2, a decimal number with a base of 10, and a hexadecimal number is represented with a base of 16. Here are a few examples for these number systems:

  • A binary number = \[22_{2}\]

  • A decimal number = \[100_{10}\]

  • A hexadecimal number = \[40_{16}\]

When we solve an octal number, each place is a power of eight.

  • \[124_{8}\]  = 1 × \[8^{2}\] + 2 × \[8^{1}\] + 4 × \[8^{0}\]


  etc … … … \[8^{3}\] \[8^{2}\] \[8^{1}\] \[8^{0}\]  .  \[8^{-1}\] \[8^{-2}\] \[8^{-3}\]... … etc

                                 ↑

                       Octal Point


To convert a number from octal to binary, the conversion has to be done by converting each number from the octal digit to the binary digit. Every digit has to be converted to a 3-bit binary number and the resultant will be the binary equivalent of an octal number.

Conversion From Octal To Binary 

                    Octal Symbol

                  Binary Equivalent

                              0

                                000

                              1

                                001

                              2

                                010

                              3

                                011

                              4

                                100

                              5

                                101

                              6

                                110

                              7

                                111



Example 1) Convert \[(145056)_{8}\] to binary.


Solution 1) To convert from octal to binary and vice versa we will need this conversion table. According to the table, the octal value \[(145056)_{8}\] can be converted to binary as

\[(001\: 100\: 101\: . 101\: 110)_{2}\]

Conversion From Binary To Octal

We can use the same table to convert a binary number to octal number. And for that, we first have to group the binary number into a group of three bits and write the octal equivalent of it.

Example 2) Convert the binary number \[(11001111)_{2}\] to octal  


Solution 2) The three bit group of binary numbers can be written as 011,001,111 because we have to add a zero before each number to complete the grouping in the form of three binary digits. Therefore, the octal numbers will be 3, 1, 7 i.e., \[(317)_{8}\]


Conversion From Octal To Decimal

The method that we use to convert an octal number into its decimal equivalent is extremely simple. We just need to expand the number in the base of eight with its positional weight and done! The resultant value will be a decimal number.

Example 3) Convert \[(317)_{8}\] into decimal. 


Solution 3) The conversion from octal to decimal can be done in the following way:

 

\[(317)_{8}\] = 3 × \[8^{2}\] + 1  × \[8^{1}\] + 7 × \[8^{0}\] = 207


Conversion From Decimal To Octal

We can convert a number from decimal to octal by dividing the number by 8 using a repeated division method that is usually known as the double dabble method. We do the repeated division and find the remainder. Here is an example.


Example 4) Convert the decimal number 158 into octal. 


Solution 4) We have to divide the number 158 by 8 like this:

                                    

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Therefore, in octal, the value will be \[236_{8}\]

FAQ (Frequently Asked Questions)

1. What are the advantages and disadvantages of the octal number system?

The advantages of the octal system are: 

  1. The octal number system is almost one-third of the binary number system. 

  2. The process of conversion from binary to octal and vice versa is extremely simple. 

  3. In the octal form, it is easier to handle the input and the output. 

The disadvantage of the octal number system is that the computer does not understand the octal number system so there has to be a requirement of an additional circuitry known as octal to binary converters before we apply it to a digital system or a computer.


2. What is the difference between decimal and hexadecimal number systems?

The major difference between decimal value and hexadecimal value is their base. The decimal number system uses the base 10 whereas the hexadecimal number uses 16 as its base. The elements of the decimal number system are:

(1, 2, 3, 4, 5, 6, 7, 8, 9, 0) and the elements of a hexadecimal number system are:

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, f). The decimal number system is also known as the Hindu-Arabic or Arabic number system. The other two number systems are the binary number system and the octal number system.