Octal Number System has a base of eight and uses the number from 0 to 7. The octal numbers, in the number system, are usually represented by binary numbers when they are grouped in pairs of three. For example, 12₈ is expressed as 001010₂, where 1 is equivalent to 001 and 2 is equivalent to 010.

Apart from octal number system, there are other number systems in Maths, such as:

• Binary Number System
• Hexadecimal Number System
• Decimal Number System

Definition

A number system which has its base as ‘eight’ is called an Octal number system. It uses numbers from 0 to 7. Let us take an example, to understand the concept. As we said, any number with base 8 is an octal number like 24₈, 109₈, 55₈, etc.

Like Octal number is represented with base 8, in the same way, a binary number is represented with base 2, decimal number with base 10 and the hexadecimal number is represented with base 16. Examples for these number systems are:

• 22₂ is a binary number
• 100₁₀ is a decimal number
• 40₁₆ is a hexadecimal number

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

• 124₈= 1 × 8² + 2 × 8¹ + 4 × 8⁰

Octal Numbers System Table

We use only 3 bits to represent Octal Numbers. Each group will have a distinct value between 000 and 111.

Note: Octal number system supports only digits from 0 to 7. Beyond 7, such as 8 and 9 are not octal digits. For example, 19 is not an octal number by 10 is an octal number.

Decimal to Octal Number

To convert decimal to octal number, octal dabble method is used. In this method, the decimal number is divided by 8 each time, it yields or gives a remainder. The first remainder we get is the least significant digit(LSD) and the last remainder is the most significant digit(MSD). Let us understand the conversion with the help example.

Solved Example

Problem: Suppose 560 is a decimal number. Convert it into an octal number.

Solution: If 560 is a decimal number, then,

560/8=70 and remainder is 0

70/8=8 and remainder is 6

8/8=1 and remainder is 0

And 1/8=0 and remainder is 1

So the octal number starts from MSD to LSD, i.e. 1060

Therefore, 560₁₀ = 1060₈

Problem: Convert 0.52 into an octal number.

Solution: The fraction part of the decimal number has to be multiplied by 8.

0.52 × 8=0.16 with carry 4

0.16 × 8=0.28 with carry 1

0.28 × 8=0.24 with carry 2

0.24 × 8=0.92 with carry 1

So, for the fractional octal number, we read the generated carry from up to down.

Therefore, 4121 is the octal number.

Octal to Decimal

To convert an octal number to decimal number we need to multiply each digit of the given octal with the reducing power of 8. 

Let us learn here, conversion of Octal number to Decimal Number or base 8 to base 10.

Solved Examples

Example 1: Suppose 215₈ is an octal number, then it’s decimal form will be,

215₈ = 2 × 8² + 1 × 8¹ + 5 × 8⁰

          = 2 × 64+ 1 × 8 + 5 × 1 = 128 + 8 + 5

          =  141₁₀

Example 2: Let 125 is an octal number denoted by 125₈. Find the decimal number.

125₈ = 1× 8² + 2 × 8¹ + 5 × 8⁰

          = 1 × 64 + 2 × 8 + 5 × 1 = 64+16+5

          =85₁₀

Binary To Octal Number

A binary number can be converted into an octal number, with the help of the below-given table.

Solved Example

Example: Convert (100010)₂ to octal number.

Solution: With the help of the table we can write,

100→4

and 010→2

Therefore,(100010)₂ = 42

Similarly, we can convert an octal number to binary number with the help of the table.

Octal to Hexadecimal Number

Hexadecimal numbers consist of numbers and alphabets. It is represented with base 16. 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 the octal number to hexadecimal requires two steps.

• First, convert octal number to decimal number.
• Then, convert decimal number to hexadecimal number.

Let us understood with the help of an example. We will take the same example, where we have converted octal number to decimal, such as;

(55)₈ =  (45)₁₀

Now, convert (45)₁₀ into a hexadecimal number by dividing 45 by 16 until you get remainder less than 16.

Therefore, we can write, (45)₁₀ =  (2D)₁₆

Or (55)₈ = (2D)₁₆

Octal Multiplication Table

Applications

The octal Number system is widely used in computer application sectors and also in the aviation sector to use the number in the form of code.

Based on octal number system applications, several computing systems are developed. All the modern generation computing system uses 16-bit, 32-bit or 64-bit word which is further divided into 8-bit words. Similarly, for various programming languages, octal numbers are used to do coding or to write the encrypted language, which is only understood by the computing machine.

Also in the aviation sector or field or say aviation industry, Transponders used in the aircraft transmits a code which is expressed as four octal digit number. These codes are interrogated by ground radar.

Importance

Octal number system uses less digits (3-bits) than hexadecimal numbers (4-bits), which is one of the advantages. It is therefore, there will be less computations and possibility of occurrence of error will degrade.

Because of less digits, it is also easy to convert octal to any other number system and vice-versa. 

One of the disadvantages is that computers do not understand the octal numbers in a direct way and hence it has to be converted into binary numbers first.

Problems and Solutions

Q.1: Convert 275₈ to decimal number.

Solution: 2 x 8² + 7 x 8¹ + 5 x 8⁰

= 64×2 + 8×7 + 1×5

= 128 + 56 + 5 

= 189

Therefore, 275₈ = 189₁₀

Q.2: Convert decimal number 139 into equivalent octal number.

Solution: We need to divide 139 by 8 repeatedly, till the quotient is 0.

Now take the remainders from down to up to get the equivalent octal number.

139₁₀ = 213₈

Q.3: Convert the octal number 540 into binary number.

Solution: Given, 540 is the octal number

By the octal number table, we can get the binary equivalent of each digit.

5 → 101

4 → 100

0 → 000

Therefore,

540₈ = 101100000₂

Practice Questions

Convert the following octal numbers into its equivalent decimal number.

• 23₈ (Answer: 19₁₀)
• 770₈ (Answer: 504₁₀)
• 152₈ (Answer: 106₁₀)

Convert the following octal numbers into hexadecimal numbers.

• 23₈ (Answer: 13₁₆)
• 770₈ (Answer: 1F8₁₆)
• 152₈ (Answer: 6A₁₆)

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