Octal Number System

Octal Number System has a base of eight and uses the number from 0 to 7. It is a classification of number system apart from Binary Numbers System, Decimal Numbers System, Hexadecimal Number System.

In Mathematics, you must have learned about the Number system, which is a way of expressing numbers or numerals. They are Natural Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers and Prime Numbers.

The octal symbol is used to represent the numbers with base 8.

 

Octal Number System
Base – 8
Octal Symbol – 0, 1, 2, 3, 4, 5, 6 and 7.

The octal number has various applications and importance. It is commonly used in computer basics. Here, in this article, we will learn about conversion octal number to decimal number. Binary to octal conversion is also an easy method, where we will first convert binary to decimal and decimal to octal.  Let us discuss octal number with its definition, uses, and application.

Octal 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

Representation of Octal Numbers

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

Octal Digital Value

Binary Equivalent
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111

Octal to Decimal Number System Conversion

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

Example: Suppose 215is an octal number, then it’s decimal form will be,

2158 = 2 × 82 + 1 × 81 + 5 × 80

          = 2 × 16 + 1 × 8 + 5 × 1

          =  4510

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

1258 = 1× 82 + 2 × 81 + 5 × 80

          = 1 × 16 + 2 × 8 + 5 × 1

          =3710

Decimal to Octal Number System Conversion

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;

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 8/0=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.

Binary To Octal Number System Conversion

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

Octal Number

Equivalent Binary Number

0

0

1

1

2

10

3

11
4

100

5

101

6

110

7

111

 

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 System

Hexadecimal number 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;

(215)8 =  (45)10

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

Decimal to Hexadecimal Number

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

Or (215)8 = (2D)16

Octal Multiplication Table

* 0 1 2 3 4 5 6 7
0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7
2 0 2 4 6 10 12 14 16
3 0 3 6 11 14 17 22 25
4 0 4 10 14 20 24 30 34
5 0 5 12 17 24 31 36 43
6 0 6 14 22 30 36 44 52
7 0 7 16 25 34 43 52 61

Uses and Applications of Octal Number System

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.

Also, study-related topics on number systems by downloading BYJU’S -The Learning App.

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