**Introduction**

In the previous section, we already learned about Decimal, binary and Hexa decimal number systems. This number system is very similar to that of Hexa decimal system. We know that the decimal system has base of 10 , as it uses the digits 0 – 9, the base of binary system is 2, as it uses the digits 0 & 1 and the hexa decimal number system has base 16, so that number system uses 16 digits i.e. from 0 – 15. Similarly, the “Octal number system” uses only 8 numbers to represent the numbers, so it has the name “Octal”. (0-7).

**Octal Number System**

In the hexa decimal number system, we represent the binary digits as a set of 4 digits (2^{4} = 16), in octal numbering system we represent the binary numbers as a set of 3 digits (2^{3}= 8). The octal number system uses 8 numbers from 0 – 7. They are 0, 1, 2, 3, 4, 5, 6 and 7.

So each digit of octal number is formed from 0 to 7 digits in them. The main advantage of the octal number system compared to other number systems is that , it is more easy to write the number in octal number form than to write in binary number system, when we are working with computers. Especially, when we are working with large string of binary numbers, it is suggested to group them as set of three digits hence it has less chance to occur error. Other advantage of octal number system is conversion of octal to binary and binary to octal number system is very simple compared to other conversions.

This number system has the base of 8 in their representation. Ex: (501)_{8} , (480)_{8}

Weight of the value of a digit will increase with power of 8. It is shown below.

Let’s see an example to understand the representation of octal number as a set of 3 digits.

So the number 100011010 is represented in octal as (432)_{8}.

**Conversion of Octal Numbers**

Octal numbers can be converted into binary and decimal number systems and also to Hexa decimal numbers.Some of them are explained below.

**Conversion From Binary to Octal Numbers**

To convert a binary number to octal number, first we should divide the binary string into set of 3 binary numbers each. Writing the corresponding number to each set will give the octal number of the binary.

Ex 1: convert 110111100010 to octal.

Dividing the binary number into set of 3 digits

110 111 100 010

6 7 4 2

(110111100010)_{2} is equal to (6742)_{8}

**Conversion From Octal Numbers to Binary**

Converting of octal numbers into binary is the reverse process of binary to octal conversion. That is each digit of the octal number should be written in its binary form & combining all the binary digits will result in our required binary number.

##### Ex 1:

Convert (43628)_{8} into Binary

Writing the equivalent binary number to each digit

4 3 6 2 8

100 011 110 010 100

So (43628)_{8} is equal to (100011110010100)_{2}

**Conversion From Decimal to Octal Numbers**

A decimal number can be converted to octal number by repeated division by 8 method. The reminder at each stage will give the required octal number.

Observe the example shown below.

##### Ex 1:

Convert (159)_{10} into Octal.

159/8 ————-> Quotient 19 Reminder 7—–LSB

19/8 ————-> Quotient 2 Reminder 3

2/8 ————-> Quotient 0 Reminder 2——MSB

So (159)_{10} = (237)8

##### Ex 2:

Convert (80)_{10} into Octal.

80/8 ————-> Quotient 9 Reminder 8—–LSB

9/8 ————-> Quotient 1 Reminder 1

1/8 ————-> Quotient 0 Reminder 1——MSB

So (80)_{10} = (118)_{8}

**Conversion From Octal Numbers to Decimal**

Octal numbers can be converted to decimal numbers by multiplying each digit by its position value. That means each digit is multiplied by power of 8 with its position.

Let’s see an example

##### Ex 1:

convert (51)_{8} to decimal

Position weight 8^{1} 8^{0}

Position value 8 1

Octal number 5 1

Equivalent decimal number = 5 x 81 + 1 x 80

= 40 + 1

= 41

Therefore (51)_{8 }= (41)_{10}

Similarly one can convert the octal number to any other number system.Below given table shows the equivalent values to other number systems.

**Representation of an Octal Number**

The octal numbers are represented with base 8, because they use only 8 digits, as explained above.Weight of each bit in an octal number is shown below.

Octal numbers are represented similarly to other number systems .In the set of octal number system given below

10 means not Ten, I t means {(1×8) + (0×8) } & 20 mean not Twenty, it means {( 2×8) + (0×8)} and so on.

**Summary**

We will represent 3 binary digits to equivalent of 1 octal digit as shown above. In the same way, the highest two digit octal-number (77_{8}) can represent 63 binary digits. Similarly, the highest three digit octal-number (777_{8}) can represent 511 binary digits. The highest three digit octal-number (7777_{8}) can represent 4095 binary digits.

- The octal number system uses 8 numbers from 0 – 7. (0, 1, 2, 3, 4, 5, 6 and 7)
- In octal number system the weight of the value of a digit will increase with power of 8.
- The decimal number can be converted to octal number by repeated division by 8 method .

Debarshi Das says

Nice article. Clear and easy to understand.

Rem97 says

Hey, I think your Binary colum of “Conversion From Octal Numbers to Decimal” isn’t completely right. I was calculating 1111 quick and found 15 instead of 17. I think I might want to up the bit by two since the Octal system skips those as well 😉

Please let me know if I see it wrong.

Kind regards.

Karthik says

Hey, The example used in conversion of octal to binary is wrong as the octal system omits usage of 8. Please correct it. Thanks!