Octal Number System is a number system with base 8 as it uses eight symbols (or digits) namely 0, 1, 2, 3, 4, 5, 6, and 7. For example, 228, 138, 178, etc. are octal numbers. This number system is mainly used in computer programming as it is a compact way of representing binary numbers with each octal number corresponding to three binary digits.
In this article, we will discuss Octal Number System, Octal Number System Conversions, Octal Number System Examples, and Others in detail.
What is Number System?
A number system is a method of expressing numbers. It contains sets of symbols (or digits) combined with a set of rules to represent a particular quantity. The number system is mainly classified into four types:
What is an Octal Number System?
The octal number system is a base-8 system using digits 0-7, where each position represents a power of 8. It is commonly used in computing for easy conversion to binary.
'OCTAL' is derived from the Latin word 'OCT' which means Eight. The number system with base 8 and symbols ranging between 0-7 is known as the Octal Number System. Each digit of an octal number represents a power of 8. It is widely used in computer programming and digital systems. Octal number system can be converted to other number systems and visa versa.
For example, an octal number (10)8 is equivalent to 8 in the decimal number system, 001000 in the binary number system and 8 in the hexadecimal number system.
Octal Numbers System Table
The table added below, shows the Octal Number and Decimal Number. 3 bits of Binary Number System is equivalent to one octal numbers.
Octal Numbers | Binary Equivalent |
|---|
0 | 000 |
1 | 001 |
2 | 010 |
3 | 011 |
4 | 100 |
5 | 101 |
6 | 110 |
7 | 111 |
Now, we will learn about the conversion of octal number system to other number systems one by one. So let's get started.
Octal to Decimal Conversion
A decimal number system has a base 10 consisting of digits 0-9. We can easily convert an octal number to a decimal number by following these simple steps:
- Step 1: Write the octal number.
- Step 2: Multiply each digit of the given octal number with an increasing power of 8 starting from the rightmost digit.
- Step 3: Sum all the products obtained in step 2.
Example 1: Represent 1238 as a Decimal Number.
Solution:
1238 = 1 × 82 + 2 × 81 + 3 × 80
⇒ 1238 = 1 × 64 + 2 × 8 + 3 × 1
⇒ 1238 = 64 + 16 + 3
⇒ 1238 = 8310
Hence 8310 is decimal representation of 1238.
Decimal to Octal Conversion
To convert a decimal number to an octal number follow these simple steps:
Step 1: Divide the given decimal number by 8.
Step 2: Write down the quotient and remainder obtained.
Step 3: Divide the quotient obtained by 8.
Step 4: Repeat step 2 and step 3 until the quotient becomes 0.
Step 5: Write the obtained remainder in reverse order.
Let's Consider an example for better understanding.
Example 2: Represent 16410 as Octal Number.
Solution:
164/8 , Quotient = 20 and Remainder = 4
20/8 , Quotient = 2 and Remainder = 4
2/8 , Quotient = 0 and Remainder = 2
Now, By writing obtained remainders in reverse order we get, 244.
Hence 2448 is octal representation of 16410
The image added below shows binary to octal conversion.
Octal to Hexadecimal Conversion
A hexadecimal number system has a base 16 and it is an alphanumeric number system consisting of digits from 0 to 9 and alphabets from A to F. To convert an octal number to a hexadecimal number: First convert the octal number to the decimal number; Then convert the obtained decimal number to the hexadecimal number.
Steps to Convert Octal Number to Decimal Number
- Step 1: Write the octal number.
- Step 2: Multiply each digit of the given octal number with an increasing power of 8 starting from the rightmost digit.
- Step 3: Sum all the products obtained in step 2.
Steps to Convert Decimal Number to Hexadecimal Number
- Step 1: Divide the decimal number by 16.
- Step 2: Write down the quotient and remainder obtained.
- Step 3: Divide the quotient obtained by 16.
- Step 4: Repeat step 2 and step 3 until the quotient becomes 0.
- Step 5: Write the obtained remainder in reverse order.
- Step 6: Convert each obtained remainder to its corresponding hexadecimal digit.
Corresponding value of 0-9 remains the same in hexadecimal and 10-15 corresponds to A-F in hexadecimal that is represented as,
Example 3: Convert 1748 to a hexadecimal number.
Solution:
Step 1: Convert 1748 to decimal
1748 = 1 × 82 + 7 × 81 + 4 × 80
1748 = 1 × 64 + 7 × 8 + 4 × 1
1748 = 64 + 56 + 4 = 124
We get 1748 = 12410
Step 2: Covert 12410 to hexadecimal
124/16, Quotient = 7, Remainder = 12
7/16, Quotient = 0, Remainder = 7
Converting the obtained remainders to corresponding hexadecimal number and writing it in reverse order we get:
12410 = 7C16
Hence we get 1748 = 7C16
Hexadecimal to Octal Number
To convert a Hexadecimal number to an Octal number we have to First convert the Hexadecimal number to a Decimal number and then the Decimal number to an Octal number.
Steps to Convert Hexadecimal Number to Decimal Number
We can use following steps to convert hexadecimal number to decimal numbers.
Step 1: Write the corresponding decimal value for the given hexadecimal number.
Step 2: Multiply each digit of the obtained number with an increasing power of 16 starting from the rightmost digit.
Step 3: Sum all the products obtained in step 2.
Steps to Convert Decimal Number to Octal Number
We can use following steps to convert decimal number to octal numbers.
Step 1: Divide the given decimal number by 8.
Step 2: Write down the quotient and remainder obtained.
Step 3: Divide the quotient obtained by 8.
Step 4: Repeat step 2 and step 3 until the quotient becomes 0.
Step 5: Write the obtained remainder in reverse order.
Let's consider an example for better understanding.
Example 4: Convert 9B16 to Octal Number.
Solution:
Step 1: First convert 9B16 to decimal number:
Corresponding decimal value of 9 and B are 9 and 11 respectively.
9B16 = 9 × 161 + 11 × 160
9B16 = 9 × 16 + 11 × 1
9B16 = 144 + 11 = 155
We get 9B16 = 15510
Step 2: Convert 15510 to Octal Number
155/8, Quotient = 19, Remainder = 3
19/8, Quotient = 2, Remainder = 3
2/8, Quotient = 0, Remainder = 2
Writing the obtained remainders in reverse order we get:
15510 = 2338
Hence we get 9B16 = 2338
Octal to Binary Number
The conversion of an octal number to a binary number is very simple, we have to simply write the corresponding binary value of each digit of the given octal number. Corresponding values of octal and binary numbers are as follows:
Octal Number | Corresponding Binary Number |
|---|
0 | 000 |
1 | 001 |
2 | 010 |
3 | 011 |
4 | 100 |
5 | 101 |
6 | 110 |
7 | 111 |
Example 5: Convert 2138 to a binary number.
Solution:
Write the corresponding binary value of each digit of the given octal number:
2 ---> 010
1 ---> 001
3 ---> 011
Hence we get 2138 = 0100010112
Binary to Octal Number
We can easily convert a binary number to an octal number by following these steps:
- Step 1: Split the binary number into sets of three digits, starting from the right.
- Step 2: Write the corresponding octal value of each binary triplet obtained in step 1.
Example 6: Convert 1001110012 to an octal number.
Solution:
Split 100111001 into sets of three digits and write its corresponding octal value
100 ---> 4
111 ---> 7
001 ---> 1
Hence we get, 1001110012 = 4718
The binary-to-decimal conversion is added in the image below,
Octal Multiplication Table
Octal Multiplication table is added below,
× | 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 |
|---|
Read More,
Solved Examples on Octal Numbers System
Example 1: What is Decimal Equivalent of 11218?
Solution:
11218 = 1 × 83 + 1 × 82 + 2 × 81 + 1 × 80
11218 = 1 × 512 + 1 × 64 + 2 × 8 + 1 × 1
11218 = 512 + 64 + 16 + 1 = 593
Hence 11218 = 59310
Example 2: Convert 278 into the binary number.
Solution:
Write binary equivalent of each digit of 278
2 ---> 010
7 ---> 111
Hence 278 = 0101112
Example 3: Find the octal equivalent of 10010012
Solution:
Breaking 10101111 into groups of three starting from rightmost digit and adding leading zeroes we get:
001, 001, 001
Write the octal equivalent of the groups formed
001 -> 1
001 -> 1
001 -> 1
Answer is (111)8
Practice Questions on Octal Number System
Q1: Convert 12110 to an octal number.
Q2: What is Octal Value of 1000100002?
Q3: Find the Decimal Equivalent of 558.
Q4: Convert 12F16 to Octal number.
Q5: What will be the binary value of 578?
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