Octal Number System

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, 17_8, 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:

• Decimal number system (with base 10 and symbols ranging between 0-9)
• Binary Number system (with base 2 and symbols 0 and 1)
• Hexadecimal number system (with base 16 and symbols raging between 0-9 and from A to F )
• Octal Number system (with base 8 and symbols ranging between 0-7)

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.

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:

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,

Read More,

• Whole Numbers
• Natural Numbers
• Irrational Numbers

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?

Suggested Quiz

10 Questions

What is the decimal equivalent of the octal number 57?

• A

  47

• B

  45

• C

  41

• D

  49

Explanation:

57 = (5 × 8¹) + (7 × 8⁰) = 47

How to write (8)10 in Octal.

• A

  8

• B

  9

• C

  10

• D

  0

Explanation:

Divide 8 by 8 and note the quotient and remainder:
8 ÷ 8 = 1 remainder 0
Write the result in reverse order: 10.

What is the sum of octal numbers 34 and 25?

• A

  61

• B

  59

• C

  63

• D

  57

Explanation:

34 = (28)₁₀
25 = (21)₁₀
28 + 21 = (49)₁₀ = (61)₈

How many bits are required to represent the octal number 756 in binary?

• A

  3

• B

  6

• C

  9

• D

  12

Explanation:

Each digit in octal requires 3 bits to represent in binary because 2³ = 8.

756_octal ​ = 111101110_binary​
The binary representation is 111101110, which has 9 bits.

What is the result of octal multiplication 13 × 7?

• A

  115

• B

  150

• C

  200

• D

  91

Explanation:

13 = 11(Base 10)
7 = 7(Base 10)
11 × 7 = 77(Base 10) = 115(Oct)

What will be the output of this octal operation (31 × 7) + 11?

• A

  270

• B

  257

• C

  255

• D

  2287

Explanation:

To solve:
31 = 25(Base 10)
7 = 7(Base 10)
25 × 7 = 175(Base 10) = 257(Oct)
257 = 175(Base 10)
11 = 9(Base 10)
175 + 9 = 184(Base 10) = 270(Oct)

What will be the output of the following octal division 110110 ÷ 10?

• A

  11011

• B

  1010.11

• C

  1110.10

• D

  11101

Explanation:

110110 = 36936(Base 10)
10 = 8(Base 10)
36936 ÷ 8 = 4617(Base 10) = 11011(Oct)

Covert (88AA)16 to Octal.

• A

  104252

• B

  105252

• C

  10106262

• D

  10105262

Explanation:

Convert each hex digit to 4 binary digits and then convert each 3 binary digits to octal digits (see conversion tables below):

88AA = 1000 1000 1010 1010

= 104252

Simplify (128)10 + (11A)16 + (1111)2, and give the answer in Octal.

• A

  651

• B

  681

• C

  671

• D

  612

Explanation:

Convert all the Numbers to Decimal:
(11A)₁₆ = 1 × 16² + 1 × 16¹ + 10 × 16⁰ = 256 + 16 + 10 = 282
(1111)₂ = 1 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰ = 8 + 4 + 2 + 1 = 15

Add the values
128 + 282 + 15 = 425

Convert (425)₁₀ to octal by repeated division by 8:
Write the remainders in reverse order: 651

Simplify (250)10 + (2F3)16 + (10101)2​, and give the answer in octal.

• A

  2022

• B

  2002

• C

  2020

• D

  202

Explanation:

Convert each value to decimal:

(250)₁₀​: This is already in decimal form, so no conversion is needed.
2F3)₁₆: 2 × 16² + 15 × 16¹ + 3 × 16⁰ = 512 + 240 + 3 = 755
(10101)₂: 1 × 2⁴ + 0 × 2³ + 1 × 2²+ 0 × 2¹ + 1 × 2⁰ = 16 + 4 + 1 = 21

Add the values in decimal: 250 + 755 + 21 = 1026
Convert (1026)¹⁰​ to octal by dividing repeatedly by 8:
Write the remainders in reverse order: 2002.

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