Octal Number System

The octal number system is a base-8 system using digits (0–7): 0, 1, 2, 3, 4, 5, 6, and 7.

For example: 22₈, 13₈, 17_8, etc, are octal numbers.

It is widely used in computer programming as a compact representation of binary numbers, where each octal digit corresponds to three binary digits.

In this article, we will discuss the Octal Number System, Octal Number System Conversions, Octal Number System Examples, and Others in detail.

Octal Number System Table

The table below shows the relationship between octal and binary numbers (each octal digit represents 3 bits in 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.

Now, we will learn about the conversion of octal number system to other number systems one by one. So let's get started.

Conversion from Octal to Other Number Systems

Octal Numbers are represented with digits 0-7 and with base 8. Conversion of a number system means conversion from one base to another. Following are the conversions of the Octal Number System to other Number Systems:

Octal to Decimal Conversion

Octal numbers are represented in base 8, but the decimal numbers are of base 10. Hence, to convert an octal number to a decimal number, the base of that number is to be changed.

Example: (247)₈ is to be converted to decimal:

Follow the steps given below:

• Step 1: Multiply each digit of the Octal number with the place value of that digit, starting from right to left i.e. from LSB to MSB.
• Step 2: Add the result of this multiplication and the decimal number will be formed.

Octal to Binary Conversion

Octal numbers are represented in base 8, but the binary numbers are of base 2. Hence, to convert an octal number to a binary number, the base of that number is to be changed.

Example: (247)₈ is to be converted to binary

Follow the steps given below:

• Step 1: Write each digit of the octal number separately.
• Step 2: Convert each digit into an equivalent group of three binary digits.
• Step 3: Combine these groups to form the whole binary number.

Octal to Hexadecimal Conversion

Octal numbers are represented in base 8, but the hexadecimal numbers are of base 16. Hence, to convert an octal number to a hex number, the base of that number is to be changed.

Example: (5456)₈ is to be converted to hex

Follow the steps given below:

• Step 1: We need to convert the Octal number to Binary first. For that, follow the steps given in the above conversion.
• Step 2: Now to convert the binary number to Hex number, divide the binary digits into groups of four digits starting from right to left i.e. from LSB to MSB. 
• Step 3: Add zeros before MSB to make it a proper group of four digits(if required)
• Step 4: Now convert these groups into their relevant decimal values.
• Step 5: For values from 10-15, convert it into Hex symbols i.e from A-F

Decimal to Octal Conversion

The image added below shows binary to octal conversion.

Follow the steps given below:

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.

Octal Multiplication Table

Octal Multiplication table is added below,

Also Check:

• 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

Example 4: 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.

Practice Questions on Octal Number System

Question 1: Convert 121₁₀ to an octal number.

Question 2: What is Octal Value of 100010000₂?

Question 3: Find the Decimal Equivalent of 55₈.

Question 4: Convert 12F₁₆ to Octal number.

Question 5: What will be the binary value of 57₈?

Answer Key:

• 171
• 420
• 45
• 457
• 101111

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

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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