Octal Number System

The octal number system employs digits ranging from 0 to 7 and is based on eight. The term octal refers to a number system with base eight. Octal numbers find many uses and significance in digital numbering systems as well as in computers. Octal numbers can be easily converted to and from binary, decimal, and hexadecimal systems. For instance, binary numbers can be converted directly to octal by grouping bits in sets of three.  

The octal number system uses base 8, whereas the binary system has a base 2, the decimal number system has a base 10, and the hexadecimal number system has a base 16.
 

Converting octal numbers from base-8 to base-2 is Octal to binary. Converting is easy and fast since each octal digit translates to three binary digits. Using a conversion table or first changing octal to decimal and then decimal to binary, this process can be carried out straightforwardly. Here are the steps of conversion:

Method 1: Divide the numbers in the supplied octal number.  Replace each octal numeral with its corresponding three-digit binary value from the chart. Mix together the binary groups to get the ultimate binary digit.
For example: Convert 7568 to Binary.
Firstly, separate the digits as 7, 6, 5.
Convert to binary using the format:
7 = 111
6 = 110
5 = 101

Lastly, combine 7658, which will be 1111011108.

Method 2: Indirect Conversion (Octal → Decimal → Binary)
In this method, the octal number is first converted to decimal and then to a binary number. 

Step 1: Octal to Decimal Conversion:
To convert the octal to the decimal number system, we first multiply each digit of the octal number by 8 raised to the power of its position, from right to left, starting at 0. Then add all the results to find the decimal equivalent

For Example: Convert 548 into Decimal.
548 = (5 × 81) + (4 × 80) 
= (5 × 8) + (4 × 1) 
= 40 + 4
= 4410.

Step 2: Decimal to Binary Conversion: 

Now, divide the decimal number repeatedly by 2 and list the remainders. By arranging the remainders in reverse order, we get the binary equivalent. 

44 ÷ 2 =22, remainder = 0
22 ÷ 2 = 11, remainder = 0
11 ÷ 2 = 5, remainder = 1
5 ÷ 2 = 2, remainder = 1
2 ÷ 2 = 1, remainder = 0
1 ÷ 2 = 0, remainder = 1
Reverse remainders: 1011002. 
 

Changing octal numbers to decimals is a simple process whereby the octal number is expanded using eight powers of 8. The following is a step-by-step guide, including examples to make things clear:

Step 1: Identify the Position of Each Digit

The positions are counted in an octal number system from right to left, starting at 0. Each digit is associated with a power of 8 based on its position. 

Step 2: Multiply by Powers of 8

Each octal digit is multiplied by 8 raised to the power of its position. For example, the digit in the 2nd position from the right is multiplied by 82.

Step 3: Sum the Results:

The decimal value of the octal number is the sum of all the products. 
Example: Convert 3218 into decimal.
Expand using powers of 8: (321)8 
= (3 × 82) + (2 × 81) + (1 × 80)
= (3 × 64) + (2 × 8) + (1×1)
Calculate:
=192 + 16 + 1=209
Result: 3218 = 20910

The two-step technique is for converting octal numbers to hexadecimal. Replace each digit of the octal number with its 3-bit binary equivalent first. First, convert the octal number to binary. Then group the binary digits into four sets and replace every group with its corresponding hexadecimal digit. Finally, convert the binary number into hexadecimal. Converting an octal number to hexadecimal involves using binary as an intermediary. Here’s how to do it step-by-step:

Step 1: Identify the octal number:

Begin by writing down the octal number you want to convert (remember, it's in base 8)…

For example, convert 1578  to a hexadecimal number
To convert the octal number 1578 to hexadecimal form, we first convert it to binary. A binary number system acts as a link between the octal and hexadecimal number systems. For this case:
Octal digit 1 is written as 001 in binary
Digit 5 becomes 101
Digit 7 converts to 111
By combining these, we get the binary value:
 157₈ = 001101111₂

Step 2: Next, to move from binary to hexadecimal, the binary number must be split into groups of four bits, starting from the right. Since the binary result has only 9 bits, we add extra zeros to the left side to make it a full set of 12 bits. This gives us:
 000001101111.

Step 3: Now, we divide this into three groups:
0000

0110

1111

Step 4: Each of these groups is then converted into its corresponding hexadecimal digit:
0000 is 0

0110 is 6

1111 is F

Therefore, the hexadecimal version of the octal number 157₈ is (06F)₁₆, which can also be written as 6F16  by excluding the leading zero.
 

Changing binary, decimal, and hexadecimal numbers to octal calls for different methods. Grouping digits into three, octal values are substituted for binary numbers. Dividing by 8 converts decimal numbers and inverts the remainder. At binary, their octal counterparts, hexadecimal values are first treated in threes. Conversion between these several number systems is helped by this multistep technique.

Convert a Binary Number to Octal

By arranging binary digits into sets of three from right to left, one can easily change binary to octal. Then each set is substituted with its octal counterpart, using the direct link between binary and octal numbers. Binary numbers are easily converted into octal by grouping digits into sets of three from the right.

Step 1: Break the Binary Number into Groups of 3:

Take your binary number. For example, let's say it’s 1011101.

Starting from the right, split it into groups of 3 digits:
 → 1 011 101

Since the left group only has 1 digit, add two zeros to make it 3 digits:
 → 001 011 101

Step 2: Change Each Group into an Octal Digit:

Now convert each 3-digit group into its octal form using this:

001 → 1

011 → 3

101 → 5

Step 3: Write the Final Octal Number:

Put the digits together: 135

So, binary (1011101)2 = 001 011 101 = Octal 135
 

To convert a decimal number to an octal number, we repeatedly divide the number by 8 and list out all the remainders. The octal equivalent will be the remainder in reverse order. 

For example, convert 75 to octal number

Step 1: Divide the Decimal Number by 8:

Dividing 75 by 8:

75 ÷ 8 = 9 remainder 3

Step 2: Divide the Quotient Again by 8:

Dividing 9 by 8:

9 ÷ 8 = 1 remainder 1

Step 3: Keep Dividing Until the Quotient is 0:

Now, 1 ÷ 8 = 0 remainder 1

Step 4: Write the Remainders in Reverse:

That is 1 1 3

So, decimal 75 = octal 113
 

First, expand each digit of the hexadecimal number into four bits to transform it to binary, then to octal. Next, rearrange the binary numbers in groups of three and replace them with their octal equivalents. Efficiently, this technique bridges the conversion from hexadecimal to octal systems.

Step 1: Turn Each Hex Digit into 4 Binary Digits:

Example: Hexadecimal 2F

2 → 0010

F → 1111

Together, binary = 00101111

Step 2: Group the Binary Digits into 3s:

From the right: 001 011 111

Step 3: Convert Each Group to an Octal Digit:

001 → 1

011 → 3

111 → 7

Step 4: Combine the Digits:

Final octal number = 137

So, hex 2F = octal 137

First, the hexadecimal 2F is expanded to binary and then grouped into threes for octal conversion, resulting in octal 137. The answer will be 001011111.
 

Converting a duodecimal to decimal is an easy process. To convert, multiply each digit by 12 raised to the power of its position, then add everything together. Follow these steps to convert duodecimal to decimal:

Step 1: Find the place value of each digit
Each digit’s value is based on its position, from right to left 
The first digit is multiplied by 120
The next digit is 121
Then 122 

Step 2: Compute the Decimal Value:
Multiply each digit by the power of 12 based on its position, then add the product. 
For example, take 1A3​ (where A stands for 10):
 1 × 122 + 10 × 121 + 3 × 120 
= 1×144 + 10 × 12 + 3 × 1 
= 144 + 120 + 3 
= 26710
 

This octal number is created by first dividing the decimal number by eight, keeping the remainder, and then, in reverse order, writing the leftover. Decimal numbers can be efficiently converted to octal by repeated division by 8 and recording the remainders in reverse order.

Method 1: Division Method:

Divide the decimal number by 8 and note the remainder. Continue dividing the resulting quotient by 8 until you reach zero.

Method 2: Convert into the Octal Number:

The octal value is formed by taking the remainders in reverse order. For our example, converting 267 into octal:

267 ÷ 8 = 33 with a remainder of 3.

33 ÷ 8 = 4 with a remainder of 1.

4 ÷ 8 = 0 with a remainder of 4.

Reading the remainder backward, the octal number is 4138.
 

The octal system finds practical use in computing and digital electronics. Here are a few real-life applications:

• The octal number system is used to represent memory addresses and binary data in a compact and readable form. 

• In digital electronics, the octal number system is used in digital circuits to simplify binary input/output operations. 

• In telecommunications and signal processing, the octal number system is used for protocol design and signal coding. 

• In the aviation industry, the octal number system is used in aircraft transponders to transmit identification codes.