Base Conversion

A base (also known as radix) of a number system tells us about the unique or different symbols and notations it uses to represent a value. For example, the base 8 tells us that it uses 8 different numbers i.e. 0 to 7 to represent a value. The most commonly used base in the number system is the base 10 which uses 10 different digits i.e. 0 to 10 to represent a value. Bases can be 0, positive, negative, complex, and non-integral. Other frequently used bases are base 2, base 8, and base 16. They are used in computing and are known as binary, octal, and hexadecimal respectively.

We can convert one base into the other base by following the simple conversion steps. Here, we will discuss the steps on how to convert a number from decimal to octal, octal to decimal, octal to binary, binary to octal, octal to hexadecimal, hexadecimal to octal, and so on.

Let us begin with the octal to decimal conversion

Decimal to Octal Conversion

To convert decimal to octal, we use the following steps:

The steps given below show how to convert a number from the decimal system to the octal system.

Example

Convert 560₁₀ into the octal system 

Step 1: Observe the base of the required number.

As we have to convert the given decimal number into the octal number system, the base of the required number is 8.

Step 2: Divide the given decimal number by the base of the required number and list down the quotient and remainder in the form of quotient-remainder.

Repeat the process (dividing the quotient value again by base 8)  till we get the quotient value less than the base.

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Octal to Decimal Conversion

To convert a number from an octal system to a decimal system, multiply each digit of the given octal number by the reducing power of 8.

Let us discuss octal to decimal conversion through an example:

Example:

Convert 764₈ into the decimal system 

764₈ = 7 x 8² + 6 x 8¹ +  4 x 8º

         = 7 x 64 + 6 x 8 + 4 x 1

          = 448 + 48 + 4

           =  500

Binary to Octal Conversion

To convert a number from binary system to the octal system, we use the following binary to the octal table:

Example:

Convert 1010111100₂ to octal number system

To convert binary to octal, we will group the binary digits by three, starting from the far right to left. You can add trailing zeros to the left of the last digit to complete the last group of three if required. Then replace each group of three digits with the equivalent octal digit. 

1010111100₂ in a group of 3 is represented by \[\overline{001}\] \[\overline{010}\] \[\overline{111}\] \[\overline{100}\]. Here we have added trailing zeros to the left of the last digit to complete the last group of three.

Let us now convert base₂ to base₈ 

Octal to Binary Conversion

To convert a number from the octal system to the decimal number system, we simply need to follow the 2 steps given below:

1. Convert the given octal number into the decimal number system.

2. Convert the decimal number that is obtained in Step 1 into the binary number.

Let us understand octal to binary conversion through an example:

Example: Convert 205₈ into the decimal number system.

Step 1: Convert a number from the octal system to the decimal system

205₈ = 2 x 8² + 0 x 8¹ +  5 x 8º

         = 2 x 64 + 0 x 8 + 5 x 1

          = 128 + 0 + 5

           =  133₁₀

Step 2: Convert a decimal number that is obtained in the previous step into the binary system.

To get the equivalent binary number, we will list down all the remainder values from bottom to top.

Octal to Hexadecimal Conversion

To convert octal to hexadecimal or hexadecimal to octal, we need to memorize the table given below:

Octal to Hexadecimal Conversion Steps

To convert a  number from the octal system to the decimal number system, we simply need to follow the 2 steps given below:

1. Convert the given number from the octal system into the binary number system.

2. Convert the binary number that is obtained in Step 1 into the hexadecimal number system.

Let us understand octal to hexadecimal conversion through an example.

Example: Convert 536₈ into the hexadecimal decimal number system

Step 1: Convert the given number from the octal system into the binary number system

To convert a number from the octal system to the binary number system, we simply need to follow the 2 steps given below:

1. Convert each octal digit into the equivalent three-digit binary number. Each of the digits must be treated as a decimal value.

2. Combine these groups of 3 binary values into a single binary number.

Convert each octal digit into the three-digit binary number as shown below:

 \[\overline{101}\] \[\overline{011}\] \[\overline{110}\]

Combine these groups of 3 binary values (starting from the far right)  into a single binary number.

536₈ = 101011110₂

Step 2: Convert the binary number that is obtained in the previous step into the hexadecimal number system.

101011110₂

To convert a number from binary system to the hexadecimal system, we will group the binary digits by four, starting from the far right and proceeding to the left as shown below. You can add trailing zeros to the left of the last digit to complete the last group of four if required. 

101011110₂ in a group of 4 is represented by as \[\overline{0001}\] \[\overline{0101}\] \[\overline{1110}\]. Here we have added trailing zeros to the left of the last digit to complete the last group of four.

Now, let us convert the 1101011110₂ into the base₁₆.

Hexadecimal to Octal Conversion

To perform the hexadecimal to octal conversion, we simply need to follow the below-mentioned steps.

• Write each of the given hexadecimal number digits into its equivalent binary number.

• Combine and make the groups of binary bits, each group containing 3 bits from right to left. You can add trailing 0’s to the left of the last digit if there are less than 3 digits in the last group.

• Find the equivalent hexadecimal number of each binary group. 

Let us understand hexadecimal to octal conversion, through an example.

Example: Convert (1BC)₁₆ into the octal number system

Step 1: Convert each hexadecimal digit into the four-digit binary number as shown below:

Step 2: Combine and make the groups of binary bits, each group containing 3 bits from right to left.

000, 101 , 111,  100

Step 3: Now, write down the equivalent hexadecimal number of each binary group of 3 bits.