Digital Electronics - Base Conversions

---

---

Read this chapter to get a good understanding of how to convert a number from one base to another. For example, how to convert a binary number to its decimal equivalent or how to get the octal equivalent of a binary number, etc.

What is Number System Conversion?

Number system conversion is a process of converting a number from one base to another. Number system conversion is an important concept to represent information in different forms. In digital electronics, the following number system conversions are mostly performed.

• Binary to Decimal Conversion
• Decimal to Binary Conversion
• Binary to Octal Conversion
• Octal to Binary Conversion
• Octal to Decimal Conversion
• Decimal to Octal Conversion
• Hexadecimal to Binary Conversion
• Binary to Hexadecimal Conversion
• Hexadecimal to Decimal Conversion
• Decimal to Hexadecimal Conversion
• Octal to Hexadecimal Conversion
• Hexadecimal to Octal Conversion

Let us understand each of these number system conversions with the help of examples.

Binary to Decimal Conversion

We can convert a binary number into its equivalent decimal number by using the positional weights method.

In this method of binary to decimal conversion, each digit of the given binary number is multiplied by its positional weight. Then, all the products are added to obtain the equivalent decimal number.

Decimal to Binary Conversion

A decimal number can be converted to their equivalent binary number by using the double-dabble method. In this method, the integer part of the given decimal number is successively divided by 2 and the fractional part is successively multiplied by 2.

In the integer part, the remainders read from bottom to top give the integer part of the binary equivalent. In the fractional part, the carries read from top to bottom give the fractional part of the binary equivalent.

Binary to Octal Conversion

A binary number can be converted into its equivalent octal number by mapping method.

Octal to Binary Conversion

We can also use the mapping method to convert an octal number into its equivalent binary number. In this method, we just replace each digital of the given octal number by its 3-bit binary equivalent.

Octal to Decimal Conversion

The conversion of an octal number to its equivalent decimal number is same as the binary to decimal conversion. To convert an octal number to its decimal equivalent, we multiply each digit of the octal number by its positional weight and then add all the product terms to obtain the equivalent decimal number.

Decimal to Octal Conversion

We can convert a mixed decimal number (having integer and fractional parts) to its equivalent octal number. For this, we convert the integer and fractional parts separately.

To convert the integer part of the given decimal number to octal, we divide the given decimal number successively by 8 till the quotient is 0. The octal equivalent is obtained by reading the remainders from bottom to top, where the last remainder will be the most significant digit.

To convert the fractional part of the given decimal number to octal, we multiply the given decimal fraction successively by 8 till the product is 0 or the desired accuracy is obtained. The fractional part of the equivalent octal number is obtained by reading the carries from top to bottom.

Hexadecimal to Binary Conversion

We can convert a hexadecimal number into its equivalent binary by using the mapping method. In this method, we replace each digit of the given hexadecimal number by its equivalent 4-bit binary group.

Binary to Hexadecimal Conversion

To convert a given binary number to its equivalent hexadecimal number, we create groups of 4 bits each on both sides of the binary point. Then, we replace each group of 4-bit binary by the equivalent hexadecimal digit.

Hexadecimal to Decimal Conversion

To convert a hexadecimal number to its equivalent decimal number, we multiply each digit in the hexadecimal number by its positional weight and then add all the product terms to obtain the final result.

Decimal to Hexadecimal Conversion

If a mixed decimal number is given that has integer and fraction parts. Then, to convert the given decimal number to its equivalent hexadecimal, we convert integer and fraction parts separately.

To convert the integer part, we successively divide the decimal integer by 16 till the quotient is 0. The integer part of the equivalent hexadecimal is obtained by reading the remainders from bottom to top.

To convert the fractional part, we multiply the decimal fractional number by 16 till the product is 0 or till the desired accuracy is obtained. The fractional part of the equivalent hexadecimal is obtained by reading the carries from top to bottom.

Octal to Hexadecimal Conversion

The conversion of octal to hexadecimal is very simple. We first convert the given octal number to binary and then the binary number to the hexadecimal.

Hexadecimal to Octal Conversion

The hexadecimal to octal conversion can be perform in the same way as the octal to hexadecimal as explained above. To convert a given hexadecimal number to octal number, we first convert the given hexadecimal number to binary and then the binary number to the octal.

Conclusion

We can convert a given number from one base to another. Number system conversion is one the important operations in the field of digital electronics, as it allows to represent the same information in different formats. This chapter provided just the introductions of the types of base conversions that are possible. Read the subsequent chapters in this section to get the full details of how each of these conversions work. In the next chapters, you will find relevant numerical examples for all these types of base conversions.