Converting Hexadecimal to Octal: A Step-by-Step Guide

When working with digital systems, it is often necessary to convert between different numeric bases. One common conversion is from hexadecimal (base 16) to octal (base 8). Hexadecimal and octal are both useful for representing binary numbers in a more compact form. This guide provides a clear and detailed explanation of how to convert hexadecimal numbers to octal numbers.

Understanding Hexadecimal and Octal Bases

Before we dive into the conversion process, it's important to understand the numeric bases in question. Hexadecimal (base 16) uses 16 symbols (0-9 and A-F), while octal (base 8) uses 8 symbols (0-7).

The Conversion Process

The process of converting a hexadecimal number to an octal number involves the following steps:

Convert Hexadecimal to Binary: Each hexadecimal digit can be represented by a 4-bit binary number. For example, the hexadecimal digit 7 is 0111 in binary, C is 1100, and D is 1101. Group Binary Digits: After converting the hexadecimal number to binary, group the binary digits into sets of 3 from right to left for the integer part, and from left to right for the fractional part. This is because 3 binary digits (bits) can represent 8 different values, which matches the octal system's 8 symbols (0-7). Convert to Octal: Convert each group of 3 binary digits to their octal equivalent. For example, 011 is 3 in octal, 110 is 6, and so on.

Example: Converting 7CD.2A to Octal

Let's illustrate this process with the example of converting the hexadecimal number 7CD.2A to octal.

Convert Hexadecimal to Binary: 7 is 0111 C is 1100 D is 1101 The integer part of 7CD becomes 0111 1100 1101 in binary. 2 is 0010 A is 1010 Pad 0 to make it 0000 The fractional part of 2A0 becomes 0010 1010 0000 in binary. Group Binary Digits: Integer part: 011 110 011 010 Fractional part: 001 010 100 000 The integer part is now grouped as 011 110 011 010 and the fractional part as 001 010 100 000. Convert to Octal: Integer part: 011 110 011 010 becomes 3715 in octal (since 011 is 3, 110 is 6, 011 is 3, and 010 is 2). Fractional part: 001 010 100 000 becomes 1240 in octal (since 001 is 1, 010 is 2, 100 is 4, and 000 is 0). Combining the integer and fractional parts, the final result is 3715.1240 in octal. Dropping the final nonsignificant 0 from the fractional part, the final answer is 3715.124 in octal.

Using Windows Calculator for Conversion

If you have access to the Windows Calculator app, you can use it to convert between hex, decimal, octal, and binary integer values. However, it only works for integers, so you will need to convert the integer and fractional parts separately. Remember to pad out the fractional part to a multiple of 3 digits to get the correct octal result for the last group.

TIP: Some programmable calculators also offer similar features, making manual conversion quicker and easier.

Conclusion

Converting hexadecimal to octal is a practical skill useful in digital systems and programming. By understanding the bases and using the step-by-step process described, you can convert between these numeric systems efficiently.