Converting from Octal (base-8) to Hexadecimal System (Base-16) involves an intermediary conversion through binary, which acts as a universal bridge between these two bases. Both octal and hexadecimal are powers of two (octal is base-8, and hexadecimal is base-16), making the transition through binary straightforward without direct arithmetic conversion.
Convert Octal to Binary
- Break Down the Octal Number: Start by separating the octal number into its individual digits.
- Convert Each Octal Digit to Binary: Translate each octal digit into a 3-bit binary number. This is direct because octal digits range from 0 to 7, and each can be represented by three binary digits (bits).
- 0 = 000
- 1 = 001
- 2 = 010
- ...
- 7 = 111
- Combine the Binary Digits: Group the binary digits together to form the binary equivalent of the octal number.
Convert Binary to Hexadecimal
- Group Binary Digits into Sets of Four: Starting from the right, divide the binary sequence into groups of four. If the leftmost group contains fewer than four digits, pad it with zeros.
- Convert Each Binary Group to Hexadecimal: Translate each 4-bit binary group into its corresponding hexadecimal digit.
- 0000 = 0
- 0001 = 1
- 0010 = 2
- ...
- 1111 = F
- Combine the Hexadecimal Digits: Group the hexadecimal digits together to form the hexadecimal equivalent of the binary number.
Example Conversion: Octal to Hexadecimal
Let’s convert the octal number 1752 to hexadecimal:
- Octal to Binary
- 1 = 001
- 7 = 111
- 5 = 101
- 2 = 010
- Binary to Hexadecimal
- 0011 = 3
- 1111 = F
- 1010 = A
Therefore, the octal number 1752 converts to the hexadecimal number 3EA.
Conclusion
This method demonstrates the logical and systematic approach to converting numbers between different bases, especially when they are not directly compatible, like octal and hexadecimal. Utilizing binary as an intermediary simplifies the process, leveraging the power-of-two relationship shared by these numeral systems, crucial for applications in computing and digital electronics.