Octal to Hexadecimal

When working with different number systems in mathematics and computer science, it’s common to convert numbers from one base to another. Two such systems are the octal (base 8) and hexadecimal (base 16) systems. This article will walk you through the process of converting a number from octal to hexadecimal, including a detailed explanation of the underlying mathematical concepts and step-by-step examples.

Number Systems Overview

1. Octal Number System (Base 8):

   • Uses digits from 0 to 7.
   • Each digit represents a power of 8.
   • Example: $237_8$ represents $2 \times 8^2 + 3 \times 8^1 + 7 \times 8^0 = 128 + 24 + 7 = 159$ in decimal.

2. Hexadecimal Number System (Base 16):

   • Uses digits from 0 to 9 and letters A to F, where A = 10, B = 11, …, F = 15.
   • Each digit represents a power of 16.
   • Example: $2A3_{16}$ represents $2 \times 16^2 + 10 \times 16^1 + 3 \times 16^0 = 512 + 160 + 3 = 675$ in decimal.

Conversion Process: Octal to Hexadecimal

The most efficient method to convert an octal number to hexadecimal is by first converting the octal number to binary and then converting the binary number to hexadecimal.

Step 1: Convert Octal to Binary

Each octal digit corresponds to exactly three binary digits (bits) because $2^3 = 8$. The conversion is done by replacing each octal digit with its three-bit binary equivalent.

• Binary equivalents of octal digits:
  • $0_8 = 000_2$
  • $1_8 = 001_2$
  • $2_8 = 010_2$
  • $3_8 = 011_2$
  • $4_8 = 100_2$
  • $5_8 = 101_2$
  • $6_8 = 110_2$
  • $7_8 = 111_2$

Example:

Convert $523_8$ to binary.

• $5_8 = 101_2$
• $2_8 = 010_2$
• $3_8 = 011_2$

So, $523_8$ becomes $101010011_2$.

Step 2: Group Binary Digits in Sets of Four

After converting the octal number to binary, group the binary digits into sets of four, starting from the right. If the leftmost group has fewer than four digits, add leading zeros to make a full group.

• For $101010011_2$, group as $0001 \ 0101 \ 0011_2$.

Step 3: Convert Binary to Hexadecimal

Each group of four binary digits directly corresponds to a hexadecimal digit.

• Binary to Hexadecimal conversions:
  • $0000_2 = 0_{16}$
  • $0001_2 = 1_{16}$
  • $0010_2 = 2_{16}$
  • $0011_2 = 3_{16}$
  • $0100_2 = 4_{16}$
  • $0101_2 = 5_{16}$
  • $0110_2 = 6_{16}$
  • $0111_2 = 7_{16}$
  • $1000_2 = 8_{16}$
  • $1001_2 = 9_{16}$
  • $1010_2 = A_{16}$
  • $1011_2 = B_{16}$
  • $1100_2 = C_{16}$
  • $1101_2 = D_{16}$
  • $1110_2 = E_{16}$
  • $1111_2 = F_{16}$

Example:

Convert $0001 \ 0101 \ 0011_2$ to hexadecimal.

• $0001_2 = 1_{16}$
• $0101_2 = 5_{16}$
• $0011_2 = 3_{16}$

Thus, $101010011_2$ becomes $153_{16}$.

Final Conversion Summary

The octal number $523_8$ converts to $153_{16}$ in hexadecimal.

Example: Converting Another Octal Number

Let’s convert the octal number $7462_8$ to hexadecimal.

1. Convert to Binary:

   • $7_8 = 111_2$
   • $4_8 = 100_2$
   • $6_8 = 110_2$
   • $2_8 = 010_2$

   $7462_8$ becomes $111100110010_2$.

2. Group into Fours:

   • $1111 \ 0011 \ 0010_2$

3. Convert to Hexadecimal:

   • $1111_2 = F_{16}$
   • $0011_2 = 3_{16}$
   • $0010_2 = 2_{16}$

   Therefore, $7462_8$ converts to $F32_{16}$.

Mathematical Insight

This conversion process works because both the octal and hexadecimal systems are closely related to the binary system. Octal digits map directly to three binary digits, while hexadecimal digits map directly to four. By leveraging these relationships, conversion between octal and hexadecimal is streamlined through binary.

Conclusion

Converting octal numbers to hexadecimal involves understanding the relationship between these systems through binary. By converting octal digits to binary, grouping the binary digits, and then converting to hexadecimal, the process is both efficient and easy to manage. This method is fundamental in various fields, especially in computing, where different number systems are frequently used.

References

1. Digital Fundamentals by Thomas L. Floyd: This book provides a thorough explanation of number systems and their conversions, including octal and hexadecimal systems.
2. Computer Organization and Design: The Hardware/Software Interface by David A. Patterson and John L. Hennessy: This textbook discusses binary, octal, and hexadecimal systems in the context of computer architecture.
3. Numerical Methods and Computation by C.F. Gerald and P.O. Wheatley: This book includes sections on the conversion between different numerical bases, offering practical examples and exercises.
4. Number Systems – Khan Academy: An online resource that explains different number systems, including binary, octal, and hexadecimal, and how to convert between them. Available at: Khan Academy.

These references provide foundational knowledge and practical examples related to the topics covered in this article.