Number Systems Conversion

Number systems are systems in mathematics that are used to express numbers in various forms and are understood by computers. A number is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. Numbers have various categories like natural numbers, whole numbers, rational and irrational numbers, and so on. Similarly, there are various types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.

Octal

Octal number is a number expressed in the base 8 numeral system. Octal number's digits have 8 symbols: 0,1,2,3,4,5,6,7. Each digit of an octal number counts a power of 8.

Octal number example: 627₈ = 6×8² + 2×8¹ + 7×8⁰ = 407₁₀

Hexadecimal

Hexadecimal number is a number expressed in the base 16 numeral system. Hexadecimal number's digits have 16 symbols: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. Each digit of a hexadecimal number counts a power of 16.

Hexadecimal number example: 62C₁₆ = 6×16² + 2×16¹ + 12×16⁰ = 1580₁₀

How to convert from octal to hexadecimal

To convert an octal number to a hexadecimal number, follow these steps:

Convert the octal number to binary.
Group the binary digits into sets of four, starting from the right. Add leading zeros if necessary.
Convert each group of four binary digits to its corresponding hexadecimal digit.

Example

Understanding Octal-to-Hexadecimal Conversion

Octal-to-hexadecimal conversion involves converting numbers in octal (base 8) into their equivalent hexadecimal (base 16) values. Octal uses digits 0-7, while hexadecimal uses digits 0-9 and letters A-F.

The general approach to converting octal numbers to hexadecimal includes:

Write down the octal number.
Convert each octal digit to its binary equivalent (using 3 bits for each digit).
Group the binary digits into sets of 4 bits (starting from the right).
Convert each 4-bit group to its hexadecimal equivalent.

Steps for Octal-to-Hexadecimal Conversion

Step 1: Write down the octal number.

Step 2: Convert each octal digit to its 3-bit binary equivalent.

Step 3: Group the binary digits into sets of 4 bits, starting from the right.

Step 4: Convert each 4-bit group to its hexadecimal equivalent.

Example: Converting Octal to Hexadecimal

Convert \( 345 \) (octal) to hexadecimal:

Step 1: Write the octal number \( 345 \).
Step 2: Convert each digit to binary:
- \( 3 = 011 \) (binary)
- \( 4 = 100 \) (binary)
- \( 5 = 101 \) (binary)
Step 3: Combine the binary digits: \( 011100101 \).
Step 4: Group into sets of 4 bits (from right): \( 0011 1001 0101 \).
Step 5: Convert each group to hexadecimal:
- \( 0011 = 3 \) (hexadecimal)
- \( 1001 = 9 \) (hexadecimal)
- \( 0101 = 5 \) (hexadecimal)
Final hexadecimal result: \( 395 \).

Conversion Table for Quick Reference

Here is a quick reference for octal digits and their binary and hexadecimal equivalents:

0 → 000 → 0
1 → 001 → 1
2 → 010 → 2
3 → 011 → 3
4 → 100 → 4
5 → 101 → 5
6 → 110 → 6
7 → 111 → 7

Applications of Octal-to-Hexadecimal Conversion

Octal-to-hexadecimal conversion is commonly used in:

Interpreting memory addresses in computing systems.
Understanding file permissions in Unix/Linux systems.
Working with digital systems that use base-8 and base-16 numbering systems.

Practice Problem

Convert \( 127 \) (octal) to hexadecimal:

Solution:
- \( 1 = 001 \) (binary)
- \( 2 = 010 \) (binary)
- \( 7 = 111 \) (binary)
Combine the binary digits: \( 001010111 \).
Group into sets of 4 bits: \( 0001 0101 0111 \).
Convert each group to hexadecimal:
- \( 0001 = 1 \) (hexadecimal)
- \( 0101 = 5 \) (hexadecimal)
- \( 0111 = 7 \) (hexadecimal)
Final hexadecimal result: \( 157 \).

Octal-to-Hexadecimal Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a single octal digit to its hexadecimal equivalent.	Write down the octal digit. Convert the octal digit to its 3-bit binary equivalent. Convert the 3-bit binary group to a 4-bit hexadecimal equivalent.	For `5`: ‘5’ in binary: `101`. Binary to hexadecimal: `101` = `5`. Hexadecimal: `5`.
Handling Larger Numbers	Converting a multi-digit octal number to hexadecimal.	Convert each octal digit to its 3-bit binary equivalent. Group the binary digits into sets of 4 bits. Convert each group of 4 bits to its hexadecimal equivalent.	For `275`: ‘2’ in binary: `010`. ‘7’ in binary: `111`. ‘5’ in binary: `101`. Combine: `010 111 101`. Group into 4 bits: `0010 1111 0101`. Convert to hexadecimal: `2F5`.
Verifying Conversion	Checking the accuracy of octal-to-hexadecimal conversion.	Convert the octal number to binary. Group the binary digits into sets of 4 bits. Convert each group of 4 bits to hexadecimal. Ensure the resulting hexadecimal matches the original conversion.	For `275` (octal): ‘2’ in binary: `010`. ‘7’ in binary: `111`. ‘5’ in binary: `101`. Combine: `010 111 101`. Group into 4 bits: `0010 1111 0101`. Hexadecimal: `2F5`.
Handling Special Cases	Converting octal numbers with leading zeroes or specific patterns to hexadecimal.	Ensure all leading zeroes are included for accuracy. Convert each digit individually and handle patterns carefully.	For `007`: ‘0’ in binary: `000`. ‘0’ in binary: `000`. ‘7’ in binary: `111`. Combine: `000 000 111`. Group into 4 bits: `0000 0001 1110`. Hexadecimal: `1E`.
Applications	Understanding where octal-to-hexadecimal conversion is used in digital systems and computing.	Octal and hexadecimal systems are widely used in computer programming and hardware for data representation. Common in memory addressing, machine code, and hardware design.	Example: Converting octal permissions `754` to hexadecimal: ‘7’ in binary: `111`. ‘5’ in binary: `101`. ‘4’ in binary: `100`. Combine: `111 101 100`. Group into 4 bits: `0111 1101 0100`. Hexadecimal: `7D4`.