Hexadecimal to Octal Conversion: Understanding \( (F3B1)_{16} \)

Number system conversions are a fundamental concept in digital electronics and computer science. The question asks for the octal equivalent of the hexadecimal number \( (F3B1)_{16} \). To convert a number from hexadecimal (base 16) to octal (base 8), the most common and straightforward method is to first convert the hexadecimal number to its binary (base 2) equivalent, and then convert the binary number to its octal equivalent.

Step-by-Step Conversion of \( (F3B1)_{16} \) to Octal

Step 1: Convert Hexadecimal to Binary

Each hexadecimal digit corresponds to a unique 4-bit binary number. We will convert each digit of \( (F3B1)_{16} \) into its 4-bit binary representation:

• F in hexadecimal is \( (1111)_2 \) in binary.
• 3 in hexadecimal is \( (0011)_2 \) in binary.
• B in hexadecimal is \( (1011)_2 \) in binary.
• 1 in hexadecimal is \( (0001)_2 \) in binary.

Now, combine these binary representations in the same order:

\( (F3B1)_{16} = (1111 \ 0011 \ 1011 \ 0001)_2 \)

Step 2: Group Binary Bits for Octal Conversion

To convert a binary number to octal, we group the binary digits into sets of 3, starting from the rightmost digit. If the leftmost group has fewer than 3 bits, we add leading zeros to complete the group.

Our binary number is \( 1111001110110001_2 \). Let's group it into sets of 3 bits from right to left:

\( 1 \ 111 \ 001 \ 110 \ 110 \ 001 \)

The leftmost group has only one bit (\( 1 \)). We add two leading zeros to make it a 3-bit group:

\( 001 \ 111 \ 001 \ 110 \ 110 \ 001 \)

Step 3: Convert Binary Groups to Octal Digits

Each 3-bit binary group corresponds to a single octal digit (from 0 to 7). Convert each group:

• \( (001)_2 = (1)_8 \)
• \( (111)_2 = (7)_8 \)
• \( (001)_2 = (1)_8 \)
• \( (110)_2 = (6)_8 \)
• \( (110)_2 = (6)_8 \)
• \( (001)_2 = (1)_8 \)

Combine these octal digits in order:

\( 171661_8 \)

Result: Octal Equivalent of \( (F3B1)_{16} \)

By converting the hexadecimal number \( (F3B1)_{16} \) to binary and then grouping the binary bits into sets of three and converting each group to octal, we find that the octal equivalent is \( (171661)_8 \).

Revision Table: Common Number System Conversions

Additional Information on Number Systems

Number systems are ways of representing numbers. Different number systems use different bases or radix. Common number systems include:

• Decimal (Base 10): Uses digits 0-9. This is the system we use daily.
• Binary (Base 2): Uses digits 0 and 1. Fundamental in computers as it represents electrical states (on/off).
• Octal (Base 8): Uses digits 0-7. Sometimes used as a compact way to represent binary numbers (since 8 is \(2^3\)).
• Hexadecimal (Base 16): Uses digits 0-9 and letters A-F (representing 10-15). Commonly used in computing to represent binary numbers concisely (since 16 is \(2^4\)).

Converting between number systems is important in various fields, especially when working with low-level computing concepts.