Converting Octal to Hexadecimal Explained

The question asks us to convert the octal number (1235)₈ into its equivalent hexadecimal representation.

To convert an octal number to a hexadecimal number, a common method is to first convert the octal number to its binary equivalent, and then convert the binary number to its hexadecimal equivalent.

Step 1: Convert Octal to Binary

Each digit in an octal number (base 8) can be represented by exactly 3 binary digits (bits), since \(2^3 = 8\). We convert each octal digit of (1235)₈ into its 3-bit binary equivalent:

• Octal digit 1 is 001 in binary.
• Octal digit 2 is 010 in binary.
• Octal digit 3 is 011 in binary.
• Octal digit 5 is 101 in binary.

Concatenating these binary representations, we get:

1 \(\rightarrow\) 001

2 \(\rightarrow\) 010

3 \(\rightarrow\) 011

5 \(\rightarrow\) 101

So, the binary equivalent of (1235)₈ is 001010011101.

Step 2: Convert Binary to Hexadecimal

Each digit in a hexadecimal number (base 16) can be represented by exactly 4 binary digits (bits), since \(2^4 = 16\). To convert the binary number 001010011101 to hexadecimal, we group the binary digits in sets of 4, starting from the rightmost digit. If the leftmost group has fewer than 4 digits, we add leading zeros to complete the group.

Our binary number is 001010011101.

Grouping in sets of 4 from right to left:

1101

0011

0100

001

The leftmost group '001' has only 3 digits. We add a leading zero to make it 4 digits: '0001'. (Wait, let's re-check the grouping from the original binary 001010011101)

Let's group correctly from right:

Binary: 0010 1001 1101

Group 1 (rightmost): 1101

Group 2: 1001

Group 3 (leftmost): 0010

Now, we convert each 4-bit group into its hexadecimal equivalent:

• 1101 in binary is 13 in decimal. In hexadecimal, 13 is represented by the digit D.
• 1001 in binary is 9 in decimal. In hexadecimal, 9 is represented by the digit 9.
• 0010 in binary is 2 in decimal. In hexadecimal, 2 is represented by the digit 2.

Combining these hexadecimal digits from left to right (corresponding to the binary groups from left to right), we get 29D.

So, the hexadecimal equivalent of the binary number 001010011101 is (29D)₁₆.

Summary of Conversion

Therefore, (1235)₈ is equivalent to (29D)₁₆.

Comparing this result with the given options, we see that option 1 is (029D)₁₆. A leading zero does not change the value of a number, so (29D)₁₆ is the same as (029D)₁₆.

Final Answer Derivation

The conversion process yields (29D)₁₆. Option 1 is (029D)₁₆, which is the same value. Thus, option 1 is the correct equivalent.

Revision Table: Number System Conversions

Additional Information: Bases and Place Values

Number systems use different bases to represent quantities. The base indicates the number of unique digits used in the system, including zero.

• Decimal (Base 10): Uses digits 0-9. Place values are powers of 10 (\(10^0, 10^1, 10^2, ...\)).
• Binary (Base 2): Uses digits 0 and 1. Place values are powers of 2 (\(2^0, 2^1, 2^2, ...\)). This is fundamental in computing.
• Octal (Base 8): Uses digits 0-7. Place values are powers of 8 (\(8^0, 8^1, 8^2, ...\)). Often used as a shorthand for binary because 8 is a power of 2 (\(8 = 2^3\)).
• Hexadecimal (Base 16): Uses digits 0-9 and letters A-F (where A=10, B=11, C=12, D=13, E=14, F=15). Place values are powers of 16 (\(16^0, 16^1, 16^2, ...\)). Often used as a shorthand for binary because 16 is a power of 2 (\(16 = 2^4\)).

Understanding the relationship between bases, especially how bases 8 and 16 relate to base 2, simplifies conversions between these systems.