State the octal equivalent of hexa decimal number (B34) 16 .

Question

Prepp Team · Accepted Answer

Hexadecimal to Octal Conversion Explained

Understanding how to convert numbers between different bases is a fundamental concept in digital electronics and computer science. This solution will guide you through the process of converting a hexadecimal number to its octal equivalent, using (B34)₁₆ as an example.

Number Systems Basics

Hexadecimal (Base-16): Uses 16 distinct symbols: 0-9 and A, B, C, D, E, F. Each hexadecimal digit corresponds to 4 binary bits.
Octal (Base-8): Uses 8 distinct symbols: 0-7. Each octal digit corresponds to 3 binary bits.
Binary (Base-2): Uses 2 distinct symbols: 0 and 1. This is an intermediate step for converting between hexadecimal and octal.

Step-by-Step Conversion of Hexadecimal to Octal

To convert the hexadecimal number (B34)₁₆ to its octal equivalent, we follow a two-step process:

Hexadecimal to Binary Conversion.
Binary to Octal Conversion.

Hexadecimal to Binary Conversion

Each hexadecimal digit can be directly converted into its 4-bit binary equivalent. Let's break down (B34)₁₆:

Hexadecimal Digit	Decimal Value	4-bit Binary Equivalent
B	11	1011
3	3	0011
4	4	0100

Concatenating these binary equivalents gives us the binary representation of (B34)₁₆:

$$(\text{B34})_{16} = (1011 \ 0011 \ 0100)_2$$

Binary to Octal Conversion

Now that we have the binary equivalent, we group the binary digits into sets of three, starting from the rightmost digit. If the leftmost group has fewer than three digits, add leading zeros to complete the group. Then, convert each 3-bit group into its corresponding octal digit.

Our binary number is $$(101100110100)_2$$

Let's group the digits from right to left:

Group 1 (Rightmost): $$100_2$$
Group 2: $$110_2$$
Group 3: $$100_2$$
Group 4 (Leftmost): $$101_2$$

Now, convert each 3-bit binary group to its octal equivalent:

3-bit Binary Group	Octal Equivalent
$$101_2$$	$$5_8$$
$$100_2$$	$$4_8$$
$$110_2$$	$$6_8$$
$$100_2$$	$$4_8$$

Combining these octal digits in order gives us the final octal equivalent:

$$(\text{B34})_{16} = (101100110100)_2 = (5464)_8$$

Conclusion

The octal equivalent of the hexadecimal number (B34)₁₆ is (5464)₈. This step-by-step method ensures an accurate conversion between hexadecimal and octal number systems by using binary as an intermediate step.