Analyzing Statements on Bits and Data Storage

The question asks us to evaluate two statements related to the number of bits required to store data and the size of a Peta Byte in bits. Let's analyze each statement carefully.

Statement I: Minimum Bits for a 3-Digit Decimal Number

Statement I says the minimum number of bits required to store any 3-digit decimal number is 10. A 3-digit decimal number can range from 000 to 999. The total number of unique values in this range is $999 - 000 + 1 = 1000$.

To determine the minimum number of bits needed to represent $N$ unique values, we need to find the smallest integer $n$ such that $2^n \ge N$. In this case, $N = 1000$.

• $2^9 = 512$ (This is less than 1000)
• $2^{10} = 1024$ (This is greater than or equal to 1000)

Since $2^{10}$ is the smallest power of 2 that is greater than or equal to 1000, we need a minimum of 10 bits to represent all 1000 possible 3-digit decimal numbers. Therefore, Statement I is correct.

Statement II: Peta Byte Size in Bits

Statement II claims that one Peta Byte is equal to $2^{50}$ bits. Let's examine the standard units of digital data storage:

• 1 Byte (B) = 8 bits
• 1 Kilo Byte (KB) = $1024$ Bytes = $2^{10}$ Bytes
• 1 Mega Byte (MB) = $1024$ KB = $2^{10}$ KB = $2^{10} \times 2^{10}$ Bytes = $2^{20}$ Bytes
• 1 Giga Byte (GB) = $1024$ MB = $2^{10}$ MB = $2^{10} \times 2^{20}$ Bytes = $2^{30}$ Bytes
• 1 Tera Byte (TB) = $1024$ GB = $2^{10}$ GB = $2^{10} \times 2^{30}$ Bytes = $2^{40}$ Bytes
• 1 Peta Byte (PB) = $1024$ TB = $2^{10}$ TB = $2^{10} \times 2^{40}$ Bytes = $2^{50}$ Bytes

So, one Peta Byte is equal to $2^{50}$ Bytes. To convert Bytes to bits, we multiply by 8 (since 1 Byte = 8 bits):

$1 \text{ PB} = 2^{50} \text{ Bytes} = 2^{50} \times 8 \text{ bits}$

Since $8 = 2^3$, we have:

$1 \text{ PB} = 2^{50} \times 2^3 \text{ bits} = 2^{50+3} \text{ bits} = 2^{53} \text{ bits}$

Therefore, one Peta Byte is equal to $2^{53}$ bits, not $2^{50}$ bits as stated in Statement II. Statement II is false.

Conclusion

Based on our analysis:

• Statement I: Minimum number of bits required to store any 3-digit decimal number is 10. (True)
• Statement II: One Peta Byte is equal to $2^{50}$ bits. (False, it is $2^{53}$ bits)

Thus, Statement I is correct, but Statement II is false.

Comparison with Options

Let's compare our findings with the given options:

• Option 1: Both Statement I and Statement II are true (Incorrect)
• Option 2: Both Statement I and Statement II are false (Incorrect)
• Option 3: Statement I is correct but Statement II is false (Correct)
• Option 4: Statement I is incorrect but Statement II is true (Incorrect)

Revision Table: Bits and Data Units

Additional Information: Data Representation and Storage

Digital computers store and process data in binary form, using bits (0s and 1s). The number of bits determines the number of distinct values that can be represented. For example, 1 bit can represent 2 values ($2^1$), 2 bits can represent 4 values ($2^2$), and $n$ bits can represent $2^n$ values.

Larger units of data storage, such as Bytes, Kilobytes, Megabytes, Gigabytes, Terabytes, and Petabytes, are based on powers of 2, specifically $2^{10} = 1024$. This is because memory addressing and bus architectures in computers often align well with powers of 2. Understanding these units and their relationship to bits is fundamental to understanding how data is stored and managed in digital systems.

The difference between the decimal prefixes (like kilo=1000, mega=1,000,000) and the binary prefixes (like Kibi=1024, Mebi=1024^2) is important in computing. While KB often means $2^{10}$ Bytes in computing contexts, sometimes the decimal meaning (1000 Bytes) is used, especially in networking or disk drive marketing. However, for units like MB, GB, TB, PB in relation to memory or OS reporting, the binary interpretation ($2^{10}$ progression) is standard.