Understanding Number Systems and Bit Requirements

Number systems are different ways of representing numerical values. Each number system has a base, which determines the number of unique digits used to represent numbers. The most common number systems encountered in computing are Binary (base 2), Octal (base 8), Decimal (base 10), and Hexadecimal (base 16).

The question asks us to arrange these number systems in ascending order based on the number of bits needed to represent the same number. While the intrinsic value of a number requires a specific minimum number of bits (determined by $\lceil \log_2(N+1) \rceil$), the way a number is represented in a particular base affects the number of digits required. The question appears to relate the "number of bits needed" to the efficiency of the number system in terms of the count of digits required to represent a number of a given magnitude.

Relationship Between Base and Number of Digits

For a given numerical value, the number of digits required to represent it in a particular base is inversely proportional to the size of the base. A larger base means each digit can represent a wider range of values, and therefore, fewer digits are needed to represent the same large number compared to a system with a smaller base.

Let's consider the bases of the given number systems:

• Binary (A): Base 2
• Decimal (B): Base 10
• Hexadecimal (C): Base 16
• Octal (D): Base 8

Arranging the bases in ascending order:

2 (Binary), 8 (Octal), 10 (Decimal), 16 (Hexadecimal)

Since the number of digits needed is inversely proportional to the base, the number of digits required to represent the same number will be in the descending order of the bases:

Hexadecimal (Base 16) < Decimal (Base 10) < Octal (Base 8) < Binary (Base 2)

This order represents the systems from requiring the fewest digits (Hexadecimal) to requiring the most digits (Binary) for the same numerical value.

Connecting Number of Digits to Bits Needed

Although the question asks about "bits needed", in contexts comparing number systems representation, this often correlates with the number of digits. For instance, if we were to store the string representation of a number, a representation with fewer digits would generally require fewer bits. Alternatively, if we consider standard encodings like using 4 bits per decimal digit (BCD), 3 bits per octal digit, and 4 bits per hexadecimal digit, the total number of bits required is the number of digits multiplied by the bits per digit. While this calculation can be complex and dependent on the specific number, the fundamental relationship that higher bases require fewer digits holds true and appears to be the basis for the ordering in the options.

Based on the principle that fewer digits are needed for higher bases, the order from the system needing the fewest digits (and thus, arguably, fewest bits for its representation) to the most digits is:

1. Hexadecimal (C) - Base 16: Fewest digits needed.
2. Decimal (B) - Base 10: More digits than Hexadecimal. ($\log_{10} N > \log_{16} N$)
3. Octal (D) - Base 8: More digits than Decimal. ($\log_8 N > \log_{10} N$)
4. Binary (A) - Base 2: Most digits needed. ($\log_2 N > \log_8 N$)

Therefore, arranging the number systems in ascending order based on the number of bits (related to the number of digits) needed to represent the same number results in the order: C, B, D, A.

The final ascending order based on the number of bits needed (corresponding to the number of digits) is Hexadecimal, Decimal, Octal, Binary.

Revision Table: Number Systems and Representation

Review the key characteristics of different number systems:

• Binary (Base 2): Uses digits 0 and 1. Fundamental for digital circuits.
• Octal (Base 8): Uses digits 0-7. Often used as a compact representation of binary numbers (3 bits per octal digit).
• Decimal (Base 10): Uses digits 0-9. The everyday number system.
• Hexadecimal (Base 16): Uses digits 0-9 and A-F (representing 10-15). Widely used in computing as a compact representation of binary numbers (4 bits per hexadecimal digit).

Additional Information: Number Representation and Efficiency

While the question focuses on the number of bits related to the representation's length (number of digits), it's important to note that the underlying numerical value itself requires a minimum number of bits regardless of the base it's written in. For example, the number 255 requires 8 bits because $2^7 < 255 ≤ 2^8-1$. This is true whether 255 is written as $(11111111)_2$, $(377)_8$, $(255)_{10}$, or $(FF)_{16}$. However, storing the representation as a string would take varying amounts of memory depending on the number of digits and the character encoding used.

The efficiency discussed in the solution relates to the conciseness of the representation in terms of digits. Higher bases offer more compact representations.

Understanding the relationship between a number's value, its base representation, and the number of digits needed is crucial in computer science and digital systems.