Range of a fixed-width unsigned binary value: What is the maximum decimal value representable by a 4-bit binary number (using all bits for magnitude)?

Introduction / Context: Understanding representable ranges for a given bit width is essential for sizing registers, counters, and fields in digital designs. This question focuses on the maximum value a 4-bit unsigned binary number can represent when all bits are used for magnitude (no sign bit).

Given Data / Assumptions:

• Bit width N = 4 bits.
• Unsigned interpretation (no sign or bias).
• All 4 bits can take values 0 or 1 independently.

Concept / Approach: For an unsigned N-bit number, the value range is 0 through (2^N − 1). Substituting N = 4 gives 0 through 15. In binary, the maximum is 1111 which equals 8 + 4 + 2 + 1 = 15. This rule generalizes to any bit width and underpins field sizing in hardware description and microarchitecture design.

Step-by-Step Solution:

Verification / Alternative check: Count all 4-bit patterns (16 of them) from 0000 to 1111 → values 0..15 inclusive; the highest is 15.

Why Other Options Are Wrong: 16 requires 5 bits; 14 corresponds to 1110; 7 corresponds to 3 bits; 31 requires 5 bits (11111).

Common Pitfalls: Confusing the number of distinct patterns (2^N) with the maximum value (2^N − 1); mixing unsigned with signed two’s-complement ranges.

Final Answer: Correct (maximum value is 15)