In digital electronics, what is the highest-value seven-bit (7-bit) binary number when interpreted as an unsigned quantity?
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A128
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B127
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C126
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D125
Answer
Correct Answer: 127
Explanation
Introduction:Seven-bit binary numbers are common in legacy character encodings and basic digital counters. The question asks for the highest-value number representable with 7 bits when interpreted as an unsigned quantity (no sign bit).
Given Data / Assumptions:
- Unsigned binary representation with exactly 7 bits.
- Bit positions range from b6 (MSB) to b0 (LSB).
- No sign or bias; pure base-2 magnitude.
Concept / Approach:For an n-bit unsigned integer, the minimum value is 0 and the maximum value is 2^n - 1. Setting all n bits to 1 yields the maximum value. For n = 7, the maximum is 2^7 - 1 = 128 - 1 = 127.
Step-by-Step Solution:
Let n = 7.Max unsigned value = 2^n - 1.Compute 2^7 = 128.Subtract 1: 128 - 1 = 127.Binary pattern 1111111₂ equals 127 in decimal.Verification / Alternative check:Write the place values: 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127. This matches 2^7 - 1.
Why Other Options Are Wrong:
- 128: Requires 8 bits (10000000₂); exceeds 7-bit range.
- 126: One less than maximum; binary 1111110₂.
- 125: Two less than maximum; binary 1111101₂.
Common Pitfalls:Confusing signed with unsigned. In signed 7-bit (two's complement is defined for 8-bit, not 7), ranges differ. Here we clearly use unsigned 7-bit magnitude.
Final Answer:127