Digital electronics: Convert the decimal number 151.75 into its binary representation. Be sure to show the correct integer part and the fractional part (to two binary fractional places as appropriate).
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A10000111.11
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B11010011.01
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C00111100.00
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D10010111.11
Answer
Correct Answer: 10010111.11
Explanation
Introduction / Context:Binary conversion is a foundational skill in computer engineering and digital electronics. This problem asks for converting a mixed decimal number (an integer plus a fractional part) into binary, which reinforces the two complementary procedures: repeated division by 2 for the integer portion and repeated multiplication by 2 for the fractional portion. Mastering both methods is essential for interpreting numeric data, fixed-point formats, and low-level representations used in microcontrollers and digital signal processing.
Given Data / Assumptions:
- Decimal value: 151.75.
- Target base: base 2 (binary).
- Precision: fraction has exactly 0.75, which is exactly representable in two binary fractional bits.
- No rounding is required because 0.75 is an exact dyadic fraction.
Concept / Approach:Convert the integer part using repeated division by 2 and noting remainders. Convert the fractional part using repeated multiplication by 2 and recording integer carries. The final binary number is integer_bits.fraction_bits. Since 0.75 equals 3/4, which is 0.11 in binary, the fractional conversion is straightforward.
Step-by-Step Solution:
Integer part: 151 / 2 → remainders give bits from LSB to MSB.151 = 128 + 16 + 4 + 2 + 1 → bits at 128, 16, 4, 2, 1 set → 10010111.Fractional part: 0.75 * 2 = 1.5 → take 1, keep 0.5.0.5 * 2 = 1.0 → take 1, fraction becomes 0 → fractional bits = .11.Combine: 151.75 → 10010111.11 (binary).Verification / Alternative check:Convert back: 10010111.11 = 128 + 16 + 4 + 2 + 1 + 1/2 + 1/4 = 151 + 0.75 = 151.75. The round-trip confirms the correctness without rounding error.
Why Other Options Are Wrong:
- 10000111.11: Integer part 135, not 151.
- 11010011.01: Represents a different integer and a non-matching fractional part.
- 00111100.00: Equals 60 decimal with zero fraction, not 151.75.
Common Pitfalls:Reversing remainder order for the integer conversion, misplacing the binary point, or confusing 0.75 with 0.3 repeating in binary. Remember that fractions with denominators that are powers of 2 convert exactly to a finite number of binary fractional bits.
Final Answer:10010111.11