1. The problem is to represent the decimal number 57.625 in binary 16-bit floating-point format, showing the sign bit, exponent, and mantissa. 2. First, determine the sign bit: since 57.625 is positive, the sign bit is $0$. 3. Convert the integer part 57 to binary: $$57_{10} = 111001_2$$ 4. Convert the fractional part 0.625 to binary: Multiply by 2: $0.625 \times 2 = 1.25$ (integer part 1) $0.25 \times 2 = 0.5$ (integer part 0) $0.5 \times 2 = 1.0$ (integer part 1) So, $0.625_{10} = 0.101_2$ 5. Combine integer and fractional parts: $$57.625_{10} = 111001.101_2$$ 6. Normalize the binary number to the form $1.xxxxx \times 2^n$: Move the binary point 5 places left: $$111001.101_2 = 1.11001101 \times 2^5$$ 7. For 16-bit floating-point (half precision), the format is: - 1 sign bit - 5 exponent bits (bias 15) - 10 mantissa bits 8. Calculate the biased exponent: $$\text{exponent} = 5 + 15 = 20$$ In binary: $$20_{10} = 10100_2$$ 9. Mantissa is the fractional part after the leading 1: $$11001101$$ plus zeros to fill 10 bits: $$1100110100$$ 10. Assemble the 16-bit representation: - Sign bit: $0$ - Exponent: $10100$ - Mantissa: $1100110100$ Final 16-bit binary representation: $$0\ 10100\ 1100110100$$ This corresponds to the binary string: $$0101001100110100$$