1. **Problem statement:** Convert the 8-bit binary string $10101101_2$ into a floating-point number. 2. **Given:** 3 bits for exponent, 4 bits for mantissa, and 1 bit for sign (since 8 bits total: 1 sign + 3 exponent + 4 mantissa). 3. **Step 1: Identify the parts of the binary string:** - Sign bit: $1$ (first bit) - Exponent bits: $010$ (next 3 bits) - Mantissa bits: $1101$ (last 4 bits) 4. **Step 2: Interpret the sign bit:** - Sign bit $1$ means the number is negative. 5. **Step 3: Calculate the exponent:** - Exponent bits $010_2 = 2_{10}$ - Assuming bias for 3-bit exponent is $2^{3-1} - 1 = 3$ (bias = 3) - Actual exponent $= 2 - 3 = -1$ 6. **Step 4: Calculate the mantissa (fractional part):** - Mantissa bits $1101$ represent the fractional part after the leading 1 (normalized form). - Mantissa value $= 1 + \frac{1}{2} + \frac{1}{4} + 0 + \frac{1}{16} = 1 + 0.5 + 0.25 + 0 + 0.0625 = 1.8125$ 7. **Step 5: Calculate the floating-point number:** - Formula: $$\text{value} = (-1)^{\text{sign}} \times \text{mantissa} \times 2^{\text{exponent}}$$ - Substitute values: $$(-1)^1 \times 1.8125 \times 2^{-1} = -1.8125 \times 0.5 = -0.90625$$ 8. **Final answer:** The floating-point number represented by $10101101_2$ is **$-0.90625$**.