1. **Problem statement:** Given the real numbers $X=25.125$ and $Y=39.625$, express them using fixed-point binary and hexadecimal coding, then perform binary arithmetic operations $X+Y$ and $X-Y$. 2. **Fixed-point binary coding:** Fixed-point representation splits the number into integer and fractional parts, representing each in binary. 3. **Convert $X=25.125$ to binary:** - Integer part 25 in binary: $25_{10} = 11001_2$ - Fractional part 0.125 in binary: $0.125 \times 2 = 0.25$ (bit 0), $0.25 \times 2 = 0.5$ (bit 0), $0.5 \times 2 = 1.0$ (bit 1) - Fractional bits: 001 - So, $X = 11001.001_2$ 4. **Convert $Y=39.625$ to binary:** - Integer part 39 in binary: $39_{10} = 100111_2$ - Fractional part 0.625 in binary: $0.625 \times 2 = 1.25$ (bit 1), $0.25 \times 2 = 0.5$ (bit 0), $0.5 \times 2 = 1.0$ (bit 1) - Fractional bits: 101 - So, $Y = 100111.101_2$ 5. **Fixed-point hexadecimal coding:** Group binary digits in 4 bits from the decimal point. 6. **Convert $X=11001.001_2$ to hex:** - Integer part: $11001_2 = 1 1001$ pad to 8 bits: $00011001_2 = 19_{16}$ - Fractional part: $001_2$ pad to 4 bits: $0010_2 = 2_{16}$ - So, $X = 19.2_{16}$ 7. **Convert $Y=100111.101_2$ to hex:** - Integer part: $100111_2 = 0010 0111_2 = 27_{16}$ - Fractional part: $101_2$ pad to 4 bits: $1010_2 = A_{16}$ - So, $Y = 27.A_{16}$ 8. **Binary arithmetic operations:** Align fractional bits to 3 places. 9. **Add $X+Y$ in binary:** $11001.001_2 + 100111.101_2$ Convert to decimal for verification: $25.125 + 39.625 = 64.75$ Add binary: $11001.001_2 = 25.125$ $100111.101_2 = 39.625$ Sum: $1000000.11_2$ Check $1000000.11_2$: $1000000_2 = 64$, fractional $0.11_2 = 0.75$ Sum = $64.75$ 10. **Subtract $X-Y$ in binary:** $11001.001_2 - 100111.101_2$ Decimal: $25.125 - 39.625 = -14.5$ Represent negative result in two's complement or signed fixed-point (not shown here). 11. **Summary:** - $X=11001.001_2 = 19.2_{16}$ - $Y=100111.101_2 = 27.A_{16}$ - $X+Y=1000000.11_2 = 40.75_{10}$ - $X-Y = -14.5$ (requires signed representation)