1. **Problem statement:** Given real number $X = 4015.9375$ (decimal), express it in base 16, then deduce its base 2 and base 8 forms. 2. **Formula and rules:** - To convert the integer part to base 16, divide by 16 repeatedly and record remainders. - To convert the fractional part to base 16, multiply by 16 repeatedly and record the integer parts. - Base 2 and base 8 can be deduced from base 16 by converting each hex digit to 4 bits (base 2) or grouping bits for base 8. 3. **Convert integer part 4015 to base 16:** - $4015 \div 16 = 250$ remainder $15$ (F) - $250 \div 16 = 15$ remainder $10$ (A) - $15 \div 16 = 0$ remainder $15$ (F) So integer part in hex is $\text{FAF}$. 4. **Convert fractional part 0.9375 to base 16:** - $0.9375 \times 16 = 15.0$ integer part $15$ (F) So fractional part in hex is $F$. 5. **Therefore, $X$ in base 16 is:** $$X = (\text{FAF}.F)_{16}$$ 6. **Convert $X$ to base 2:** - Each hex digit to 4 bits: - F = 1111 - A = 1010 - F = 1111 - .F = .1111 So, $$X = (1111\ 1010\ 1111.1111)_2$$ 7. **Convert $X$ to base 8:** - Group bits in 3s from binary: Binary: 111110101111.1111 Group integer part: 1 111 101 011 111 Pad left with zeros: 001 111 101 011 111 Octal digits: 1 7 5 3 7 Fractional part: .1111 = .111 100 (pad right) Fractional octal digits: 7 4 So, $$X = (17537.74)_8$$ 8. **Problem 1b:** Given $Y = (50.1)_{16}$ 9. **Convert $Y$ to decimal:** - Integer part: $5 \times 16 + 0 = 80$ - Fractional part: $1 \times 16^{-1} = 1/16 = 0.0625$ So, $$Y = 80.0625$$ 10. **Calculate $Z = X + Y$ in base 16:** - $X = (FAF.F)_{16} = 4015.9375$ decimal - $Y = (50.1)_{16} = 80.0625$ decimal - Sum in decimal: $Z = 4015.9375 + 80.0625 = 4096$ 11. **Convert $Z=4096$ to base 16:** - $4096 \div 16 = 256$ remainder 0 - $256 \div 16 = 16$ remainder 0 - $16 \div 16 = 1$ remainder 0 - $1 \div 16 = 0$ remainder 1 So, $$Z = (1000)_ {16}$$ 12. **Convert $Z$ to base 2:** - $4096 = 2^{12}$ - So, $$Z = (1\ 0000\ 0000\ 0000)_2$$ 13. **Calculate $\sqrt{Z}$ and $\sqrt[3]{Z}$ in base 10 and base 2:** - $\sqrt{4096} = 64$ - $64$ in base 2 is $1000000_2$ - $\sqrt[3]{4096} = \sqrt[3]{2^{12}} = 2^{4} = 16$ - $16$ in base 2 is $10000_2$ 14. **Find largest natural number $n$ such that $\sqrt[n]{Z} > 1$:** - $\sqrt[n]{4096} = 4096^{1/n} = 2^{12/n} > 1$ - Since $2^{12/n} > 1$ for all $n$ where $12/n > 0$ - The largest $n$ is when $2^{12/n} > 1$ but $2^{12/(n+1)} \leq 1$ - Since $2^{0} = 1$, $n$ can be arbitrarily large but must be natural. - So largest $n$ is $\infty$ theoretically, but practically any $n$. 15. **Problem 2a:** Given $A = (FAF.F)_{16}$ express as $A = \pm 1,M \times 2^{E}$ 16. **Convert $A$ to decimal:** - $FAF.F_{16} = 4015.9375$ decimal (from above) 17. **Express in normalized binary form:** - $A = 4015.9375 = 111110101111.1111_2$ - Normalize: move decimal point left 11 places: $$A = 1.1111010111111111_2 \times 2^{11}$$ 18. **So, $M = 1111010111111111_2$ (fractional part after 1), $E=11$** 19. **Problem 2b:** IEEE-754 single precision (32 bits): - Sign bit: 0 (positive) - Exponent: $E + 127 = 11 + 127 = 138 = 10001010_2$ - Mantissa: fractional bits after leading 1, take 23 bits: $11110101111111110000000$ 20. **Binary representation:** $$0\ 10001010\ 11110101111111110000000$$ 21. **Problem 2c:** Convert binary to hex: - Sign + exponent: $0100\ 0101\ 0 = 0x45$ - Mantissa: $1111 0101 1111 1111 0000 0000$ grouped: - 1111 = F - 0101 = 5 - 1111 = F - 1111 = F - 0000 = 0 - 0000 = 0 So mantissa hex: $F5FF00$ 22. **Full hex representation:** $$0x45F5FF00$$ 23. **Problem 2d:** For $B = -A$: - Sign bit flips to 1 - Exponent and mantissa same - Hex representation changes first hex digit sign bit: $$0xC5F5FF00$$ **Final answers:** - $X = (FAF.F)_{16} = (111110101111.1111)_2 = (17537.74)_8$ - $Y = (50.1)_{16} = 80.0625_{10}$ - $Z = X + Y = (1000)_{16} = 4096_{10} = (1 0000 0000 0000)_2$ - $\sqrt{Z} = 64_{10} = 1000000_2$ - $\sqrt[3]{Z} = 16_{10} = 10000_2$ - Largest $n$ with $\sqrt[n]{Z} > 1$ is any natural number - $A = +1.1111010111111111_2 \times 2^{11}$ - IEEE-754 single precision binary: $0 10001010 11110101111111110000000$ - Hex representation of $A$: $45F5FF00$ - Hex representation of $B = -A$: $C5F5FF00$