1. **Problem 1: Convert real numbers to IEEE-754 32-bit floating-point format** The IEEE-754 standard represents a real number in 32 bits as follows: - 1 bit for sign (0 for positive, 1 for negative) - 8 bits for exponent (with bias 127) - 23 bits for fraction (mantissa) after the leading 1 in normalized form **Steps to convert a number:** - Convert the number to binary scientific notation: $\pm 1.m \times 2^e$ - Calculate exponent bits: $e + 127$ - Write mantissa bits (fractional part after the leading 1) --- **For $X=70.75$:** 1. Convert to binary: $70 = 1000110_2$, $0.75 = 0.11_2$, so $70.75 = 1000110.11_2$ 2. Normalize: $1000110.11_2 = 1.00011011 \times 2^6$ 3. Exponent bits: $6 + 127 = 133 = 10000101_2$ 4. Mantissa (23 bits after decimal): $00011011000000000000000$ 5. Sign bit: 0 (positive) **IEEE-754 code for X:** $0\ 10000101\ 00011011000000000000000$ --- **For $Y = -250.5$:** 1. Convert to binary: $250 = 11111010_2$, $0.5 = 0.1_2$, so $250.5 = 11111010.1_2$ 2. Normalize: $11111010.1_2 = 1.11110101 \times 2^7$ 3. Exponent bits: $7 + 127 = 134 = 10000110_2$ 4. Mantissa (23 bits): $11110101000000000000000$ 5. Sign bit: 1 (negative) **IEEE-754 code for Y:** $1\ 10000110\ 11110101000000000000000$ --- **For $Z=0.375$:** 1. Convert to binary: $0.375 = 0.011_2$ 2. Normalize: $0.011_2 = 1.1 \times 2^{-2}$ 3. Exponent bits: $-2 + 127 = 125 = 01111101_2$ 4. Mantissa (23 bits): $10000000000000000000000$ 5. Sign bit: 0 (positive) **IEEE-754 code for Z:** $0\ 01111101\ 10000000000000000000000$ --- 2. **Problem 2: Decode IEEE-754 binary to real number** Given $X = (11011000011010110000000000000000)_2$ 1. Extract bits: - Sign bit: 1 (negative) - Exponent bits: next 8 bits $10110000_2 = 176_{10}$ - Mantissa bits: $11010110000000000000000$ 2. Calculate exponent: $e = 176 - 127 = 49$ 3. Mantissa (fractional part): $1 + \sum_{i=1}^{23} b_i 2^{-i}$ Calculate mantissa: Bits: 1 1 0 1 0 1 1 0 ... Positions with 1s: 1st, 2nd, 4th, 6th, 7th bits Value: $1 + 2^{-1} + 2^{-2} + 2^{-4} + 2^{-6} + 2^{-7} = 1 + 0.5 + 0.25 + 0.0625 + 0.015625 + 0.0078125 = 1.8359375$ 4. Calculate real number: $(-1)^1 \times 1.8359375 \times 2^{49}$ This is approximately: $-1.8359375 \times 5.6295 \times 10^{14} = -1.033 \times 10^{15}$ (very large negative number) --- 3. **Graph description:** The circle is given by: $$x^2 + y^2 = 1$$ The triangle vertices are at $(-1,0)$, $(1,0)$, and $(0,1)$. The triangle is right-angled with base along the x-axis from $-1$ to $1$ and height $1$. The shaded triangle lies inside the circle. --- **Summary:** - Problem 1: IEEE-754 codes for $X=70.75$, $Y=-250.5$, $Z=0.375$ given above. - Problem 2: Decoded $X$ from binary to approximately $-1.033 \times 10^{15}$. - Graph: Circle $x^2 + y^2 = 1$ with shaded triangle vertices $(-1,0)$, $(1,0)$, $(0,1)$.