a) 85.125 in IEEE 754 single precision floating point representation:

Step 1: Convert the integer part to binary. 85 in binary is 1010101.

Step 2: Convert the fractional part to binary. 0.125 in binary is 0.001.

Step 3: Combine the integer and fractional parts. So, 85.125 in binary is 1010101.001.

Step 4: Normalize the binary number. The normalized form is 1.010101001 x 2^6.

Step 5: The exponent is 6. In IEEE 754, the exponent needs to be biased by adding 127. So, the biased exponent is 6 + 127 = 133. In binary, 133 is 10000101.

Step 6: The mantissa is the part after the decimal point in the normalized form, which is 010101001. In IEEE 754, the mantissa is 23 bits. So, we pad it with zeros on the right to make it 23 bits: 01010100100000000000000.

Step 7: Combine the sign bit, exponent, and mantissa. The sign bit is 0 (since the number is positive), the exponent is 10000101, and the mantissa is 01010100100000000000000. So, the IEEE 754 representation of 85.125 is 0 10000101 01010100100000000000000.

b) -4.75 in IEEE 754 single precision floating point representation:

Step 1: Convert the integer part to binary. 4 in binary is 100.

Step 2: Convert the fractional part to binary. 0.75 in binary is 0.11.

Step 3: Combine the integer and fractional parts. So, 4.75 in binary is 100.11.

Step 4: Normalize the binary number. The normalized form is 1.0011 x 2^2.

Step 5: The exponent is 2. In IEEE 754, the exponent needs to be biased by adding 127. So, the biased exponent is 2 + 127 = 129. In binary, 129 is 10000001.

Step 6: The mantissa is the part after the decimal point in the normalized form, which is 0011. In IEEE 754, the mantissa is 23 bits. So, we pad it with zeros on the right to make it 23 bits: 00110000000000000000000.

Step 7: Combine the sign bit, exponent, and mantissa. The sign bit is 1 (since the number is negative), the exponent is 10000001, and the mantissa is 00110000000000000000000. So, the IEEE 754 representation of -4.75 is 1 10000001 00110000000000000000000.

Question

a) 85.125 in IEEE 754 single precision floating point representation:

Step 1: Convert the integer part to binary. 85 in binary is 1010101.

Step 2: Convert the fractional part to binary. 0.125 in binary is 0.001.

Step 3: Combine the integer and fractional parts. So, 85.125 in binary is 1010101.001.

Step 4: Normalize the binary number. The normalized form is 1.010101001 x 2^6.

Step 5: The exponent is 6. In IEEE 754, the exponent needs to be biased by adding 127. So, the biased exponent is 6 + 127 = 133. In binary, 133 is 10000101.

Step 6: The mantissa is the part after the decimal point in the normalized form, which is 010101001. In IEEE 754, the mantissa is 23 bits. So, we pad it with zeros on the right to make it 23 bits: 01010100100000000000000.

Step 7: Combine the sign bit, exponent, and mantissa. The sign bit is 0 (since the number is positive), the exponent is 10000101, and the mantissa is 01010100100000000000000. So, the IEEE 754 representation of 85.125 is 0 10000101 01010100100000000000000.

b) -4.75 in IEEE 754 single precision floating point representation:

Step 1: Convert the integer part to binary. 4 in binary is 100.

Step 2: Convert the fractional part to binary. 0.75 in binary is 0.11.

Step 3: Combine the integer and fractional parts. So, 4.75 in binary is 100.11.

Step 4: Normalize the binary number. The normalized form is 1.0011 x 2^2.

Step 5: The exponent is 2. In IEEE 754, the exponent needs to be biased by adding 127. So, the biased exponent is 2 + 127 = 129. In binary, 129 is 10000001.

Step 6: The mantissa is the part after the decimal point in the normalized form, which is 0011. In IEEE 754, the mantissa is 23 bits. So, we pad it with zeros on the right to make it 23 bits: 00110000000000000000000.

Step 7: Combine the sign bit, exponent, and mantissa. The sign bit is 1 (since the number is negative), the exponent is 10000001, and the mantissa is 00110000000000000000000. So, the IEEE 754 representation of -4.75 is 1 10000001 00110000000000000000000.

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