13.3 Floating-point numbers, representation and manipulation (3)
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1.
Consider a decimal number 65536. Convert this number to its binary floating-point representation using the single-precision (32-bit) IEEE 754 format. Show each step of the conversion, including the calculation of the exponent and mantissa.
1. Convert 65536 to binary: 65536 in decimal is 10000000000 in binary.
2. Normalize the binary number: The normalized form is 1.10000000000000000000000 (we've added an implicit leading '1').
3. Determine the sign bit: Since 65536 is positive, the sign bit is 0.
4. Calculate the exponent: The exponent in the normalized form is 10. The biased exponent is (10 - 127) = -117. The exponent in binary is 01110101.
5. Determine the mantissa: The mantissa is 10000000000000000000000. Since we are using 23 bits for the mantissa, we only take the first 23 bits: 10000000000000000000000.
6. Combine the bits: The complete 32-bit representation is: 0 01110101 10000000000000000000000
7. Represent in hexadecimal: The binary representation is 0x40000000.
2.
Explain the process of normalising floating-point numbers. Include a discussion of the benefits of normalisation in terms of representation and precision. Consider the implications of using different exponents and mantissas.
Normalising floating-point numbers involves adjusting the representation of a number to ensure that the mantissa (significand) has a leading '1' (except for denormalised numbers). This is achieved by shifting the binary point until only one non-zero digit is to the left of the binary point. The exponent is then adjusted accordingly to maintain the original value of the number.
Benefits of Normalisation:
- Improved Representation: Normalisation allows for a more compact and efficient representation of numbers. It avoids having many leading zeros in the mantissa, which would waste storage space.
- Increased Precision: By ensuring a consistent leading '1' in the mantissa, normalisation improves the precision of floating-point numbers. This is because it allows for a more accurate representation of numbers with varying magnitudes.
- Avoidance of Denormalisation: Normalisation helps to avoid the issue of denormalisation, where the exponent is too small to represent a significant number, leading to a loss of precision.
Implications of Exponent and Mantissa:
The exponent determines the magnitude of the number, while the mantissa determines the precision. A larger exponent represents a larger number, and a larger mantissa represents higher precision. Normalisation ensures that the relationship between the exponent and mantissa is consistent, allowing for accurate representation of numbers across a wide range of magnitudes. Incorrectly handling the exponent and mantissa can lead to significant errors in floating-point calculations.
3.
Describe the format of a single-precision (32-bit) binary floating-point real number according to the IEEE 754 standard. Include details of the number of bits allocated to each component (sign, exponent, and mantissa) and their respective representations.
A single-precision (32-bit) binary floating-point real number, as defined by the IEEE 754 standard, is structured into three main components:
- Sign Bit (1 bit): The most significant bit (leftmost) represents the sign of the number. 0 indicates a positive number, and 1 indicates a negative number.
- Exponent (8 bits): The next 8 bits represent the exponent. These bits are biased by 127. The actual exponent is calculated as (value in exponent bits) - 127. This bias allows for the representation of both positive and negative exponents.
- Mantissa/Fraction (23 bits): The remaining 23 bits represent the fractional part of the number (also known as the significand). The mantissa is normalized, meaning it has an implicit leading '1' bit (except for denormalized numbers). This allows for a greater precision in representing fractional values.
In summary, the 32-bit format is: