Final answer:
The conversion of given numbers into their closest single precision IEEE 754 floating point representation involves determining the sign bit, normalizing the number to get the mantissa, calculating the biased exponent, and then assembling these parts into a 32-bit binary format which is converted into hexadecimal. Zero and negative infinity are represented by special patterns in IEEE 754 format.
Step-by-step explanation:
The IEEE 754 standard for floating-point representation is a common way to represent real numbers in computers, which allows for a wide range of values to be stored in a fixed amount of space. The single-precision floating-point format uses 32 bits, consisting of a sign bit, an 8-bit biased exponent, and a 23-bit fraction (mantissa).
Let's convert the given numbers into their IEEE 754 representations:
- 50.4375: This is a positive number, so the sign bit is 0. The number is expressed as 1.5809375 x 2^5 in binary format, making the exponent 5 + 127 (bias) = 132, or '10000100' in binary. The mantissa is the fractional part after removing the implicit leading one, which is '00111100011100000000000'. The entire IEEE representation is '0 10000100 00111100011100000000000', and its hexadecimal value is 0x4247C000.
- 0.0: This is a special case that represents zero. All bits are set to zero, giving a hexadecimal value of 0x00000000.
- -Infinity: Negative infinity has a sign bit of 1, all exponent bits set to 1, and all fraction bits set to 0. Its hexadecimal value is 0xFF800000.
- 1.0000001: The sign bit is 0, with the number represented as 1.0000001 x 2^0. The exponent is 0 + 127 = 127, or '01111111' in binary. The mantissa is the fractional part after the leading one, which is very close to '00000000000000000000000', considering the precision limits. The hexadecimal value is 0x3F800002.
Remember to always round the mantissa when beyond 23 bits of precision, and adjust the exponent accordingly when normalizing the number.