Exercises

True. Floating point:

• Provides support for a wide range of values. (Both very small and very large)

• Helps programmers deal with errors in real arithmetic because floating point can represent $+\infty$, $-\infty$, $\text{NaN}$ (Not a Number)

• Keeps high precision. Recall that precision is a count of the number of bits in a computer word used to represent a value. IEEE 754 allocates a majority of bits for the significand, allowing for the use of a combination of negative powers of two to represent fractions.

False. Floating Point can represent infinities as well as NaNs, so the total amount of representable numbers is lower than Two’s Complement, where every bit combination maps to a unique integer value.

True. The uneven spacing is due to the exponent representation of floating point numbers. There are a fixed number of bits in the significand. In IEEE 32-bit storage there are $23$ bits for the significand, which means the LSB represents $2^{−23}$ times 2 to the exponent. For example, if the exponent is zero (after allowing for the offset) the difference between two neighboring floats will be $2^{−23}$. If the exponent is $8$, the difference between two neighboring floats will be $2^{−15}$ because the mantissa is multiplied by $2 ^{8}$. Limited precision makes binary floating-point numbersdiscontinuous; there are gaps between them.

False. Because of rounding errors, you can find Big and Small numbers such that: (Small + Big) + Big != Small + (Big + Big)

FP approximates results because it only has 23 bits for the significand.

A non-zero digit is required prior to the radix in scientific notation, and since the only non-zero digit in base-2 is 1, the normalized value will always start with a 1.