Question: $8.$ Suppose we have a $7-$bit computer that uses IEEE floating$-$point arithmetic where a floating point number has $1$ sign bit, $3$ exponent bits, and $3$ fraction bits. All of the bits in the hardware work properly. Recall that denormalized numbers will have an exponent of $000,$ and the bias for a $3-$bit exponent is $23-11=3.($a$)$ For each of the following, write

$8.$ Suppose we have a $7-$bit computer that uses IEEE floating$-$point arithmetic where a floating point number has $1$ sign bit, $3$ exponent bits, and $3$ fraction bits. All of the bits in the hardware work properly.

Recall that denormalized numbers will have an exponent of $000,$ and the bias for a $3-$bit exponent is

$23-11=3.$

$($a$)$ For each of the following, write the binary value and the corresponding decimal value of the $7-$bit floating point number that is the closest available representation of the requested number. If rounding is necessary use round$-$to$-$nearest. Give the decimal values either as whole numbers or fractions. The first few lines are filled in for you.

$($b$)$ The associative law for addition says that a $+($b $+$ c$)=($a $+$ b$)+$ c$.$ This holds for regular arithmetic, but does not always hold for floating$-$point numbers. Using the $7-$bit floating$-$point system described above, give an example of three floating$-$point numbers a$,$ b$,$ and c for which the associative law does not hold, and show why the law does not hold for those three numbers.