The floating point representation of a non$-$zero real number can take the form x $=$

$\backslash$pm $(0.$a$1$a$2...$ a$$n$)\backslash$beta $\backslash$beta e$,$ where a$1=0,$M $<=$ e $<=$ M $.($This is a slight variation on

what we$$ve seen in the book. It allows a$1=0$ to signal a $$float$$ of zero or some other

symbol. I believe this is $($therefore$)$ more realistic than what we$$ve seen.$)$ The meaning

is $\backslash$pm a$1\backslash$beta e$1+$ a$2\backslash$beta e$2++$ a$$n$\backslash$beta e$$n$.$ Suppose that $\backslash$beta $=2,$ n $=4,$ and M $=5,$ where

$\backslash$beta $,$ n$,$ M are fixed for the platform and the leading sign, a$1...$ a$$n$,$ and e are associated

with a particular user number.

$($a$)$ Find the smallest and largest positive numbers that can be represented in this floating

point system. Give your answers in decimal form.

$($b$)$ Find the floating point number in this system that is closest to $3.($Hint: Bisection

search by hand in base $2$ is a lot easier than it sounds!$)$