The floating point representation of a real number has the form x $=\backslash$pm $(0.$d$1$d$2...$ dn$)\backslash$beta $\backslash$beta $^$e$,$where d$1=0,0<=$ di $<=\backslash$beta $1,$M $<=$ e $<=$ M$.$

a$)$ Suppose $\backslash$beta $=2,$ n $=4,$ M $=3.$ How many different numbers can be represented in this system?

b$)$ Find fl$(2),$ the floating point representation of $2$ in this system. Apply the usual rules for rounding numbers up and down. Then express the final result in decimal form. How large is the roundoff error?

c$)$ In class we saw that xmax $=7\mathrm{.}5,$ xmin $=0\mathrm{.}0625($these are the largest and smallest positive numbers that can be represented in this system$).$ Suppose we switch to a system with n $=5.$ Find the new values of xmax, xmin.