$1\mathrm{.}3.$ For this exercise, you want to recall how to

write integers in binary $($base$-2)$ notation. Exam$-$

ple: the $($decimal$)5$ in binary is written as $101.$

Also, $lgx$ means "logarithm base$-2$ of $x."$

$($a$)$ What is $|$~$lg35$~$|?$

$($b$)$ What is ${|}_{{?}_{?}}lg{35}_{{?}_{?}}|?$

$($c$)$ Write $35$ in binary $($base$-2$ notation$).$

$($d$)($True$/$False$)$ The number of bits used to

represent $35$ in binary is equal to $|$~$lg35$~$|.$

$($e$)($True$/$False$)$ The number of bits used to

represent $32$ in binary is equal to $|$~$lg32$~$|.$

$1\mathrm{.}4.$ Doing exercise $1\mathrm{.}3$ first will help you think

this one through. You may assume $n$ and $k$ are

non$-$negative integers.

$($a$)($True$/$False$)$ If $n=2^k,$ then $|$~$lgn$~$|=k.$

$($b$)($True$/$False$)$ If $n=2^k,$ then ${|}_{{?}_{?}}lg{n}_{{?}_{?}}|=k.$

$($c$)($True$/$False$)$ Any integer $n$ can be written as

sandwich any $\{$:$n?)$

$($d$)$ How many bits does it take to write a number

$n$ such that

$($e$)$ In exercise $1\mathrm{.}3($a$)$ you concluded that

$|$~$lg35$~$|=6,$ and in exercise $1\mathrm{.}3($d$)$ you con$-$

cluded that the number $35$ is six bits long: $35$

in binary is $100011.$

So$,$ is it true that $|$~$lgn$~$|$ is the length of $n$ writ$-$

ten in binary? Explain your answer clearly.