$3.(20$ points$)$ A standard way for computers to represent non$-$negative and negative integers using

binary fixed$-$width N is the $2$s complement method TN $($x$).$ The idea is to have different rules for

non$-$negative and negative integers.

$($a$)$ Non$-$negative integers x with $0<=$ x $<=2$N $11$ are represented using ordinary fixed$-$width binary

$($e$.$g$.$ Using binary fixed$-$width $8,$ T$8(30)=(00011110)).$

$($b$)$ Negative integers x with $2$N $1<=$ x $<0$ are represented by flipping the bits of $|$x$|$ and adding $1.$

$($e$.$g$.$ Using binary fixed$-$width $8,$

T$8(30)=$F lipBits$(00011110)+1$

$=(11100001)+1$

$=(11100010)$

$)$

$($a$)(4$ points$)$ According to the definition above, explain why each negative integer in binary fixed

width N $2$s complement must start with $1$