Consider the floating point number system of base $\backslash$beta $=2.$

$\backslash$pm $($d$0.$d$1$d$2...$ dt$1)\backslash$beta $\backslash$times $\backslash$beta

$\backslash$pm $($e$)\backslash$beta

$.$

Note that the notation $($

$)\backslash$beta means the number is represented in base $\backslash$beta system.

$16$ bits were used to represent numbers in this floating point number system:

$1$ bit is reserved for the sign of the number $[0$ for $+$ and $1$ for $].$

$9$ bits are reserved for the mantissa d$1,...,$ dt$1.$

$1$ bit is reserved for the sign of the exponent $[0$ for $+$ and $1$ for $].$

$5$ bits are reserved for the exponent e$.$

$$ Chopping is used for rounding.

$($a$)$ What is machine for this floating point number system? Provide both in binary and

decimal values.

$1$

$($b$)$ What is the largest number that can be represented? Provide both in binary and

decimal values.

$($c$)$ What is the smallest number that can be represented? Provide both in binary and

decimal values.

$($d$)$ What is the decimal representation of x $=1101110110001011?$

$($e$)$ With x $=1031$ what is fl$($x$)?$ Provide both in binary and decimal values. Calculate

the relative error, $\backslash$delta x$,$ of representing x using fl$($x$)$ and verify that $\backslash$delta x$<=$ machine.