The nonzero binary floating$-$point numbers can be expressed as

$+-(1+f)2^e$

where $f$ is the mantissa $($the fractional part of the significand$)$ and $e$ is the exponent. In

IEEE single precision arithmetic, $1$ bit is for the sign, $8$ bits are for the signed exponent

and $23$ bits are for the mantissa.

$($a$)$ Find the smallest normalized number, closest to zero, in IEEE single precision format.

$($b$)$ Machine epsilon $$ is the smallest number that when added to $1$ gives a number greater

than $1.$ Find the machine epsilon in IEEE single precision arithmetic.