Chapra Text:

Chapter $3$: $,3\mathrm{.}1$

$3\mathrm{.}1$ Convert the following base$-2$ numbers to base$-10$: $($a$)101101,$

$($b$)101\mathrm{.}011,$ and $($c$)0\mathrm{.}01101.$

Additional Questions:

Consider a $16-$bit binary floating point number representation system:

The first bit of the exponent is dedicated to its sign. Assume that the mantissa must start with a $\text{'}1\text{'}.$ Use

this system to answer the following questions:

What is the smallest $($magnitude$)$ number that can be represented with this system?

Calculate the relative round$-$off error introduced when this system tries to represent

$e=2\mathrm{.}718281828459045235360287$dots

using rounding.

Suppose a calculator using this storage system is used to perform the calculation:

$2\mathrm{.}2+1\mathrm{.}3$

What number would it output? What is the relative error of this output to the true value?