for each problem, document all steps you took to solve the problem. This can be handwritten, but must be legible for credit. If the problem states $\text{'}$By hand', do not use any program$/$function to solve the problem.

Problem $1$

By hand, convert the binary number $1001\mathrm{.}010$ into a decimal number.

Problem $1$ required submission:

Handwritten $($or printed$)$ all steps to obtain the answers, incl. the final answers.

Problem $2$

By hand, convert the decimal number $21\mathrm{.}625$ to

a$)$ a binary number;

b$)$ a binary floating point number in engineering notation $($keep the exponent as a decimal number$).$

Problem $2$ required submission:

Handwritten $($or printed$)$ all steps to obtain the answers, incl. the final answers.

Problem $3$

a$)$ Develop a Matlab function decimal$2$binary that converts a decimal number $x$ into a binary floating point number in engineering notation. The function shall take as input the decimal number $x$ and the number of bits to be used in the mantissa $N($excluding the hidden bit$)$ and shall output the vector $m$ containing the $N$ mantissa bits $($excluding the hidden bit$),$ the exponent $p$ in decimal of the binary floating point number in engineering notation, and the remainder $r$ of the transformation. Do not print out results to screen within the function.

b$)$ Using the function decimal$2$binary convert the decimal number $0\mathrm{.}1$ to a binary floating point number in engineering notation using $10,50,$ and $60$ mantissa bits and report the remainders.

c$)$ Why does Matlab report the remainder as $0$ when $60$ mantissa bits are used, even though infinitely many bits would be required to convert the decimal number $0\mathrm{.}1$ into a binary floating point number?

Problem $3$ required submission:

$$ Binary floating point numbers in engineering notation and their remainders;

$$ Handwritten $($or printed$)$ answer to c$)$;

Problem $4$

Using the function decimal$2$binary from problem $3,$ determine if a computer can tell the numbers $1\mathrm{.}23456$ and $1\mathrm{.}2345$ apart if bfloat$16$ numbers are used.

Problem $4$ required submission:

Printout of well commented script you coded included in your Gradescope submission;

Printout of supporting evidence;