college/Fall-2024/CS-3843/Assignments/3/Assignment.typ

#import "@preview/tablex:0.0.4": tablex, rowspanx, colspanx, hlinex, vlinex

#show link: set text(blue)
#set text(font: "Calibri")
#show raw: set text(font: "Fira Code")

#set page(margin: (x: .25in, y: .25in))
#let solve(solution) = [
  #let solution = align(
    center,
    block(
      inset: 5pt,
      stroke: blue + .3pt,
      fill: rgb(0, 149, 255, 15%),
      radius: 4pt,
    )[#align(left)[#solution]],
  )
  #solution
]

#let note(content) = [
  #align(
    center,
    block(
      inset: 5pt,
      stroke: luma(20%) + .3pt,
      fill: luma(95%),
      radius: 4pt,
    )[#align(left)[#content]],
  )
]

#align(center)[
  = CS 3843 Computer Organization
  HW 3\
  #underline[Price Hiller] *|* #underline[zfp106]
]
#line(length: 100%, stroke: .25pt)

+ Assuming $w = 4$, we can assign a numeric value to each possible hexadecimal digit, assuming either an unsigned or a two's-complement interpretation. Fill in the following table according to these interpretations by writing out the nonzero powers of 2 in the summations shown in the equations below:

  $ \B2T_w (accent(x, arrow)) ≐ -x_(w-1)2^(w-1) + sum_(i=0)^(w-2) x_i 2^i $
  $ \B2U_w (accent(x, arrow)) ≐ sum_(i=0)^(w-1) x_i 2^i $

  #solve[#tablex(
    columns: 4,
    column-gutter: 15pt,
    align: center + horizon,
    auto-vlines: false,
    auto-hlines: false,
    stroke: .5pt,
    header-rows: 2,

    /* --- header --- */
    colspanx(2)[$accent(x, arrow)$], colspanx(2)[], colspanx(2)[], colspanx(2)[],
    hlinex(start: 0, end: 2, stop-pre-gutter: true, stroke: 1pt),
    [Hexadecimal], [Binary], [$\B2U_4(accent(x, arrow))$], [$\B2T_4(accent(x, arrow))$],
    hlinex(start: 0, stroke: 1pt),
    /* -------------- */
    [0xA], [1010], [$2^3 + 2^1 = 10$], [$-2^3 + 2^1 = -6$],
    hlinex(start: 0),
    [0x1], [0001], [$2^0 = 1$], [$2^0 = 1$],
    hlinex(start: 0),
    [0xB], [1011], [$2^3 + 2^1 + 2^0 = 11$], [$2^3  + 2^1 + 2^0 = -5$],
    hlinex(start: 0),
    [0x2], [0010], [$2^2 = 2$], [$2^2 = 2$],
    hlinex(start: 0),
    [0x7], [0111], [$2^2 + 2^1 + 2^0 = 7$], [$2^2 + 2^1 + 2^0 = 7$],
    hlinex(start: 0),
    [0xC], [1100], [$2^3 + 2^2 = 12$], [$-2^3 + 2^2 = -4$],
  )]

  #v(1em)
+ We have a machine of 16-bits. This 16-bits machine can be used to represent $2^16$ or $65,536$ different values. We have another machine of 32-bits. This 32-bits machine can be used to represent $2^32$ or $4,294,967,296$ different numbers. So, if you wanted to store a value of $100,000$ then which machine should you choose? The 16-bits machine or 32-bits machine? Mention and explain your choice. (It's Ok to use a calculator for this problem)

  #solve[
    Generally, we would want to choose the 32-bit system to store a value of $100,000$. The 16-bit system has a max value it can store of $65,535$ which is $< 100,000$ meaning we can't store the $100,000$ value directly in a single word. Whereas, the 32-bit system can store a max value of $4,294,967,295$ which is $>= 100,000$, meaning the value will fit within a single word.

    #note[Now _technically_ we can store $100,000$ on the 16-bit system by spreading the value across multiple words and creating a *lot* of procedures to handle it. This is how naive Big Integer/Big Number libraries work in the wild (notice I mentioned _naive_, there are much, _*much*_, better methods of doing this).]

    Again though, ignoring the aforementioned technicality, we'd want to use the 32-bit system to store the value of $100,000$.
  ]

  #v(1em)
+ We have an 8-bits machine. But we don't know if the machine follows unsigned representation or the two's complement/signed representation. A hexadecimal value is given as `0xF1`. Find out the unsigned representation of this hexadecimal value in decimal. Find out the two's complement/signed representation of this hexadecimal value in decimal.
  #solve[
    + `0xF1` to binary: $1111 0001$
    + Unsigned representation (_ugh_): $1 ⋅ 2^7 + 1 ⋅ 2^6 + 1 ⋅ 2^5 + 1 ⋅ 2^4 + 0 ⋅ 2^3 + 0 ⋅ 2^2 + 0 ⋅ 2^1 + 1 ⋅ 2^0 = 241$
    + Signed representation (_*ugh*_): $-1 ⋅ 2^7 + 1 ⋅ 2^6 + 1 ⋅ 2^5 + 1 ⋅ 2^4 + 0 ⋅ 2^3 + 0 ⋅ 2^2 + 0 ⋅ 2^1 + 1 ⋅ 2^0 = -15$
      #note[
        - Alternative way of grabbing the signed value via one's complement:
          + Start with `0xF1` to decimal: $1111 0001$
          + Recognize that the most significant bit is 1, so we'll multiply our conversions at the end by $-1$
          + $~1111 0001 = 0000 1110$
          + Add $1$ and pray I understood this right: $0000 1110 + 1 = 00001111$
          + $0000 1111$ to decimal is $15$
          + Now multiply by $-1$ (and still pray I understood this right): $-1 ⋅ 15 = -15$
          + Hallelujah? Do let me know if you have time, I am most curious if I did that right.
      ]

    #note[
      Some `c` code to grab those values (and for my sane validation):
      ```c
      #include <stdio.h>

      int main(int argc, char *argv[]) {
        unsigned int val = 0xF1;
        printf("Hex Value: 0x%X\n", val);
        printf("Unsigned Decimal value: %d\n", (unsigned char)val);
        printf("Signed Decimal Value %d\n", (signed char)val);
      }
      ```
    ]
  ]

#v(1em)
#text(red)[*Instructions:*]
+ Show your work
+ You should solve all the problems manually. Then you can use calculator to check your answers.
+ You can submit a typed version of your HW or handwritten (scan it/take a photo, then convert it to pdf version. But you need to submit a pdf file.)
+ This is an individual HW.
+ Late submissions are not allowed.
+ Please include your Full Name and abc123 in your HW.