From 6c291be12f9be99fed7f06cb03bd9517e28eecc8 Mon Sep 17 00:00:00 2001 From: Price Hiller Date: Tue, 24 Sep 2024 23:34:35 -0500 Subject: [PATCH] cs-3843: add HW3 --- .../CS-3843/Assignments/HW-3/Assignment.typ | 123 ++++++++++++++++++ 1 file changed, 123 insertions(+) create mode 100644 Fall-2024/CS-3843/Assignments/HW-3/Assignment.typ diff --git a/Fall-2024/CS-3843/Assignments/HW-3/Assignment.typ b/Fall-2024/CS-3843/Assignments/HW-3/Assignment.typ new file mode 100644 index 0000000..75586d6 --- /dev/null +++ b/Fall-2024/CS-3843/Assignments/HW-3/Assignment.typ @@ -0,0 +1,123 @@ +#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], [$0 ⋅ 2^3 + 0 ⋅ 2^2 + 0 ⋅ 2^1 + 1 ⋅ 2^0 = 1$], [$-0 ⋅2^3 + 0 ⋅ 2^2 + 0 ⋅2^1 + 1 ⋅2^0 = 1$], + hlinex(start: 0), + [0xB], [1011], [$1 ⋅ 2^3 + 0 ⋅ 2^2 + 1 ⋅ 2^1 + 1 ⋅ 2^0 = 11$], [$-1 ⋅ 2^3 + 0 ⋅ 2^2 + 1 ⋅ 2^1 + 1 ⋅ 2^0 = -5$], + hlinex(start: 0), + [0x2], [0010], [$0 ⋅ 2^3 + 1 ⋅ 2^2 + 0 ⋅ 2^1 + 0 ⋅ 2^0 = 2$], [$-0 ⋅ 2^3 + 1 ⋅ 2^2 + 0 ⋅ 2^1 + 0 ⋅ 2^0 = 2$], + hlinex(start: 0), + [0x7], [0111], [$0 ⋅ 2^3 + 1 ⋅ 2^2 + 1 ⋅ 2^1 + 1 ⋅ 2^0 = 7$], [$-0 ⋅ 2^3 + 1 ⋅ 2^2 + 1 ⋅ 2^1 + 1 ⋅ 2^0 = 7$], + hlinex(start: 0), + [0xC], [1100], [$1 ⋅ 2^3 + 1 ⋅ 2^2 + 0 ⋅ 2^1 + 0 ⋅ 2^0 = 12$], [$-1 ⋅ 2^3 + 1 ⋅ 2^2 + 0 ⋅ 2^1 + 0 ⋅ 2^0 = -4$], + )] + + #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 + + 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.