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a/Homework 11/bonus.tex b/Homework 11/bonus.tex new file mode 100644 index 0000000..0cc6621 --- /dev/null +++ b/Homework 11/bonus.tex @@ -0,0 +1,280 @@ +\documentclass[12pt]{article} +\title{Computer Architecture -- Assignment 11 Bonus} +\author{Claudio Maggioni \and Tommaso Rodolfo Masera} +\date{} + +\begin{document} +\maketitle + +\section{Bonus Question 1} + +\subsection{Decimal to Hexadecimal} + +\textbf{All the numbers are going to be first converted to binary and then to + hexadecimal for an easier calculation} + +\begin{itemize} + +\item[a)] \textbf{\large 11}; + +First we find how 11 is expressed in binary which is 1011 and then we convert it to +scientific base 2 notation: $1011 = 1.\textbf{011 } \times 2^3$. + +The sign bit is 0 since $11>0$ and therefore the number is positive. + +To determine the exponent we take the single precision bias (127) and we add to it +the amount of times we moved a bit past the floating point: $127 + 3 = 130$. + +We then convert 130 to decimal by successive halving:\\ +\begin{tabular}[h]{l|c} +\textbf{Fraction} & \textbf{Rest}\\ +\hline \\ +$130/2 = 65$ & 0 \\\\ +$65/2 = 32$ & 1 \\ +$32/2 = 16$ & 0 \\ +$16/2= 8$ & 0 \\ +$8/2 = 4$ & 0 \\ +$4/2 = 2$ & 0 \\ +$2/2 = 1$ & 0 \\ +$1/2 = 0$ & 1 \\ +\end{tabular} + +By reading from most significant bit to least significant bit we get 10000010 for our +exponent. + +We then finally assemble the floating point binary number with the parts we have +converted: + +$0\ 10000010\ 01100000000000000000000 = 01000001001100000000000000000000$ + +And we convert to hexadecimal by splitting the number in 4 bit groups and converting +each group to its equivalent hexadecimal digit: + +$0100\ 0001\ 0011\ 0000\ 0000\ 0000\ 0000\ 0000 = 4\ 1\ 3\ 0\ 0\ 0\ 0\ 0\ = 41300000$. \\\\ + +\textbf{From now on the passages are going to include only the calculations to avoid verbosity. Nonetheless, the same reasoning will be applied for each conversion.} + +\item[b)] \textbf{\large 5/64}; + +Sign bit = 0 + +$\frac{5}{64} = 0.078125$ + +Conversion to binary:\\ +\begin{itemize} +\item $0.078125 \times 2 = \textbf{0} + 0.15625$; +\item $0.15625 \times 2 = \textbf{0} + 0.3125$; +\item $0.3125 \times 2 = \textbf{0} + 0.625$; +\item $0.625 \times 2 = \textbf{1} + 0.25$; +\item $0.25 \times 2 = \textbf{0} + 0.5$; +\item $0.5 \times 2 = \textbf{1} + 0$; +\end{itemize} + +Result of conversion: $0.000101$. + +Normalization to scientific notation: $1.\textbf{01} \times 2^{-4}$. + +Exponent bits in decimal = $127 - 4 = 123$ + +Conversion of exponent:\\ +\begin{tabular}[h]{l|c} +\textbf{Fraction} & \textbf{Rest}\\ +\hline \\ +$123/2 = 61$ & 1 \\\\ +$61/2 = 30$ & 1 \\ +$30/2 = 15$ & 0 \\ +$15/2= 7$ & 1 \\ +$7/2 = 3$ & 1 \\ +$3/2 = 1$ & 1 \\ +$1/2 = 0$ & 1 \\ +\end{tabular} + +Exponent bits in binary = $01111011$ + +Final binary number = $0\ 01111011\ 01000000000000000000000 =$ + + $00111101101000000000000000000000$ + +Conversion to hexadecimal: + +$0011\ 1101\ 1010\ 0000\ 0000\ 0000\ 0000\ 0000 = 3$ D A $0\ 0\ 0\ 0\ 0 = 3$DA$00000$. + +\item[c)] \textbf{\large -5/64}; + +With this being the negative counterpart of the previous number we just need to flip +the sign bit of $\frac{5}{64}$. + +With that we get: $10111101101000000000000000000000$. + +Which, converted to hexadecimal, gives: + +$1011\ 1101\ 1010\ 0000\ 0000\ 0000\ 0000\ 0000 =$ B D A $0\ 0\ 0\ 0\ 0 =$ BDA$00000$. + +\item[d)] \textbf{\large 6.125}; + +Sign bit = 0 + +Conversion to binary: + + Integer part = $6_{10} = 110_{2}$ + + Fractional part = $0.125_{10}$: + \begin{itemize} +\item $0.125 \times 2 = \textbf{0} + 0.25$; +\item $0.25 \times 2 = \textbf{0} + 0.5$; +\item $0.5 \times 2 = \textbf{1} + 0$; + \end{itemize} + = $001_2$. + +Normalized binary = $110.001 = 1.\textbf{10001} \times 2^2$ + +Exponent bits in decimal = $127 + 2 = 129$. + +\begin{tabular}[h]{l|c} +\textbf{Fraction} & \textbf{Rest}\\ +\hline \\ +$129/2 = 64$ & 1 \\\\ +$64/2 = 32$ & 0 \\ +$32/2 = 16$ & 0 \\ +$16/2= 8$ & 0 \\ +$8/2 = 4$ & 0 \\ +$4/2 = 2$ & 0 \\ +$2/2 = 1$ & 0 \\ +$1/2 = 0$ & 1 \\ +\end{tabular} + +Exponent bits in binary = $10000001$ + +Final binary number = $0\ 10000001\ 10001000000000000000000 =$ + + $01000000110001000000000000000000$. + +Conversion to hexadecimal: +$0100\ 0000\ 1100\ 0100\ 0000\ 0000\ 0000\ 0000 = 4\ 0$ C $4\ 0\ 0\ 0\ 0 = 40$C$40000$. + +\end{itemize} + +\subsection{Hexadecimal to Decimal} + +\textbf{All the numbers are going to be first converted to binary and then to +decimal for an easier calculation} + +\begin{itemize} + +\item[a)] \textbf{\large 42E48000$_{16}$} + +Conversion to binary: + +$42$E$48000 = 0100\ 0010\ 1110\ 0100\ 1000\ 0000\ 0000\ 0000 =$ + +$ 0\ 10000101\ 11001001000000000000000 = $ + +$ 01000010111001001000000000000000$ + +From this we can observe that the sign bit is 0 (i.e. positive). + +Conversion of exponent from binary to decimal: + +$10000101 = 2^7 + 2^2 + 2^0 = 128 + 4 + 1 = 133$ + +To find how many bits have been moved beyond the floating point we now find the +exponent for our notation: + +$ 133 - 127 = 6$ + +And we then take the mantissa to multiply it by $2^6$ (where 6 is the number we found through the last passage) to get the final binary number to convert to decimal: + +$1.11001001000000000000000 \times 2^6 = 1.11001001 \times 2^6 = 1110010.01$ + +Conversion to decimal: + +$1110010.01 = 2^6 + 2^5 + 2^4 + 2 + 2^{-2} = 64 + 32 + 16 + 2 + 0.25 = 114.25$ + +\item[b)] \textbf{\large 3F880000$_{16}$} + +Conversion to binary: + +$3$F$880000 = 0011\ 1111\ 1000\ 1000\ 0000\ 0000\ 0000\ 0000\ =$ + +$0\ 01111111\ 00010000000000000000000 =$ + +$00111111100010000000000000000000$ + +Sign bit = 0 + +Conversion of exponent from binary to decimal: + +$01111111 = 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2 + 1 = 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127$ + +Power of 2 = $127 - 127 = 0$ + +Final binary number: + +$1.0001 \times 2^0 = 1.0001$ + +Conversion to decimal: + +$1.0001 = 2^0 + 2^{-4} = 1 + 0.0625 = 1.0625$ + +\item[c)] \textbf{\large 00800000$_{16}$} + +Conversion to binary: + +$00800000 = 0000\ 0000\ 1000\ 0000\ 0000\ 0000\ 0000\ 0000 =$ + +$0\ 00000001\ 000\ 00000000000000000000 =$ + +$00000000100000000000000000000000$ + +Sign bit = 0 + +Conversion of exponent from binary to decimal: + +$00000001 = 2^0 = 1$ + +Power of 2 = $1 - 127 = -126$ + + Final binary number: + + $1.00000000000000000000 \times 2^{-126}$ + + Conversion to decimal: + +$1.00000000000000000000 \times 2^{-126} = 2^{-126} = 1.1754944 \times 10^{-38}$ + +\item[d)] \textbf{\large C7F00000$_{16}$} + +Conversion to binary: + +C$7$F$00000 = 1100\ 0111\ 1111\ 0000\ 0000\ 0000\ 0000\ 0000 =$ + +$1\ 10001111\ 11100000000000000000000 =$ + +$11000111111100000000000000000000$ + +Sign bit = 1 + +Conversion of exponent from binary to decimal: + +$10001111 = 2^7 + 2^3 + 2^2 + 2^1 + 2^0 = 128 + 8 + 4 + 2 + 1 = 143$ + +Power of 2 = $143 - 127 = 16$ + +Final binary number: + +$1.11100000000000000000000 \times 2^{16} = 1.111 \times 2^16 =$ + +$11110000000000000.0$ + +Conversion to decimal: + +$11110000000000000.0 = 2^16 + 2^15 + 2^14 + 2^13 = 65536 + 32768 + 16384 + +8192 = 122880 $ and, since the sign bit is 1 (i.e. negative), $= -122880$. + +\end{itemize} + +\section{Bonus Question 2} + + + +\end{document} \ No newline at end of file diff --git a/Homework 11/hw11_ex3.ods b/Homework 11/hw11_ex3.ods index bcdaf0731a1fc1c0d57d0ed3f05b18d5567daccf..2ff32f67ae941807137b4eea3a99689abe294902 100644 GIT binary patch delta 8554 zcmZvCbx>SQv-jfei+gbQ;1=AS;10pvZIR#-U~zW>1PidZy9aj(1b4RpU-G>7t^3@n zZ~BjOX8PCNQ>VK7Ojl3Brzbm9RRtJWAOL^}01Ue(qN+m=yVjWmp#FJ)H&H1-Y8u_MM1+=C@kZJ`sB z=aEXDb7edlId!AjYM4yuE>ungAr!La2wVqOr_LKsRuAMObN~+ z8UqH)c0*@pXI=fhc;-#rL&~zE<*mAa0|b6}Ko;5Ls4y+2ls;EihvZcGPIjev>u)Eu zzP|^T)y1xwB51r59puM;$~L-5M12!AL(HqPDz5VDu=)%~hQrRE4|N>}npbMMm{(FP 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