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‎AR/06_subtraction/06_subtraction.tex‎

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% Source: CPTR 280 Computer Organization and Assembly Language Fall 2021
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% File: "06 Subtraction (key).pdf"
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% Author: James Foster, pogil@jgfoster.net
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% comment out for student version
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% \ifdefined\Student\relax\else\def\Teacher{}\fi
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\documentclass[12pt]{article}
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\title{Activity 6: Subtraction}
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\author{James Foster}
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\newcommand{\activityeditor}{James Foster}
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\newcommand{\activitysource}{\url{pogil@jgfoster.net}}
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\date{Fall 2021}
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\input{../../cspogil.sty}
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\usepackage{graphicx}
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\usepackage{tabularx}
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\begin{document}
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\begin{center}
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\maketitle
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\rolenames
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\end{center}
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\keyquestions{
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\item Model 1, Question \#2
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\item Model 2, Question \#7
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\item Model 3, Question \#12
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\item Model 3, Question \#16
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\item Model 4, Question \#21, a--c
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}
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\newpage
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\maketitle
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We said earlier that most math operations can be simplified to addition. In this lesson we will build on
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earlier exercises dealing with binary and with digital logic gates to understand how computers do subtraction.
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\guides{
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\item Use nine's complement to subtract without borrowing; and,
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\item Represent negative numbers in binary using two's complement.
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}{
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\item Reflect on how what how the team could work and learn more effectively.
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}{
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No additional notes
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}{
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\item odometer \href{https://search.creativecommons.org/photos/57c70ba8-18f2-482a-87ab-285606463446}{link here}
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\item clock: \url{https://pngio.com/images/png-a1553797.html}
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}
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\input{subtraction_without_borrowing.tex}
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\newpage
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\input{wrap_around.tex}
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\newpage
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\input{negative_numbers.tex}
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\newpage
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\input{encoding_in_binary.tex}
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\end{document}

‎AR/06_subtraction/encoding_in_binary.tex‎

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\model{Encoding in Binary}
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While we said that there isn't anything special about decimal (base ten), there is something special about binary:
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it is the simplest number system possible and forms the foundation of all digital computers. Anything that can be mapped to
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discrete values can be encoded in binary; the only requirement is that everyone who uses the code must understand the mapping.\\
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{\it\large Refer to Model 4 above as your team develops consensus answers
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to the questions below.}
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\quest{15 min}
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\Q ``Hang a lantern aloft in the belfry arch / Of the North Church tower as a special light, --- / One, if by land, and two. if by sea \ldots''
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Map Paul Revere's code to binary.
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\vspace{10pt}
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\begin{center}
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\begin{tabular}{|c|l|}
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\hline
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\textbf{Code} & \textbf{Meaning} \\
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\hline
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00 & \ans[1in]{``no invasion''} \\
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\hline
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01 & \ans[1in]{``by land''} \\
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\hline
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10 & \ans[1in]{``by land''} \\
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\hline
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11 & ``by sea'' \\
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\hline
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\end{tabular}
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\end{center}
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\Q The WWU CS department has four professors, Preston Carman, James Foster, John Foster, and Natalie Smith-Gray.
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Assign each a unique ID using just two bits.
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\vspace{10pt}
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\begin{center}
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\begin{tabular}{|c|l|}
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\hline
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\textbf{Code} & \textbf{Professor} \\
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\hline
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\ans[0.4in]{00} & Preston Carman \\
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\hline
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\ans[0.4in]{01} & James Foster \\
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\hline
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\ans[0.4in]{10} & John Foster \\
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\hline
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\ans[0.4in]{11} & Natalie Smith-Gray \\
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\hline
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\end{tabular}
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\end{center}
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\Q How many bits would be required to encode the following:\key\\[-2.5mm]
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\begin{enumerate}
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\item The letter grades A, B, C, D, F? (Hint: not 5!)
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\begin{answer}[0.5in]
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3 bits are needed to encode 5 values (2 bits give only 4 values)
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\end{answer}
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\item The 30 students enrolled in CPTR 280?
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\begin{answer}[0.5in]
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5 bits are needed to encode 30 values (4 bits give only 16 values)
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\end{answer}
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\item All 34 students if we admitted everyone on the wait list?
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\begin{answer}[0.5in]
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6 bits are needed to encode 34 values (5 bits give only 32 values)
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\end{answer}
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\end{enumerate}
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\vspace{-20pt}
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\Q If you were using four bits to represent a non-negative integer (to keep things simple), what is the range of numbers you could encode (that is, what would they convert to in decimal)?
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\begin{answer}[0.5in]
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4 bits can represent 16 values, from 0 to 15
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\end{answer}

‎AR/06_subtraction/figures/clock.png‎

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‎AR/06_subtraction/figures/odometer.png‎

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‎AR/06_subtraction/negative_numbers.tex‎

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\model{Negative Numbers}
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\quest{15 min}
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\Q Consider a traditional analog clock face.
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\vspace{10pt}
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\begin{center}
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\includegraphics[width=0.25\textwidth]{figures/clock.png}
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\end{center}
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\begin{enumerate}
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\item How many unique hours can it represent?
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\hfill\ans{12}
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\item What is the range of hours it represents?
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\hfill\ans{1 to 12}
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\item If we think of 3:00 as ``three hours after noon'' (or ``+3''), what hour position on the clock is represented by ``three hours before noon'' (``-3'')?
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\begin{answer}[0.5in]
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9
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\end{answer}
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\item What range of integers would support the same number of negative clock-face ``hours'' as nonnegative clock-face ``hours''?
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\begin{answer}[0.5in]
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-6 to +5
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\end{answer}
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\end{enumerate}
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\vspace{-20pt}
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\Q Complete the following table to map the typical clock face hours (line 1) to the hours before and after noon (line 2).
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(Hint: start at each end and work to the middle.)
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\vspace{10pt}
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\begin{center}
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\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
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\hline
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12 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\
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\hline
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\ans[0.3in]{0} & \ans[0.3in]{1} & \ans[0.3in]{2} & \ans[0.3in]{3} & \ans[0.3in]{4} & \ans[0.3in]{5} & \ans[0.3in]{-6} & \ans[0.3in]{-5} & \ans[0.3in]{-4} & \ans[0.3in]{-3} & \ans[0.3in]{-2} & \ans[0.3in]{-1} \\
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\hline
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\end{tabular}
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\end{center}
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\Q With this relabeled clock, answer the following questions:
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\begin{enumerate}
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\item If we start at 2 and add 2, what do we get?
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\hfill\ans{+4}
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\item If we start at 2 and subtract 1, what do we get?
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\hfill\ans{+1}
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\item If we start at 2 and subtract 2, what do we get? (Hint: not 12)
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\hfill\ans[1.5in]{0}
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\item If we start at 2 and subtract 5, what do we get?
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\hfill\ans{-3}
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\item If we start at -2 and subtract 2, what do we get?
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\hfill\ans{-4}
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\item If we start at -2 and add 6, what do we get?
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\hfill\ans{+4}
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\item So far this maps very nicely to ``normal'' arithmetic. But if we start at 5 and add 3, what do we get? (Hint: not 8!)
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\begin{answer}[0.5in]
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-4
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\end{answer}
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\item If we start at -5 and subtract 4, what do we get? (Hint: not -9!)
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\hfill\ans[1.5in]{+3}
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\end{enumerate}
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\Q Using a similar approach, fill in the following table to map selected unsigned values [0, 255] to signed values [-128, 127]. This is called a two's complement.
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\vspace{10pt}
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\begin{center}
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\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
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\hline
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0 & 1 & 2 & \ldots & 126 & 127 & 128 & 129 & 130 & \ldots & 254 & 255 \\
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\hline
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\ans[0.2in]{0} & \ans[0.2in]{1} & \ans[0.2in]{2} & \ldots & \ans[0.25in]{126} & \ans[0.25in]{127} & \ans[0.3in]{-128} & \ans[0.3in]{-127} & \ans[0.3in]{-126} & \ldots & \ans[0.2in]{-2} & \ans[0.2in]{-1} \\
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\hline
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\end{tabular}
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\end{center}
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\Q Convert some of these binary values to unsigned and signed decimal:\key\\[-2.5mm]
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\vspace{10pt}
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\begin{center}
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\begin{tabular}{|c|c|c|}
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\hline
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\textbf{Binary} & \textbf{Unsigned Decimal} & \textbf{Signed Decimal} \\
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\hline
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0000 0000 & \ans[0.4in]{0} & \ans[0.4in]{0} \\
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\hline
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0000 0001 & \ans[0.4in]{1} & \ans[0.4in]{1} \\
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\hline
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0111 1110 & \ans[0.4in]{126} & \ans[0.4in]{126} \\
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\hline
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0111 1111 & \ans[0.4in]{127} & \ans[0.4in]{127} \\
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\hline
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1000 0000 & \ans[0.4in]{128} & \ans[0.4in]{-128} \\
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\hline
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1000 0001 & \ans[0.4in]{129} & \ans[0.4in]{-127} \\
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\hline
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1111 1110 & \ans[0.4in]{254} & \ans[0.4in]{-2} \\
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\hline
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1111 1111 & \ans[0.4in]{255} & \ans[0.4in]{-1} \\
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\hline
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\end{tabular}
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\end{center}
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\Q How does the high-order (left-most) bit correlate with the sign?
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\hfill\ans[1.5in]{0 $\Rightarrow$ positive; 1 $\Rightarrow$ negative}
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\newpage
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\Q Perform some addition of signed binary numbers and note how the sign works:
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\vspace{10pt}
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\begin{center}
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\begin{tabular}{|c|c|c|c|}
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\hline
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11 & 0000 1011 & -118 & 1000 1010 \\
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+ 5 & + 0000 0101 & + 44 & + 0010 1100 \\
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\hline
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\ans[1in]{16} & \ans[1in]{0001 0000} & \ans[1in]{-74} & \ans[1in]{1011 0110} \\
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\hline
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106 & 0110 1010 & -22 & 1110 1010 \\
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+ -63 & + 1100 0001 & + -86 & +1010 1010 \\
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\hline
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\ans[1in]{43} & \ans[1in]{1 0010 1011} & \ans[1in]{-204} & \ans[1in]{1 1001 0100} \\
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\hline
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\end{tabular}
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\end{center}
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This suggests that another way to do subtraction is just to add the negative value. So how do we get a negative? Compare a few examples and
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note that in each case, if you added the values you would get zeros in the right-most (low order) 8 bits (which is, of course, correct).
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\vspace{10pt}
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\begin{center}
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\begin{tabular}{ccc}
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1 = 0000 0001 & & -1 = 1111 1111 \\
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2 = 0000 0010 & & -2 = 1111 1110 \\
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126 = 0111 1110 & & -126 = 1000 0010 \\
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127 = 0111 1111 & & -127 = 1000 0001 \\
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\end{tabular}
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\end{center}
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\Q For each positive value above (1, 2, 126, and 127), write out its one's complement or inverse (replace 0 with 1, 1 with 0).
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\vspace{10pt}
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\begin{center}
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\begin{tabular}{cccc}
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\ans[1in]{1111 1110} & \ans[1in]{1111 1101} & \ans[1in]{1000 0001} & \ans[1in]{1000 0000} \\
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\end{tabular}
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\end{center}
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\vspace{-20pt}
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\Q How does the inverse compare with the negative value? What would you add\key\\[-2.5mm] to the inverse to get the negative? Learn this useful rule for computing the negative of a number!
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\begin{answer}[0.5in]
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The inverse is one less than the negative. To get a negative, invert and add 1.
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\end{answer}

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