\documentclass[12pt]{article} \usepackage{amssymb,amsmath,graphicx,hyperref} \hypersetup{ pdfnewwindow=true, pdffitwindow=false } \pagestyle{empty} \begin{document} \begin{enumerate} \item (\textbf{do, but don't turn in}) Consider this model of the integers. Every integer is an IOU, either a credit or debt in dollars. So $20$ means we are owed \$20 (a \$20 credit), and $-20$ means we owe \$20 (a \$20 debt). Addition means putting together and subtracting something means to remove it. Explain in words using this model what is a reasonable value for: \begin{enumerate} \item 20 + (-20) \item 20 - (-20) \item (-20) + (-20) \item -20 + 0 \end{enumerate} \item (\textbf{do, but don't turn in except for (\ref{minustimesminusiou})}). Modeling multiplication is more complex. Let's model $M \times N$ as follows. For a positive $N$, we say it's getting $N$ IOUs each worth $\$M$. For a negative $N$, we say it's losing $N$ IOUs each worth $\$M$. Explain in words using this model what is a reasonable value for: \begin{enumerate} \item $20 \times 1$ \item $1 \times 20$ \item $20 \times 0$ \item $0 \times 20$ \item $20 \times -3$ \item $(20 \times 3) + (20 \times (-3))$ \item $-20 \times 3$ \item $-20 \times -1$ \item \label{minustimesminusiou}\textbf{(turn in)} $-20 \times -3$ \end{enumerate} \item \textbf{Defining the Integers from the Natural Numbers: Addition}. Suppose we have constructed the natural numbers to our satisfaction along with a binary addition operation. Note there is no ``subtraction'' operation or ``opposite'' operation yet, so this is an approach independent from the ``opposite adjoining'' in the last homework. Here we will use what is called the \emph{Grothendieck Construction}. This is a very general construction which actually applies to any structure with a commutative, associative binary operation and an identity element. It takes such a structure and embeds it in a larger structure that extends the operation so that every element now has an inverse (i.e. embeds it in an abelian group). The set of natural numbers fits these preconditions, and we'll discuss this specific case, but this exact argument works for many other sets. One defines new numbers of the form ``$a - b$" where $a$ and $b$ are natural numbers. It is tempting to just see ``3-2'' as ``1'' because of habit, but we're not assuming we know anything about subtraction yet. So for now we just regard ``$a-b$'' as an ordered pair written with panache. And yet $3-2$ really ought to be the same as $7-6$. Thus, we declare ``$a-b$" to be equivalent to ``$c-d$" if and only if $a+d=b+c$ (notice how we only use addition in the definition). We write this $a-b \sim c-d$ In fact, this is a genuine equivalence relation, so each ordered pair is really an equivalence class of expressions ``$a - b$". These ordered pairs are called the integers and the natural numbers are embedded in the integers as the classes ``$a - 0$". \begin{enumerate} \item (Do, don't turn in) Prove to yourself that the relation defined above is really an equivalence relation. \item What is the natural way to define an addition on these objects? I.e. $(a - b) + (c - d) = ?$, where your answer is another difference pair. \item Verify that your addition is well-defined. That is, if $(a-b) \sim (a'-b')$ and $(c-d) \sim (c'-d')$ then $(a - b) + (c - d) \sim (a' - b') + (c' - d')$. \item What is the additive identity? Prove it. \item What is the opposite of $(a-b)$? Prove it. \item Give a clean description of the set of negative numbers (the opposites of the natural numbers)? % Addition is what you'd expect: $(a - b) + (c - d) = (a+c - (b+d) )$. \end{enumerate} \item \textbf{Defining the Integers from the Natural Numbers: Multiplication}. Once the integers are constructed with an addition, it's natural to want to extend multiplication. \begin{enumerate} \item What is the natural way to define multiplication on pairs of integers? I.e. $(a - b)(c - d) = ?$. \item Check that your multiplication is well-defined, that is if $(a-b) \sim (a'-b')$ then $(a-b)(c-d) \sim (a'-b')(c-d)$. Technically you should also check that $(a-b)(c-d) \sim (a-b)(c'-d')$ but I don't need to see that (similar) calculation. \item Prove that this new integer multiplication is commutative assuming natural number multiplication is commutative. \item Check that $0$ times anything is $0$. \item Check that two negative numbers multiply to a positive product. \end{enumerate} % (ac + bd - (bc + ad))$. One does need to check that these operations are well-defined in that they do not depend on the choice of representative for the equivalence class. Details are available in your local abstract algebra text. % % Given this definition, the product of two negative numbers $(0 - b)(0 - d)$, where $b$, $d$ are (nonzero) natural numbers, is $(0 + bd - (0 + 0)) = (bd - 0)$, which is the natural number $bd$. Thus, the product of two negative numbers is positive. \item (don't turn in) Divide 128 by 7 using relaxed repeated subtraction. \item Explain why in class we called our long division algorithm abbreviated uptight repeated subtraction. Explain this using the example of 128 divided by 7. \item Describe 128 divided by 7 using \begin{enumerate} \item a partitive model (sharing) \item a measurement model (taking out bundles) \end{enumerate} \item Henry spends 3/14 of his monthly income for rent and 2/11 of what is left on food. If he has \$540 left, what is his monthly income ? (Hint: Pictures really help). \end{enumerate} \end{document}