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\begin{enumerate}
 \item (don't turn in) Symbolically check that American, European, Make It Nice and Negative Numbers methods work in base 4 using $311_4 - 112_4$ as an example.
 \item (don't turn in) Demonstrate \textbf{why} Lattice Multiplication works using the example $78 \times 23$.

\item \textbf{French Hand Multiplication}. In Butterworth's \emph{What Counts}, p.205, he writes:

\emph{To this day [about 1930], the peasant of central France (Auvergne) uses
 a curious method for multiplying numbers above 5. If he wished to multiply $9 \times 8$
 he bends down 4 fingers on his left hand (4 being the excess of 9 over 5)
 and 3 fingers on his right hand ($8-5=3$) . Then the number of bent-down fingers
 gives him the tens of the result ($4+3=7$) while the product of the unbent fingers
 gives him the units ($1 \times 2=2$).}

Does this method work for multiplying any two numbers between 6 and 9? Justify why, or give a counter-example.

\item \textbf{Doubling/Halving Multiplication}. I showed you in class how to multiply two numbers $A \times B$, by making two columns. In the first column, halve A (discarding the remainder), the next column, double B. Go to the next line and repeat the process. Stop when you halve A down to 1. Then you circle each line with an odd A. Add up all the numbers in the B column.
 Demonstrate why this works.
 \item Take the natural numbers as the set we know and love equipped with addition and multiplication (and for simplicity, we count 0 as a natural number). For each natural number $A$ let us define a new mysterious opposite number, $-A$, with the property $A + (-A) = 0 = (-A) + A $, i.e. $-A$ is an additive inverse of $A$. The union of the naturals and all their opposites we'll now call the integers. The nonzero natural numbers are called positive integers, and their opposites, negative integers.
 \begin{itemize}
 \item Prove from the definition that the opposite of $(-X)$ is $X$ (or that ``minuses cancel'') and therefore every integer has an opposite.
 \end{itemize}

\item We would like to extend addition and multiplication to these new opposites in a way that extends essential properties of natural number arithmetic. We ask that
 \begin{itemize}

\item they obey the distributive law

\item addition and multiplication are associative and commutative

\item $0$ is an additive identity

\item $1$ is a multiplicative identity

\end{itemize}

It can be checked that the integers can be assigned these properties in a way that doesn't lead to contradictions. You can now deduce a lot about the integers using \textbf{only these basic properties}.

\begin{enumerate}
 % \setcounter{enumii}{b}
 \item Prove that $0$ is the unique additive identity (i.e. prove that any additive identity has to equal $0$).
 \item Prove that every integer has a unique opposite.
 \item Prove zero times anything is zero.
 \item Prove the product of $-A$ and $-B$ is $AB$ and therefore ``negative times negative is positive''. (Hint: The slickest way I know proves first that $(-A)(-B)$ and $(A)(-B)$ are opposites, then that $(A)(-B)$ and $AB$ are opposites, then wraps things up.)
 \end{enumerate}

\end{enumerate}


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