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\centerline{\Huge Introduction to Bases}

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As you may know, our normal way of representing numbers is called ``base ten'',
because it uses ten digits.  We count from 0 to 9, and then we write 10,
representing 1 group of ten and 0 leftover units.  The next number, 11,
represents 1 group of ten and 1 unit.  This can continue indefinitely; the
number 43,507 represents 4 groups of ten thousand, 3 groups of one thousand, 5
groups of one hundred, 0 groups of ten, and 7 units.

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\noindent 1.  Perhaps the best way to get a feel for other bases is to practice
grouping.  Take fifty three chips.  Those can be grouped into 5 groups of ten
and 3 units.  That's why this number is written 53 (base ten).  But they could
also be grouped into 1 group of thirty six (which is six times six), 2 groups of six, and 5 units.  So we could write this
number as 125 (base six), or $125_{six}$.  By grouping your counters, write
this number in the bases two through twelve:

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\noindent\begin{tabular}{lll}
Base two: \hspace*{1in} & Base six: 125\hspace*{1in} & Base ten: 53\hspace*{1in}\\
& & \\
Base three: & Base seven: & Base eleven: \\
& & \\
Base four: & Base eight: & Base twelve: \\
& & \\
Base five: & Base nine: & \\
\end{tabular}

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\noindent 2.  Now that you are good at grouping, it's fairly easy to add
numbers.  Take thirty-seven and fifteen, and put both in base ten groupings
(we'll start by adding in base ten).  Now, start by adding the units.  There
are 7 units and 5 units, for a total of twelve.  This is too many for our base,
so we take a group of ten and move them over.  Now we add the tens.  There are
$3+1+1$ groups of tens, for a total of 5.  This number is less than our base,
so we leave it.  We now have 5 groups of ten and 2 units, so $37+15=52$ in base
ten.

Once you understand this idea, do the same (adding thirty-seven and fifteen)
in the following bases:

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(a) base eight \hfill (b) base four \hfill (c) base two \hfill \ 

