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\centerline{MONT 107Q -- Thinking About Mathematics}
\centerline{Problem Set 3, {\bf due:} 5:00pm on Monday, March 27.}
\centerline{Some problems from Al-Khwarizmi}
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I.  Solve the following quadratic equations using the methods described in 
Chapters I through VI of ``Book of Algebra and Almucabola"
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\item{A.}  Six squares equal 486 units.  Find the root.   (See Chapter II.)
\item{B.}  A square and 8 roots equal 20 units.  Find the root.  (See Chapter IV.)
\item{C.}  A square and 35 units equal 12 roots.  Find the root.  (See Chapter V.)
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II.  Why does Al-Khwarizmi treat problems like the ones in B and C from question I
as different types?  Explain using modern algebraic notation.
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III.  Illustrate how the geometric reasoning Al-Khwarizmi gives for the ``squares and
roots equals numbers'' case applies to the problem from I. B.  (See pages 77 and 79.)
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IV.  Here's a translation of one of the inheritance problems from Al-Khwarizmi's
book:  A man dies, leaving two sons behind him, and bequeathing one-third of his capital
to a stranger.  He leaves ten dihrams of property and a claim of 10 dirhams upon 
one of the sons.  [JL:  That is, one of the sons owes the father 10 dihrams.  The question is
to determine how much of the debt the son who owes the 10 dihrams must pay to the estate 
so that everyone would come out ``even'' according to the terms of the inheritance when
everything was distributed.  Because of the size of the debt you can see that that one son is
not going to get anything back -- he'll pay something less than 10 dihrams and he'll
be ``even'' -- the amount he pays in will be the same as the amount he would 
receive from the estate if the transactions were done in two stages -- that is, if he
paid first and then the estate was distributed.  The purpose of the calculation
is to do the transaction all in one step.]  
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Computation:  You call the sum which is taken out of the debt {\it thing}.  [JL:  This
means that the son who owes the 10 dihrams only has to pay {\it thing} out of that debt.]
Add this to the capital, which is 10 dihrams.  The sum is ten and {\it thing}.  Subtract
one-third since he has bequeathed one-third of his property, that is, three dihrams
and one-third, plus one-third of {\it thing}.  The remainder is six dihrams and two-thirds
plus two-thirds of {\it thing}.  Divide this between the two sons.  The portion of each
of them is three dihrams and one-third, plus one-third of thing.  This is equal
to the {\it thing} which was sought for.  Reduce it, by removing one-third from {\it thing}, on account
of the other third of {\it thing}.  There remain two-thirds of {\it thing}, equal to three dihrams
and one-third.  It is then only required that you complete the {\it thing}, by adding to it
as much as one half of the same; accordingly you add three and one-third as much as
one half of them.  This gives five dihrams, which is the {\it thing} that is taken out of 
the debt.  
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Explain the logic behind this solution, using modern algebra as appropriate.
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