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\centerline{MONT 109N -- Mathematics Across Cultures}
\centerline{Discussion/Problem Set 5 -- Indian Mathematics}
\centerline{April 8, 2011}
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\noindent
{\it Background and Assignment}
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In this assignment, we want to look at a sample of Indian
mathematics from the ``classical'' period (roughly
400 to 1200 C.E.).   Group writeups are due in class on Friday, April 15.  
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\noindent
{\it Questions}
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\noindent
A.  The following problems appear in a text by the mathematician 
Mahavira (circa 850 C.E.).  Solve them using our modern notation
and methods.  (The colorful directions are a semi-literal translation
of the original :) )
\begin{enumerate}
\item[1.]   Of a collection of mangoes, the king took $1/6$, the queen
$1/5$ of the remainder, the chief princes $1/4, 1/3, 1/2$ of the successive
remainders, and the youngest child took the remaining $3$ mangoes.  O you who
are clever in miscellaneous problems on fractions, give the measure of that
collection of mangoes.
\item[2.]  The total price of $9$ citrons and $7$ fragrant wood apples 
is $107$.  Again, the total price of $7$ citrons and $9$ fragrant wood apples
is $101$.  O you arithmetician, tell me quickly the price of a citron and
a wood apple here, having distinctly separated those prices well.
\item[3.]  One fourth of a herd of camels was seen in the forest; twice
the square root of that herd had gone to the mountain slopes; and three
times five camels remained on the river bank.  O smiling maiden with the 
shining eyes, what is the numerical
measure of that herd of camels? 
\end{enumerate}
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\noindent
B.  Bhaskaracharya (circa 1150 C.E.) developed the following identities for
the square roots of numbers of the form $a \pm \sqrt{b}$ ($a$,$b$ integers
or rational numbers more generally):
$$\sqrt{a \pm \sqrt{b}} = \sqrt{(a + \sqrt{a^2 - b})/2} \pm 
\sqrt{(a - \sqrt{a^2 - b})/2}.$$
\begin{enumerate}
\item[1.]  Show that the formula is valid (square both sides and simplify).
\item[2.]  Use the formula above to express $\sqrt{17 + \sqrt{240}}$ 
as a sum of two square roots.
\end{enumerate}
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\noindent
C.  Probably the high point of Indian geometry is a formula for the area
of a {\it cyclic quadrilateral} found by Brahmagupta (circa 630 C.E.).  A convex
quadrilateral $PQRS$ is said to be {\it cyclic} if all four vertices $P,Q,R,S$ lie on
one circle (this is not automatic for sets of four points in the plane).  Then Brahmagupta's
formula gives a way to compute the {\it area} $A$ of the quadrilateral if the 
lengths of the sides are known.  Say the four sides of the cyclic quadrilateral
are $a,b,c,d$ in some order.  Also, let $s = \frac{a + b + c + d}{2}$ be
the ``semiperimeter.''  Then the area $A$ is given by 
\begin{equation}
\label{Brahma}
A = \sqrt{(s - a)(s - b)(s - c)(s - d)}.
\end{equation}
(Some of you may know a very similar formula for triangles called {\it Heron's formula}.
Heron was an Alexandrian mathematician and engineer who lived roughly 10 - 70 C.E.  The result
known as Heron's formula says that
the area of a {\it triangle} with sides $a,b,c$ and semiperimeter $s$ is 
$$A = \sqrt{s(s - a)(s - b)(s - c)}.$$
It is thought that Brahmagupta was probably familiar with that result.  And indeed,
it is possible to see Heron's formula {\it as a limiting case of Brahmagupta's} if you 
let two of the vertices of the cyclic quadrilateral coincide, so $d \to 0$, for instance(!))

\begin{enumerate}
\item[1.]  Show that if a quadrilateral has perpendicular diagonals
then the sums of the squares of the lengths of opposite pairs of sides are equal.  That is, 
if the two pairs of opposite sides are $a,c$ and $b,d$, then $a^2 + c^2 = b^2 + d^2$.
\item[2.]  Conversely, show that if $a^2 + c^2 = b^2 + d^2$, then the diagonals
of the quadrilateral are perpendicular.
\item[3.]  Brahmagupta showed that if $\alpha^2 + \beta^2 = \gamma^2$ 
and $A^2 + B^2 = C^2$ are two Pythagorean triples, then a quadrilateral
having the lengths $\alpha C,\gamma B, \beta C, \gamma A$
for consecutive sides has perpendicular diagonals.  Show this.
\item[4.]  Suppose $(\alpha,\beta,\gamma) = (3,4,5)$ and $(A,B,C) = (5,12,13)$.
Find the sides, diagonals, and the area of the quadrilateral formed as in part 3.     
(Hint:  You can use equation (\ref{Brahma}) for the area; do you see why?)  
\end{enumerate}   
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