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\begin{document}
\begin{center}
{\bf Learning Mathematics with the Spreadsheet}
Jen-chung Chuan
Department of Mathematics
National Tsing Hua University
Hsinchu, Taiwan 300
\end{center}
\section{Introduction}
Spreadsheet program provides an alternative to conventional programming
languages in developing educational software. It offers the user an easy way
to create an interactive and user-friendly instructional program. It has
several built-in features such as graphic capability and data base
management capability. The program developed on the spreadsheet format has
the 'transparent black box' feature which is considered to be very useful
for educational purposes. Programming with spreadsheet program can save a
considerable amount of development time. In what follows we would like to
share our experience on the following topics:
1) how a formula discovered by Newton becomes alive in the spreadsheet;
2) how the spreadsheet reveals a common thread of two seemingly unrelated
topics in calculus;
3) how to generate the Catalan numbers with the spreadsheet after their
natural combinatorial properties;
4) how the spreadsheet may be used to assist the understanding of an
algorithm in approximation theory.
\section{Newton's Formula}
Suppose that $\alpha ,\beta ,\gamma $ are the zeros of the cubic equation%
$$
z^3-pz+q=0.
$$
Problem: express%
$$
\omega _n=\alpha ^n+\beta ^n+\gamma ^n
$$
as a function of $p$ and $q$ for $n=1,2,3,\cdots .$ Solution: working out as
a special case of Newton's formulae for sums of zeros of an algebraic
equation [3], we obtain the recursive relation%
$$
\omega _{n+3}=p\omega _{n+1}-q\omega _n
$$
with the initial values%
$$
\omega _{-1}=\frac pq,\omega _0=3,\omega _1=0.
$$
From this we obtain successively%
$$
\begin{array}{c}
\omega _2=2p,\omega _3=-3q,\omega _4=2p^2,\omega _5=-5pq, \\
\omega _6=2p^3+3q^2,\omega _7=-7p^2q,\omega _8=2p^4+8pq^2, \\
\cdots
\end{array}
$$
The coefficients occurring in $\omega _n$ form an interesting pattern, which
can be generated in a spreadsheet:
\verbatimfile{/bir/tmp.prn}
What do these numbers represent? It is seen that the $\omega $ with an even
index $n$ and the $\omega $ with an odd index behaves differently. For
example, the row consisting of%
$$
\begin{array}{ccccccc}
2 & 0 & 80 & 0 & 175 & 0 & 20
\end{array}
$$
indicates that%
$$
\omega _{20}=2p^{10}+80p^7q^2+175p^4q^4+20pq^6
$$
while the row formed by%
$$
\begin{array}{cccccccc}
0 & -23 & 0 & -276 & 0 & -322 & 0 & -23
\end{array}
$$
represents the fact%
$$
\omega _{23}=-23p^{10}q-276p^7q^3-322p^4q^5-23pq^7.
$$
This example shows that computations with the spreadsheet, in some sense,
preserves the habit of abstract thinking which is so very much treasured in
developing one's mathematical maturity.
\section{Successive Derivatives and Indefinite Integrals}
Let $g(x)$ be a function of the form%
$$
g(x)=\frac{f(x)}{x+1},
$$
where $f(x)$ is an infinitely-differentiable function. The first few
successive derivatives of $g(x)$ take the form%
$$
g^{\prime }(x)=\frac{f^{\prime }(x)}{x+1}-\frac{f(x)}{(x+1)^2},
$$
$$
g^{\prime \prime }(x)=\frac{f^{\prime \prime }(x)}{x+1}-\frac{2f^{\prime
}(x) }{(x+1)^2}+\frac{2f(x)}{(x+1)^3},
$$
$$
g^{^{\prime \prime \prime }}(x)=\frac{f^{\prime \prime \prime }(x)}{x+1}-
\frac{3f^{\prime \prime }(x)}{(x+1)^2}+\frac{6f^{\prime }(x)}{(x+1)^3}-\frac{%
6f(x)}{(x+1)^4},
$$
$$
g^{(IV)}(x)=\frac{f^{(IV)}(x)}{x+1}-\frac{4f^{\prime \prime \prime }(x)}{%
(x+1)^2}+\frac{12f^{\prime \prime }(x)}{(x+1)^3}-\frac{24f^{\prime }(x)}{%
(x+1)^4}+\frac{24f(x)}{(x+1)^5},
$$
$$
\cdots
$$
The coefficients, when displayed in reversed order, appear as
\verbatimfile{/bir/p3.prn}
This number pattern suggests that:%
$$
\begin{array}{c}
\text{If }g^{(n-1)}(x)=\sum_{k=0}^{n-1}(-1)^{(k+n-1)}\frac{c_kf^{(k)}(x)}{%
(x+1)^{n-k}},c_{n-1}=1 \\ \text{then }g^{(n)}(x)=\sum_{k=0}^n(-1)^{(k+n)}
\frac{c_k^{\prime }f^{(k)}(x)}{(x+1)^{n+1-k}}, \\ \text{with }c_k^{\prime
}=-nc_k,0\le k\le n-1,c_n^{\prime }=1,
\end{array}
$$
a rule to be verified readily. Guided by this formidable formula, the actual
implementation in a spreadsheet, however, is quite straightforward: just
assign the number $1$ to cells along the diagonal and make every cell on row
$n$ the product of $-n$ with the cell located directly above it.
The same number pattern also appear in the formulae of indefinite integrals
[4]:%
$$
\int x^0e^x\,dx=e^x,
$$
$$
\int x^1e^x\,dx=e^x(x-1),
$$
$$
\int x^2e^x\,dx=e^x(x^2-2x+2),
$$
$$
\int x^3e^x\,dx=e^x(x^3-3x^2+6x-6),
$$
$$
\int x^4e^x\,dx=e^x(x^4-4x^3+12x^2-24x+24),
$$
$$
\cdots
$$
Is this set of formulae a consequence of the former, or vice versa?
Curiosity in creating number-pattern with the spreadsheet could therefore
lead to interesting and perhaps deep mathematical investigations.
\section{Catalan Numbers}
The Catalan numbers
$$
C_k=\frac 1{k+1}\binom{2k}k=\binom{2k}k-\binom{2k}{k-1}
$$
arise naturally in many problems of discrete mathematics [2]. This sequence
of numbers can be obtained by going down the center column of the Pascal's
triangle and from each number subtract the adjacent number. The same
sequence can be generated using the spreadsheet when $C_k$ is viewed as the
solution of this interesting combinatorial problem:
Consider chessboards of sides 2,3,4,$\cdots .$ All squares south and west of
the main diagonal are deleted. We are to move the rook from the lower right
corner to the upper left corner, and its only allowed movements are north or
west. For a board of size $k,$ how many different paths can the rook make?
Think of the entries of the spreadsheet as the squares of the chessboard. If
numbers are filled among entries lying on and above the main diagonal
indicating the number of different paths reaching the A1 entry, obeying the
same restrictions as above, one would analyze the situation and set up the
spreadsheet this way:
(1) if the rook starts from anywhere on the first row, there is only one way
of reaching the A1 entry (by moving westerly): thus all entries on the top
row are assigned 1;
(2) if the rook starts anywhere below the first row and above the main
diagonal, its first move may either going north or west: thus each such
entry is given a formula reflecting the sum of entries directly above and to
its left;
(3) if the rook start anywhere on the main diagonal below the first row, its
only allowed first move is by going north: thus such entry should reflect
the number directly above it.
The actual implementation of these three rules is carried out by applying
the COPY command given in the spreadsheet. The following tables show the
resulting formulae and the resulting numerical values. We see that the
Catalan numbers occur along the diagonal entries and the entries directly
above it.
\verbatimfile{/bir/pp4.prn}
\verbatimfile{/bir/p4.prn}
Can one find a better computing environment highlighting the Catalan numbers?
\section{Sequence of Polynomials}
Let the sequence of polynomials $Q_k$ be given recursively by%
$$
\begin{array}{c}
Q_0=1,Q_1(x)=x-a_1, \\
Q_n(x)=(x-a_n)Q_{n-1}(x)-b_nQ_{n-2}(x)\text{ for }n\ge 2.
\end{array}
$$
Then a linear combination of $Q_k$ of the form%
$$
f(x)=\sum_{k=0}^nc_kQ_k(x)
$$
can be computed by means of the formula%
$$
\begin{array}{c}
d_{n+2}=0,d_{n+1}=0, \\
d_k=c_k+(x-a_{k+1})d_{k+1}-b_{k+2}d_{k+2}, \\
f(x)=d_0.
\end{array}
$$
Such linear combination occurs frequently in the theory of orthogonal
polynomials.[1] With an appropriate arrangement of a spreadsheet, the
process of computing $d_k,$ hence $f$ itself, can be vividly visualized. The
principle behind the implementation in a spreadsheet is that of the ``method
of detached coefficients'' to be found in algebra texts before computers
became personal.
To illustrate this method, we consider the following concrete set of data:%
$$
\begin{array}{rrrrrrr}
k & 0 & 1 & 2 & 3 & 4 & 5 \\
a_k & & -3 & 6 & 5 & -1 & 3 \\
b_k & & & -2 & -2 & 1 & -2 \\
c_k & 2 & 1 & 3 & 4 & 2 & -1
\end{array}
$$
and the polynomials $Q_k$ given by%
$$
\begin{array}{c}
Q_0=1,Q_1(x)=x-a_1, \\
Q_2(x)=(x-a_2)Q_1(x)-b_2Q_0(x) \\
Q_3(x)=(x-a_3)Q_2(x)-b_3Q_1(x) \\
Q_4(x)=(x-a_4)Q_3(x)-b_4Q_2(x) \\
Q_5(x)=(x-a_5)Q_4(x)-b_5Q_3(x)
\end{array}
$$
Once the given data are placed on the first three columns, we need only to
place one formula in one cell and apply the COPY command to make the other
appropriate cells obey the same rule.
\verbatimfile{/bir/p5.prn}
The required linear combination%
$$
f(x)=2Q_0(x)+Q_1(x)+3Q_2(x)+4Q_3(x)+2Q_4(x)-Q_5(x)
$$
is to be read-off from the table:%
$$
f(x)=639+342x-143x^2-25x^3+12x^4-x^5.
$$
\begin{center}
References
\end{center}
1. E.W. Cheney, Introduction to Approximation Theory, Chelsea, 1982.
2. M. Gardner, Catalan Numbers, Time Travel and other Mathematical
Bewilderments, Freeman, Chapter 20, 253-266.
3. T.S. Nanjundiah, A note on an identity of Ramanujan, Amer. Math. Monthly,
100 (5), 485-487.
4. B.O. Peirce, A Short Table of Integrals, Blaisdell, 1956.
\end{document}