This abstract refers to a Teaching Module mentioned in the article Development of Teaching Modules on the Internet by Keith Tognetti and Ian Doust in this issue of the Gazette.
The Teaching Module is stored at the Australian Mathematical Society's
electronic site at
http://www.austms.org.au/Modules/ which
also contains information for presenting Modules and contributing to the
associated Commentaries.
This Module is written as a self contained introduction to e, bringing together the main theorems and important properties of this fundamental constant of natural growth processes. It assumes only an elementary understanding of integration and is intended for the serious student in mathematics who wishes to begin a deeper understanding, and also perhaps, colleagues who are returning to this area after a long break. Some history and anecdotage are also included - in particular, Nobel Prize winning physicist Richard Feynman showing how he can use his feel for numbers to outdo a calculator. Euler's gamma constant also comes into the main theorem and this highlights the intimate interconnection between e, the area under the curve 1/x, and the the truncated Harmonic series. Although e is the base of Natural logarithms it is not, as commonly supposed, the base of Naperian logarithms - in fact, as is shown, the Naperian approach led to one of the few dead ends in mathematics. And yes the absurdity of expecting sustainable interest rates is highlighted through an example based on the Dutch buying Manhatten Island from the Indians.
The layout of the Module is as follows:
An Introduction based on a simple example is used to demonstrate how e becomes the limit as we increase the number of periods of compound interest in a year. `` Growth Models" is a simplified treatment of intrinsic rate of increase and
`` Malthusian" growth - it develops the formula for doubling time which is
approximately 70 divided by the % interest rate. A short history of e is
coupled with an appendix outlining Naperian logarithms.
The axiomatic development begins with the definition of a function which is the area under the curve 1/t from t =1 to t =x. It is then shown that this function has all of the properties we normally associate with the logarithmic function . From this we define the inverse function and e and then demonstrate that this inverse function has all of the properties that we associate with exp(x). Using only properties of the log function, the main theorem proves that e is indeed the limit of ( 1 + 1/n )^n , as we had suspected in our introduction. It also gives as by-product, a bound for the Euler Gamma constant and an entry into Harmonic numbers.
A collection of proofs is then given which are selected only on the basis of their elegance.
Department of Mathematics, University of Wollongong, NSW, 2522