Various ways to introduce e

Mary Coupland, School of Mathematical Sciences, University of Technology, Sydney

What is this e number? How is it related to the way things grow? We look at two ways to introduce e without differentiation by first principles: a graphical approach considering  and  and a compound interest approach.

A little history

Do you ever wonder whether mathematics is discovered or invented? Regularities in the physical world that lead to numbers like  and e seem to lend weight to the former proposition, while reading the history of mathematics, and finding out how various mathematical creations were constructed to meet particular needs of the times, I am led to think that the latter proposition has much merit.

It seems that many European mathematicians in the sixteenth and early seventeenth centuries occupied themselves with the search to make calculations easier, in particular to find ways to replace multiplications by additions. This led to Napier's invention of logarithms, which is well described in Lecture 17 of 'Great Moments in Mathematics' by Howard Eves (Eves, 1983). He explains that Napier's logarithms were not originally written about in terms of indices at all, and were not, as is often assumed, logarithms in base e. Henry Briggs, a London professor of mathematics, visited Napier in 1614 and together they decided that logarithms would be easier to use if the logarith of 1 would be 0 and the logarithm of 10 would be a power of 10. Briggs, and also Adrieaen Vlacq, a Dutch bookseller and publisher, worked out these new 'common' logarithms and published tables of their values. The tables were not superseded until 1924! (Eves, 1983, pp.184-5).

Vlacq is mentioned in the following extract from an article written in 1727 or 1728 by the Swiss mathematician Leonhard Euler (1707-1783). This article was not printed until 1862 but Euler must have promoted the use of e for the constant 2.718 2817 & in other works, because it seemed to catch on and was adopted by other writers, for example by Daniel Bernoulli in 1760 and Lambert in 1764. Apparently other symbols were suggested by other writers, for example Leibnitz used the letter b for it in 1690. (Cajori, 1929, p.13). The extract below is taken from Smith (1929, p.95).

None of the sources mentioned so far give any reason for Euler's choice. Why not use another letter of the alphabet? Perhaps we will never know why Euler chose the letter e.

Introducing e

Teachers of 2 Unit mathematics might use one or a combination of methods to introduce e and the exponential function in their calculus classes. Telling a historical anecdote might set the scene, although this lacks the surprise that the next method, a graphing approach, includes. Perhaps the most satisfactory method is the development from compound interest, since that leads naturally into the applied topics of exponential growth and decay. Finally there is the approach via the logarithm function, which is one of the methods suggested in the syllabus. The syllabus also describes quite fully the introduction of e via differentiation by first principles of the function  which I will not go into in this presentation.

The graphing approach

This could be motivated by pointing out to students that for the functions they have met so far, namely polynomials and trig functions, the derivative, or gradient function, is always different from the original function. Would this always be the case? Might there be a function which is the same as its derivative? Having disposed of polynomials and trig functions, you might investigate any other functions whose graphs are familiar to your students, such as hyperbolas and exponential functions like

For a quick introduction, sketch by sight the derivative of  and show that the general shape is the same as the original function, so that further investigation is warranted. Next, calculate the gradients at selected points on the curve and show that for  the derivative is below the curve, while for  the derivative is above the curve, but closer to it than in the previous case. This leads to the conjecture that for some number between 2 and 3, but closer to 3, there is a similar exponential function whose derivative is the same as the original function. You could tell the students that this number is called e and is approximately 2.718 28..., but it is much more valuable to have them look for it using either a graphing calculator or graphing software on a computer.

On the following pages the details of this graphing approach are shown. I suggest two alternatives: in (a) tangents are drawn; in (b) points close together are chosen and gradients of secants calculated. The second method is easily achieved with software such as A Graphing Approach to the Calculus (Tall et al., 1990).

Graphical methods

(a)  Use a graph on grid paper, draw tangents at selected points, and calculate the gradients.

 

 

(b) Choose points very close together on the curve and calculate the gradient of the secant joining those points. For example, let  and  Then the gradient of the secant joining these points is given by:

                   approximately.

       Repeat for several pairs of points on the curve, perhaps asking the students to choose a point each, and plot the results.

Graphical methods: the search for e

Having established that there is a number between 2 and 3, say a, for which the derivative of the graph of  is the same as the original graph, we can look for this number using the graphing facilities of a computer or a graphing calculator.

On the Texas Instruments TI-81, the following steps will give the value of the 'numerical derivative', that is, the slope of the secant line through the points  and  when  is 0.001 and x is 3, and  The command NDeriv is found under the MATH menu.

This could be used as an alternative to hand calculations in method (b) above, and can be extended to graphing the derivative function: the following will graph the function  and its derivative, calculated numerically.

I have shown below the graphs obtained from Mathematica (Wolfram Research Inc., 1989) of various functions (solid lines) and their derivatives (dotted lines).

Using compound interest as a starting point

It is helpful if students have first investigated the value of

for increasing values of t. Yes, it does seem to be 'out of the blue'. It is very interesting to ask for predictions (and reasons) before any calculations are done. Some people focus on the fact that  becomes very small for large values of t, and they predict that the expression will have values approaching 1. Others focus on the fact that we are finding higher and higher powers, and predict that the expression will 'just keep on growing'. With a calculator, it does not take long to settle this disagreement.

t
1	2
10	2.593 742 46...
100	2.704 813 829...
1000	2.716 923 932...
10000	2.718 145 927...
.	.
.	.

At this stage, the value of knowing a little history is plain. You can say that mathematicians have known this result for at least three hundred years, and have given a name to the number that the second column approaches: e. Note that I am avoiding using limit notation, but it could be used here if students are familiar with it. If not, we can just say that there is a number, called e, with the property that the last column in the table can be made as close as we want to e by taking a large enough number for t. It is also interesting to look at what happens to  for various values of a, and at

Now we turn to compound interest. Recall that if we invest P dollars for  n years at r % p.a. compound interest, our investment accumulates to:

          or      with  

For example, $1000 invested for ten years at 6% p.a. compounded yearly amounts to  to the nearest cent.

But what is the effect of more frequent compounding? This is shown in the table:

How often compounded	Amount	To nearest cent
Monthly		$1819.40
Weekly		$1821.49
Daily		$1822.03
Hourly		$1822.11

There seems to be an advantage in compounding over shorter and shorter time intervals but the advantage seems to be diminishing. Is there a limit to this advantage? To investigate this, we need an expression for the general case. Say we invest $P at r% p.a., compounded over c time intervals per year for n years. The expression looks neater and is easier to work with if we put R = r/100.

The accumulated amount is

We want to know what happens for shorter and shorter time intervals, that is, when the number of time intervals per year, c, increases. With the following change of variable, we introduce the expression that we investigated earlier:

       Put t = c/R, so that c = Rt.

Now for any fixed R, as c increases, so will t.

The expression now becomes

As t increases, this results gets as close as we want to

This means that with continuous compounding, the best return we could possibly get on $1000 at 6% p.a. over 10 years is

The next step would be to talk about the exponential function, plot its graph, and use either the syllabus method or the graphical approach described above to convince students that its derivative is the same function; or perhaps show this by looking at Newton's expansion for , and deriving from it the fact that  Note that Newton's expansion gives us another way to estimate e, by putting x = 1:

The advantage of the compound interest approach is that it is a good basis for the work on the applications of calculus to the physical world. The arguments used for compound interest would be applicable for any case of continuously compounding growth. For example, 1000 grams of bacteria increasing at a continuous rate of 6% per hour will grow to 1822.12 grams (approx.) over 10 hours. Compare this with  which is the amount you get if all the bacteria sit around and do nothing until, on the hour, every hour for ten hours, 6% of them reproduce. When r is negative, you get exponential decay, of course.

References

Cajori, Florian (1929). A History of Mathematical Notations, Volume 2. The Open Court Publishing Company, Chicago.

Eves, H. (1993). Great Moments in Mathematics (Before 1650). The Mathematical Association of America, USA.

Smith, David Eugene (1929/1959). A Source Book in Mathematics, Volume 1, p.95. Dover Publications. New York. (Originally published in 1929 by McGraw-Hill).

Tall, D., Blokland, P., and Kok, D. (1990). A Graphic Approach to the Calculus (Computer Program). Sunburst, Pleasantville, NY.

Wolfram Research Inc. (1989). Mathematica (Software). Steven Wolfram. Champaign, IL.

Extension

If you have time to go further with this topic, or even as an extension for Year 10 students, I recommend Unit 7, Growth and Decay, in the excellent series Investigating Change: an Introduction to Calculus for Australian Schools by Mary Barnes, published by the Curriculum Corporation and available from AAMT. This is also a good reference for applications of calculus to the physical world.

A postscript

From The Penguin Dictionary of Curious and Interesting Numbers, by David Wells, Penguin, 1986.

2.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 ..e, the base of natural logarithms, also called Napierian logarithms, though Napier had no conception of base and certainly did not use e.

It was named 'e' by Euler, who proved that it is the limit as x tends to infinity of

Newton had shown in 1665 that   from which  a series which is suitable for calculation because its terms decrease so rapidly.

By chance, the first few decimal places of e are exceptionally easy to remember, by the pattern 2.7 1828 1828 45 90 45...

The best approximation to e using numbers below 1000 is also easy to recall: 878/323 = 2.718 26...

Like , e is irrational, as Lambert proved.

Hermite proved that e is also transcendental in 1873.

e features in Euler's beautiful relationship,  and, more generally, e is related to the trigonometrical functions by

It possesses the remarkable property that the rate of change of  at  is  from which follows its importance in the differential and integral calculus, and its unique role as the base of natural logarithms.