# Engaging students: Introducing the number e

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Julie Thompson. Her topic, from Precalculus: introducing the number $e$.

What interesting word problems using this topic can your students do now?

I found a very interesting word problem involving the number e and derangements. A derangement is a permutation of a set in which no element is in its original place. The word problem I found is as follows:

“At the bohemian jazz parties frequented by aficionados of the number e, the espresso flows freely, and at the end of the evening, party-goers are just as likely to go home in someone else’s overcoat as they are in their own. After such a party, what are the chances that at least one person goes home wearing the right coat?”

To start off, we need to find out how many permutations, or how many combinations of ways the coats can be put on at random when guests leave the party. The problem asks us to identify the chance that at least one person IS wearing the right coat. In other words, we need to delete all the combinations in which nobody grabbed the correct coat. These are the derangements. Interestingly, when you divide the number of permutations by the number of derangements, you get a number extremely close to the value of e. And the ratio is always so.

Looking at a numerical example with 10 guests, the number of ways 10 people can pick up 10 coats (permutations) is 3628800, and the number of ways nobody would pick up the right coat is 1334961. Dividing, 3628800/1334961= 2.71828, which is extremely close to e. Therefore, the chance of nobody getting the right coat is about 1 in e. How interesting. I feel like this word problem would really interest students!

The number e was not discovered as ‘naturally’ as you may think. Mathematicians came close to discovering e in their calculations many times in the 17th century but thought it was just a random number without any real significance. The first person to calculate e is not documented, but historians believe it to not even be a mathematician, but a banker or trader. Why is this?

The number e is very fundamental to a financial process that took off in the 17th century. “The number e lies at the foundations of one of the most fundamental processes of finance: compound interest.” Mathematicians, including Jacob Bernoulli, would later go on to define:

. $e = \displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x$

“This is why the number e appears so often in modeling the exponential growth or decay of everything from bacteria to radioactivity.” This fact was adopted by the mathematical community and many mathematicians started collaborating and making many more discoveries on the number e, such as Euler, who estimated e correctly to 18 decimal places, gave the continued fraction expansion of e, and made a connection between e and the sine and cosine functions. The number e is one of the most beautiful and powerful number in all of mathematics and the use of it was adopted into mathematics most likely by a banker…how interesting.

How can technology be used to effectively engage students with this topic?

Any graphing technology, such as a TI calculator, Mathematica, MatLab, Desmos, etc., are great tools to use in order to engage students when discovering the number e. For instance, to convince students that the above limit is true,

$e = \displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x$,

I can have them graph the function for themselves and actually see that the function approaches the number e as x gets very large. Similarly, I can simulate numbers of e on a computer program with the expansion  1 + 1/1! + 1/2! + 1/3! + … to show the sum getting closer and closer to the value of e the more terms I add. I believe this will be really engaging because the expansion for the number e and the limit for e look like they have nothing to do with e at first glance. To make the connection between them graphically would be somewhat magical to students and hopefully make them curious for more.

References:

http://wmueller.com/precalculus/e/e6.html (this is word problem from A1)

https://brilliant.org/wiki/the-discovery-of-the-number-e/

http://mathworld.wolfram.com/e.html

http://www-history.mcs.st-and.ac.uk/HistTopics/e.html