# 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 Deanna Cravens. Her topic, from Precalculus: introducing the number $e$. The number e is a relatively newer irrational number if compared to pi. However, it first made its appearance very subtly in 1618. Napier was working on a table of natural logarithms, however it was not noted that the base was e. There were a few other appearances of e but mathematicians had not truly made a connection to it. Eventually in 1683, Jacob Bernoulli was looking at a business application dealing with continuously compounded interest and recognized that the log function and the exponential function were inverses. In 1690, a letter was written by Leibniz and e officially had a name, except it was called ‘b’ at the time. As it comes to no surprise, Euler had his hand in discovering e. He published Introductio in Analysin infinitorum in 1748 where he showed that e is the limit of $(1 + 1/n)^n$. Now Euler did not explicitly prove that e is irrational, however most people accepted it at that point, but it was indeed later proven. How could you as a teacher create an activity or project that involves your topic?
Where does the number e come from? Well, the answer is a business application dealing with continuously compounded interest. However, students in a pre-calculus class can easily discover the number e without having to use the calculus behind it. Simply give students this short activity at the beginning of class. One of the good things about this activity is that it gives a brief snippet of the history of e before students begin to calculate it. Then, students can easily use a calculator and plug in the listed values in the table into the equation $(1+1/n)^n$. As the numbers get increasingly large, students will notice that they will all appear to be getting closer to 2.718… which is now known as the number e. As a teacher it is important to note that e is like pi, it is an irrational number that goes on forever and doesn’t have any sort of repeating pattern, yet it is extremely important in mathematics. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

This video would be excellent to show students who are asking, “why is e so important or where does it come from?” The video starts out by stating what e is approximately equal to. Then it gives a brief history about e and talks about compounded interest. It does a great job at explaining compounded interest. It is executed in a way where pre-calculus students can easily understand the concept. It also uses good visual cues to show how it would work. Next it lists several applications of e. These applications include: statistics through the normal curve, biology by modeling population growth, and physics by the exponential decay of a radioactive material. Overall, it does a great job showing the importance of e in real world applications. Thus, showing the importance of e to a pre-calculus students.

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