The number $e$ has many interesting and important properties, many of which are related to calculus. How can I explain what $e$ is and why it is important to those who have not had calculus (or even, when introducing it in calculus for the first time before derivatives are defined).
$\begingroup$
$\endgroup$
5
asked Mar 18, 2014 at 12:18
-
4$\begingroup$ I motivate $e$ by talking about compound interest (as the gaps in the time tend to zero, you get the natural logarithm popping up). But I do this because I find it interesting, rather than because it is the best way... $\endgroup$– user1729Mar 18, 2014 at 12:40
-
$\begingroup$ @user1729: I started writing my answer about 5 min ago. Funny that we thought of the same example and pretty much the same time! :) $\endgroup$– ThomasMar 18, 2014 at 12:43
-
1$\begingroup$ I would like to mention that if you have no direct motivation to introduce a constant like $e$, you probably shouldn’t, as it is pretty useless on its own. Instead, I would introduce constants when they first come up – with something along the lines of: “The result is roughly 2.71… Since this number turns out to be very relevant in calculus, it has its own symbol: $e$” $\endgroup$– WrzlprmftMar 19, 2014 at 16:01
-
$\begingroup$ If they know what imaginary numbers are, you could tell them that $e^{i\pi}=-1$. Of course, explaining why $e^{i\pi}=-1$ would be hard... $\endgroup$– Mike ShulmanApr 8, 2014 at 16:32
-
1$\begingroup$ Is it important? To me, the function $x\to e^x$ is important. I don't know what's so interesting about the value of that function at $1$. $\endgroup$– Jack MApr 13, 2014 at 22:17
Add a comment
|
3 Answers
$\begingroup$
$\endgroup$
4
I think the number $e$ is inherently analytical in the sense that every definition I know of it uses a limit somewhere. So, I think from a purely mathematical standpoint, it's very difficult to explain the importance of $e$ without calculus.
Of course, there are other standpoints. The one I usually try to use with my college algebra students is to stress its utility - it comes up all over the place. For example, it arises when modeling population growth, or radioactive decay, or the temperature in a cup of a coffee, or the money in a savings account.
To back this assertion up in my college algebra classes, I usually will show how the "limit" as $n$ goes to $\infty$ of the expression $P(1+\frac{r}{n})^{nt}$ "turns into" $Pe^{rt}$. Of course, they don't really know what a limit is (and I don't use that terminology), but this is one of those cases where limits really are intuitive, and I think most of them follow it. Then they get to see for themselves where the $e$ comes from.
Of course, many of my college algebra students don't really care about where things come from, but for many of them, simply saying "it comes up everywhere" is as good enough.
answered Mar 18, 2014 at 12:45
-
4$\begingroup$ You can even somewhat nail down the inherent connection to calculus by e’s trancendency: It cannot be produced by algebraic operations only. The only other available way to produce a number that could be considered non-calculus would be “pure” geometry (which is where we got the trancendent π from), but I know of no purely geometric way to produce e. $\endgroup$ Mar 18, 2014 at 13:00
-
$\begingroup$ @Wrzlprmft: I agree, but the issue with $\pi$ caused me to be a bit reserved in my post. $\endgroup$ Mar 18, 2014 at 13:04
-
$\begingroup$ I quite liked the way that this article explained e as something that "comes up everywhere", using the definition that "e is the base rate of growth shared by all continually growing processes": betterexplained.com/articles/… $\endgroup$ Apr 14, 2014 at 10:33
-
$\begingroup$ @Wrzlprmft: "Purely geometric" is a bit open to interpretation, but $e$ is the number such that the area between the $x$-axis and the standard hyperbola from $1$ to $e$ is $1$. The problem now is that without calculus it is quite unclear what that number would ever have to do with radioactive decay, cooling objects or accumulated debt. $\endgroup$ Feb 8, 2019 at 6:09
$\begingroup$
$\endgroup$
0
This might not be what you are looking for, but if the students know about derivatives (which comes before derivatives) you could tell your students about compound interest. If an initial $P_0$ dollars invested at $100r\%$ (APR) compounded $n$ times a year then the value of the investment after $t$ years is $$ P = P_0\left(1 + \frac{r}{n}\right)^{nt}. $$ The funny thing is that if you compounded the interst more and more frequently, that is, if you let $n$ go to infinity, then you get $$ P = P_0e^{rt}. $$ This, of course uses that $$ e = \lim_{n\to \infty} \left(1 + \frac{1}{n}\right)^{n}. $$ So, this is really an example of how naturally $e$ shows up.
$\begingroup$
$\endgroup$
5
What is it? I would start by defining it as $$1+\frac{1}{1}+\frac{1}{2}+\frac{1}{6}+\cdots$$
Why is it important? 1) In almost every subject with numbers, there are formulas with $e$'s: the biologist's exponential growth, the economist's value of stock options, the physicist's Planck law of radiation, the statistician's bell curve, etc.
Why is it important? 2) The constant also appears in contexts that don't sound mathematical, like the marriage problem:
Suppose you see candidates one at a time, and want to choose the best one immediately. The optimal algorithm is to see the first $1/e$ of the candidates, and then pick the first one who is better than all of them.
-
1$\begingroup$ Another example where the calculus is hidden is the Catenary: The shape of any losely hanging rope can be described by a function of the form $f(x) = a \left ( e^{x/a} + e^{-x/a}\right)$. Unfortunately, understanding this requires exponentiation with at least rational exponentials. $\endgroup$ Mar 18, 2014 at 16:43
-
$\begingroup$ @Wrzlprmft, you can describe the curve equally well saying $f(x) = b (a^x + a^{-x})$, the $e$ is just the most calculus-natural way of expressing it. $\endgroup$– vonbrandApr 11, 2014 at 16:32
-
$\begingroup$ I'll ask, what are the next denominators (2 please) in that series? Not obvious to me. And I did not realize Black Scholes used e, but of course, it does. $\endgroup$ Dec 20, 2014 at 22:39
-
$\begingroup$ @JoeTaxpayer, the denominators are the factorials, so the next two are 24 and 120. $\endgroup$– user173Dec 20, 2014 at 22:48
-
$\begingroup$ Much appreciated. (I am old, and knew this at some point, I appreciate the reminder!) $\endgroup$ Dec 20, 2014 at 22:52