# The challenge that “e” poses

The Launch

Last week I met up with my co-teacher in Algebra 2. We’re working on our unit of exponential functions and logarithms, and we were talking about spending a short amount of time introducing “e” to our kids. Personally, this question has haunted me because when I taught Algebra 2 at the start of my career, I couldn’t ever find a motivation for it — except for interest being compounded continuously. That never quite sat well with me because you have to assume that you have an interest rate of 100%.

Like $(1+\frac{100\%}{n})^n$ as $n$ gets larger and larger.

But where are you getting an interest rate of 100%?!? It isn’t a terrible way to introduce “e” (getting kids to understand the structure of that equation above, there’s a lot of deep thinking that goes on in there). There is also the idea that there is a limiting value for that expression above — instead of the value just going up infinitely — that can be exploited and discussed.

But I never thought “e” and “ln” really belonged in Algebra 2, precisely because I couldn’t motivate them in a way that was intellectually satisfactory. So I tweeted out:

Little did I know I was going to get so many responses! I wanted to archive them here, which is why I’m writing this post, but then share where I’m landing on this whole “e” thing right now.

Ways To Introduce “e”

1. @retaneri linked to this question from “Play With Your Math” I love that site, but hadn’t worked on this problem before.

Interestingly, @bowmanimal and @averypickford came up with the same problem to share with me! Apparently the answer is to break up the number into a bunch of es … I haven’t figured out why this works yet.  But for 25, if I understand this correctly, I think it means $25=e+e+e+e+e+e+e+e+e+e+0.196986e$ which has a product of $e^{9.196986}$. (Or to be super precise: $e^{25/e}$.)
2. @mikeandallie shared with me this approach by throwing darts. Which includes this gem:

and this instantiation:
3. @jensilvermath suggested just looking at $(1+1/n)^n$ for larger and larger $n$, without reference to a limit. Have kids make predictions about what is going to happen and why, and then let them explore it. At first, I was like “hmmm, would this work?” but I love the idea of kids stumbling upon and wrestling with (1+almost zero number)^(super huge number) might be tricky. Does it have to be a huge result? What data could they collect? What would they do with it?

And @LukeSelfwalker shared this simple but stunning Desmos activity which gets kids to see how polynomials can start approximating exponential functions — a beautiful visual connection to all of this.

4. Of course @dandersod showed me a connection between Pascal’s Triangle and e, which I didn’t know about (or if I did, I totally forgot). He sent me this link:

Whaaaa?! But okay! WHOA?!
5. @roughlynormal suggests:

I have to think about this… Basically the differences relate to the derivative… But I did a quick 5 minute look with a google spreadsheet and I couldn’t make this work. I think for it to work, you need to divide the first differences by the change in x, and also divide the second differences by the change in x, and look for them to be equal. In other words, the derivative…
6. @jdyer gave this gem which I’ve never heard of before:
“You have a full glass with 1 liter of water. You take gulps from the glass; each gulp is a random real number of liters from 0 to 1. On average, how many gulps do you expect it to take before the glass is empty?”

And his discrete version is you start with 100 and kids take away a random number (generated from 0 to 100) per step.

Whoa! TOTALLY new to me. Since we can’t put up e fingers, we could play the game where we each pick a number (doesn’t have to be an integer) from 0 to 5. I had to see this for myself!
The red line is always greater than the blue line (I think… I want to prove it algebraically!). [Note: related, this set of tweets on $x^y=y^x$ which @BenjaminASmith alerted me to]. And @mathillustrated shared this amazing presentation (read it!) on scaffolding and formalizing this game with kids.
8. @CmonMattTHINK shared this fact which I LUUURVE but forgot about:
The probability of a random permutation of n objects being a derangement (no object remaining in its original position) approaches 1/e as n->oo.And @DavidKButlerUoA shared a wonderful presentation he made on where the derangement formula comes from.
9. @bobloch also shared with me a fact I don’t remember ever learning! e shows up in the harmonic series.$sum_{n=1}^{\infty} \frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...$Apparently, if we look at which n values bring this series to an additional integer, we get:

And if you go further and further, and take ratios of these n values, you get a better and better approximation for e. I calculated 227/83 and got about 2.735.

10. @mathgeek76 reminded me of the chapter in Steve Strogatz’s excellent book The Joy of X which talks about how you find the right partner, mathematically.

The answer involves $1/e$. (Spoiler here.)

And then a wonderful conversation about where “e” belongs in a high school curriculum popped up on my feed, launched off by this tweet:

I should say this was Steve’s launching tweet, but he was open to thoughts on both sides of the discussion that ensued. My favorite part of the conversation was when someone (I can’t remember who) brought up $\pi$ and when that should be introduced and why.

My conclusions

First and foremost, I knew there was a lot to $e$ — it bridges continuous and discontinuous phenomena. It is important in calculus and (to some degree) combinatorics. There’s a lot I don’t fully understand yet about what people shared with me in their tweets — and I have to work through them to see their connections. I do see lots of connections but I haven’t worked through the math of any of these things to draw them in fully.

Second, in my musings on this, I think I’ve come to recognize that why I have always found $e$ so fascinating is that is keeps on popping up in unexpected places. In my mathematical career, my jaw has hit the floor a number of times when I see it suddenly emerge when I never thought there would be a connection. I mean, think of the first time you saw $e^{i\theta}=\cos(\theta)+i\sin(\theta)$. ‘Nuff said.

So if I want my kids to see what I see when I encounter $e$, that’s what I want them to encounter. A surprise. That it can pop up in totally unexpected places, and you  might not initially know why, but it eventually can become clear.

I don’t think I have time to pull this together for this year, but here’s what I’ve decided I want to do in some future year… I want to have a period of time where kids are told “this is a problem solving day” (or set of days). And I give the class different problems that will result in them approximating $e$… but they don’t know it. Like Pascal’s triangle… they can just do the calculations and see $e$ pop out. Or the chess board experiment. Or taking random numbers away from 100. Or the harmonic series. Or the product challenge. Or the $x^y$ vs $y^x$ game.  Or the compound interest problem. And have them work on them. They all seem unrelated. Yes, this is contrived. Yes, I’m telling them what to do in many of them.

But BOOM. Soon this $e$ number that keeps popping up in all these unrelated problems.

For each of these, they are going to get numbers close to $e$. And for me, I can say

This, my friends, this is what is so beautiful about this number for me. It is a universal constant. It is like $\pi$. It pops up in so many unexpected places. There is an underlying structure to why this is all happening, why it pops up everywhere. That is going to start to be revealed in calculus, but that’s only the bare beginnings… It goes much deeper. But I wanted you to get the experience of wonderment and have something that’s you know is true that begs the question why… WHY? WHAAAA? WHYYYYYYY! Because this desire to know, to figure out why something is true when you know it must be and it feels too unbelievable to be true, that’s a feeling mathematicians get that drive them forward in their work. And making those connections, and we know they must exist, it’s awesome when it happens. So yeah… I wanted to introduce you to this important number $e$ which we’ll just take as that… an important number that we’ll get to play around with like we do $\pi$… but know it’s more than just a number. For you, now, it’s a question that’s begging an answer.

A realistic ending to an idealist exhortation from me to my students

Okay, I don’t know if I could really pull this off. But I’d love to.