I do explain e from the derivative point of view. We encounter it when we’re searching for a formula for derivating exponential functions.
We first investigate using a computer and numerical aproximation what the derivative of a^x is. We easily see that this is a^x times a given constant (c_a). Different a’s yield different constants.
Using the right questions, we come to the point where pupils ask if there’s an a which constant is 1. Meaning de derivative of the function is the function itself.
We first approximate it (trial and error). And then try to find a proper,formal definition (yielding the limit definition)

[…] Last time I visited Better Explanations, I got stuck there for hours. I resisted this time once I realized I was reading all the same articles for a second time. […]

I think the approach of starting with the rate of change works too. Pretty much any corner is good, though I’ve yet to see a nice, intuitive explanation starting from the natural log definition and working its way around to e – it just seems too indirect a starting point.

The key is seeing that exponential functions are linked because they change proportional to their current amount. e is like the unit circle where the radius is 1 – other functions are a scaled version of it.

Thanks for leading students through a path that helps build an understanding.

Great article! Sometimes I think I should find out all my math teachers from Senior School & Engineering School and make them read your articles. I wish you were one of my teachers! Though now you definitely are!

I really relish your posts and I think you are my first virtual math teacher! :)… Will definitely like to meet you sometime!

I like that explanation of e, but the notion of developing an intuitive idea behind mathematical concepts isn’t new (perhaps it’s because I have a different sense of intuition from most people).
The first thing that young children learn is counting from 1. They start at 1 instead of 0 because it’s considered more intuitive to think of something that’s there rather than something that isn’t. After that, basic arithmetic is taught by the notion of incrementing until students get a ‘feel’ for how much any given number of increments affects a particular number.
I could be wrong about that last part. I think it was that intuition that pushed me to studying math in college, and I definitely know people who never developed that intuition!
But, even when we studied concepts such as derivation- we didn’t just write down the limit definition. We made calculations of slope for very small steps.

I am a math major and this is right on the dot. There are so many facets of math that unless you do something like this to connect them all, you forget most of it very quickly.

@Prateek: Thanks for the note! Heh, if your old teachers would enjoy it, feel free to send the articles along :). And sure, feel free to drop me an email if you’re ever in the Seattle area.

@Jehan: Nope, the idea of using an intuitive approach isn’t new, but I wanted to spread the word. You’re lucky that you were able to start with slopes and work to the limit definition – some people just have limits, epsilons and deltas just thrown at them without any context.

@Samson: Thanks! I agree, if you learn a subject as a set of disconnected facts it becomes easy to forget.

[…] Nice post Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is. […] not all starting points are equal. The right perspective makes math click — and the mathematical “cavemen” who first found an idea often had an enlightening viewpoint. Let’s learn how to build our intuition. […]

Actually this is not valid only in math, but in all other topics. As an example, we learn forecasting in our production planning course, if you study the history of invention of the theorem, you can understand topic easier.

@nanotürkiye: Thanks for the comment! Yes, I very much agree – nearly any subject can be understood at a more intuitive level by looking at its context.

determining whether a property of an object is “if and only if”, ie sufficient to define that object, is an important habit to build for a mathematician. however, all of these terms have “canonical” definitions. For example, the set of all points equally distant from a common center was what the term “circle” was invented to refer to.

“we know what a circle is, but how do we define it?” is actually kind of dishonest, viewed in this context. if you know what it is, that knowledge IS the definition.

Is there a “plain English” way to explain e^(i*pi) = -1? To me, this is the most mystifying formula. No amount of staring at De Moivre’s theorem, the series expansion, etc seems to offer any real clarity.

Its always good to see things being explained from a practical point of view; however, if you intend to study math, such a luxury is not always available and so it may not be a good habit to get into.

Instead I believe what is just as instructive as studying an intuitive approach–in terms of insight gained–is showing that a number of definitions are indeed equal by whatever tools are available to you–yes that means epsilon-delta proofs may be necessary. In fact, I would opt for rigorous arguments over intuitive ones as often intuition can be just as damaging as it can be helpful in mathematics.

Instead of teaching intuition I think its much more productive to teach the logic behind the argument.

@misanthropope: Interesting point, thanks for the comment. Yes, sometimes a given property can be described as the definition of an item. However, sometimes the reason for picking that particular property can be obscured. In the circle example, I imagine it’s inventor focused on the roundness/symmetry before noticing that all points were the same distance from the center. A better way to phrase “We know what a circle is, but how do we define it” may be “We understand the concept that a circle conveys, but how is it described in math?”

@anon: I think that De Moivre’s/Euler’s identity can be understood intuitively – I’m planning on getting that one eventually :).

@matt: Thanks for the note. I think there’s a balance between intuition and rigor. Unfortunately, I think math education has skewed too far on the rigor side (symbol manipulation) while losing the deeper meaning of the meaning of what the equations are trying to convey.

I think it is a cycle though – you use intuition to formulate ideas, rigor to refine and clarify them, intuition to formulate more detailed ideas, rigor to refine those, and so on.

Great article. In the chemistry course that I’m taking, for example, no one can understand what is being taught because the teacher rarely explains why certain things are true. The only reason why I’m doing well in the class is I take some time to understand what I’ve been told.

For example the order in which electron orbitals fill was simply given to us. I never memorized the order as if it was a new alphabet… I learned the reason why it follows a certain order and produced that sequence to see that indeed I was taught correctly.

Finally, I agree that it is intuition that produces one’s love for a subject. I don’t like mathematics simply because I solve a bunch of problems for homework. I enjoy the subject because I learn more of its secrets with each question that I answer.