An Intuitive Guide To Exponential Functions & e | Hacker News Reader

An Intuitive Guide To Exponential Functions & e

betterexplained.com

93 pointsvijaydev15 comments

Hi all, thanks for the comments (writing from a train using a kindle so forgive lack of links)

google "better explained math intuition" which is an article on developing the four common definitons of e. the problem with saying "e is the function which is its own derivative" is the same as saying "a circle is the set of points where x squared plus y squared equals radius squared". i asks too much of a beginner and imo is best shown after they have an idea of the basics.

ie, here is a round shape. we call it a circle. look at this neat property where every point is the same from the middle! lets write an equation for that...

with e. what is growth? it is like interest on your bank account. why do we wait till the end of the year? month? second? intant? how can we write this as an equation? (in algebra, in calculus)

the other reason to avoid calculus definitions is that 2% of students will bother asking for clarification/more details but most high schoolers can at least follow the algebra of compounded interest.i still don't get limits/infinitesimals to the level i like despite writing about them.

anyway thanks for the comments, i love seeing what explanations work for people!

I've liked the previous posts from betterexplained, but this one seems to be extremely unintuitive to me. I was teaching this recently, and the simpler idea that e is the function which is its own derivative is a much simpler way of expressing a meaningful truth about it, and following it through, leads you to exactly the same limit as a formal definition. It also tails very neatly into why exponentials are used in physics for decay curves. That said, obvious to me and one student is not necessarily obvious to everyone! Anyone else have an opinion on this?

Well, for me this post is the only one that made sence and thanks to it I now have a clear understanding what e is. "e is the function which is its own derivative" is way less intuitive.

Perhaps this would be more intuitive if shown graphically. Something like; draw a curve where the gradient is the value. As the value goes up, so does the gradient, so does the value and so on. What you have drawn, e^x is the function where the value is the gradient - exponential growth.

That sounds pretty damn intuitive. Just imagining the drawing you're describing has given me some insight.

Thanks for this - this is exactly how you would say it to someone who doesn't know what a derivative is. Apologies for assuming anyone interested in this would know calculus - poor form indeed! Explaining in terms of derivatives definitely isn't simple if you don't know what a derivative is.

Your explanation appears more elegant, but it isn't simpler. I as an algebra student can grasp nearly all of the lesson from betterexplained. I haven't yet learned derivatives, so your explanation doesn't much sense. I very much appreciate the lower knowledge level explanation, as all my teacher would say when we covered natural log was that e was a magic number used in growth equations.

Thanks for the replies. For those who like the betterexplained explanation:

1) Did you understand the bit with the limits, or did you find the thing made enough sense without that? I would have thought that would be harder than introductory calculus, but if not, I may start using it.

2) Do you think it made it easier to get to the point parallel described? I ask because regardless of other interpretations, my students need to understand what e means in calculus.

Thanks again!

1) I'm probably missing some truths by not knowing how limits work, but his use of limits didn't impede my understanding of the larger point. He quickly moves each time from an equation with limits to a more basic algebra expression.

2) I understand what parallel is saying after looking up "gradient"..

e is:

the unique number such that for all real x, e^x >= x^e

the x-coordinate of the maximum of the function x^(1/x); for all real x, x^(1/x) < e^(1/e)

the root of the hyperbolic logarithm

...hyperbolic logarithm? Well, 1/x -- the reciprocal function -- is a hyperbola, and the area under that function turns out to be:

http://en.wikipedia.org/wiki/Natural_logarithm#Definitions

Unfortunately, it's kind of hard to try to make sense of e without going into at least some basic calculus (which betterexplained does, in the form of a limit, but glosses over). It is commonly stated that e was discovered by Bernoulli, but the first references are from Napier, who presented a table of natural logarithms without any reference to e the number itself. I'm not a terribly huge fan of the two most common explanations of e:

e = lim (1+1/n)^n as n grows

e defined such that d/dx e^x = e^x

The first definition has no obvious connection with most of the interesting properties of e, so while it makes plenty of sense it doesn't actually clarify anything about exponential functions or natural logarithms. The second definition asks students to simply assume one of the most interesting properties of e, without any clarification as to the derivation of such a number. Worse yet, students are usually introduced to both long before they are capable of understanding the connection between these definitions (this requiring L'Hopital's rule to properly understand) which basically tells them that "e is some magic that you can't understand".

My favorite treatment of the idea is the one I had the privilege of learning as a student (perhaps I'm biased), which ignores e at first and defines the natural logarithm as shown above, as a definite integral resulting from the function 1/x, and then the fundamental property that d/dx e^x = e^x is proven from the previous definition, which requires only a little basic calculus. In fact, the text did not even say it was a logarithm, merely "Consider the function L(x) such that L(x) = the integral of dt/t from 1 to x", and then went on to prove that this satisfies several properties of a logarithm, that e is the unique number such that L(e) = 1, and finally derives the fundamental property of the exponential function (dy/dx = y) and so forth.

I think that the fear that surrounds calculus and the use of its methods does many students a disservice by rejecting a more complete explanation in favor of one that requires slightly fewer uses of the word "integral". Of course, I'm a cantankerous prick.

hi, thank you for the detailed comment! i have a writeup with more details on the common definitions (including the natural log one), if you google "better explained math intuition" (on a kindle currently). would love to know what you think. my ultimate goal for e elucidation is to get people to jump from any definiton to the others as they are all linked (i.e. see them as wys to rephrase "write down the equation for perfectly continuous growth").

Well, I have a clever way to explain the relationship between areas under a hyperbola and continuously compounded interest:

dy/dx = 1/x

dx/dy = x

Hopefully that isn't too confusing. If you start with the second relation (continuously compounded interest), a reciprocal gets you to the first (area under a hyperbola) -- the connection between exponentials/logarithms and areas under hyperbolae isn't immediately obvious to most people. Implicit differentiation is kind of scary, but "draw a curve whose slope is its height" requires an art degree.

I like your essay on intuition, but I don't know if your interpretation of the factorial series is really deep enough to explain what is going on. A fully elementary treatment of Taylor's series is a really hard thing to achieve. This, however, could help:

http://en.wikipedia.org/wiki/Binomial_theorem#Series_for_e

The binomial theorem for integer exponents is easy enough to grasp, though Wikipedia's geometrical explanation is kind of hilarious:

"[...] if one sets a = x and b = Δx, interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, [...]"

I'm afraid infintesimal changes in the volume of n-dimensional hypercubes probably aren't going to make exponential functions any easier to grasp for epsilons...

Thanks for taking a look! My intuition for the dy/dx relations are "amount of growth" and "time needed to grow to the next increment". So for me, e is the amount you have after growing continuously for exactly 1 unit of time (integrate 1/x until it is 1).

I'll have to examine that link while I have time to study (vacationing now) and agree the wiki explanation wouldn't be the most enlightening :).

e is like a speed limit (like c, the speed of light) saying how fast you can possibly grow using a continuous process.

Can someone explain what he means? Doesn't that assume dx/dt <= x?

The statement is a little misleading. He's not saying you can't have a function that grows faster than e^x as you certainly can, for example the the gamma function.

I think he's saying that for a process where the rate of growth depends on the current "population" such as interest in a bank or splitting bacteria then exponential growth is limit.

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