An Intuitive Guide To Exponential Functions & e

Thank you so much for posting this material! I’m getting back into math for my engineering degree and it’s hard to get momentum on the new concepts of calculus if you don’t have an intuitive understanding of the fundamental numbers and concepts you’re using.

I will definitely be back to learn more to get a better understanding of how these concepts all make sense. A+ on the explanation and great use of diagrams!

[…] An Intuitive Guide to Exponential Functions and e  I never really understood e when I was in college.  Time value of money made perfect sense.  I was happy to discount functions for any time period using any interest rate.  But, you add a letter into the equation, and I’m wasn’t sure what to do with myself.  This is by far the best explanation that I’ve found on the concept of continually compounding interest, and it (finally!) gives a clear explanation of the topic.  I’d recommend this reading for anyone that deals with finance and discounting. […]

Hi Kalid,

There is a minor typo I just wanted to let you know.

In the sentence “1 period of 30% growth means 30 changes of 1%, but happening in a single year. So you grow for 30% a year and stop.”

I think it should say “1 period of 30% growth means 1 change of 30%, and happening in a single year. So you grow for 30% a year and stop.”

-kumar

[…] Think of e as a universal component in all growth rates, just like pi is a universal factor in all […]

thanks for your time in explaining e! It’s very clear!

The best explanation of what e actually is that I’ve ever heard. Thanks!

@Anonymous: More than happy to help – very glad the insights are working.

For the sentence “You’d think you should have $100 (original) + $50 (interest for half year).” I was trying to show the problem with naive reasoning, which might be our guess for how interest should be divided up.

If I invest $100 and expect to double my money at the end of the year, shouldn’t I have $150 halfway through?

Nope! Because if you did, it means you earned $50 in the first 6 months (starting with $100), and then $50 in the last 6 months (even though you started with $150).

As you write, you need to work out the amount so (1 + r)^n holds true :). So in the case of interest being paid every 6 months, if you were to double you’d have

(1 + r)^2 = 2

which becomes r = 41%. That means at 6 months you should have $141, so after another 6 months you get

$141 + 41% ~ $200

Of course, we could say we get interest every 3 months, or 1 month, or every day, and we end up realizing we need to pay interest based on e :).

@raj: Hah, glad you enjoyed it.

@Joe: Appreciate the comment, thanks! I agree, a poor explanation can turn us off from a subject entirely. It’s really important to be honest with ourselves about what explanations are working (or not) and always try to improve them.

This is simply great! I have always wanted to understand “e” and its origins better and this was the first time I came across something like this. Now I can explain it to my son as well. Thanks for the knowledge share.

Thanks Ravi, glad you were able to share it with your son!

I like very much math. This explonation is realy simple and very nice to undestading. Thanks for sending to me thise e-mail Kazik

Excellent explanation. Remarkable how one can complete years of higher education without encountering the like. Thank you!

Superb work! I’m going to read the rest of your series now! I’ll certainly recommend this to anyone who’s confused by e! Have you thought abut writing a book?

[…] numbers always confused me. Like understanding e, most explanations fell into one of two […]

It would be so fantastic if you could cover another article on Exponential Decay. Such as the decay of amplitude of sound etc. Would go brilliantly along with your Sine Wave intuition.

This article is brilliant by the way. Helped me have a small mental breakthrough in regards to Audio Programming. Big Thanks

Hey Kalid, thanks for the kind reply. I’ve read this article innumerable times & I was exactly with the same problem as others’ have fallen when understanding what the hell “e” is. Wiki deteriorates the whole thing more. But your site is the first things that seems like a heavenly place for me who always seeks intuitive meaning of an equation.

I’ve grasped the intuition of “e”…I also have understood that what “e^rt” stands for…all your logical statements from the first to the last…I can feel it! Continuous Compounded growth! It’s happening naturally everyday. And “e” is the one hero to exactly express them. It’s significance is obviously clear. And your insight is one hell of a thing I admit.

But still… still except one thing about the intuitive process you have come to “e” through- I can’t get it, as you’ve said,

“You’d think you should have $100 (original) + $50 (interest for half year).”

You’d “think”? Why I’d think so? What’s the intuitive/mathematical reason for “thinking” so? Just for the sake of simple dividing policy? Interest for one is $100, hence for half year it’s $50?

At first glance, I also thought so but later I can’t take this so easily.

Is this should be illogical if I think that I should get not " $50" but " $x" and the value of the “x” should be such that this “$100 + $x” would yield net $200 after a year when compounded for next half a year! So, that the equation (1 + r)^n) still holds true?!! Have I clarified myself now?

Thanks wes!

Thanks Joshua! I’d like to do a follow-up to discuss more applications of the exponential function, how we model various decay rates/growth rates with it. For a long time I just used e as an abstract tool, but it’s become more real to me over the years. I almost see it like a little machine that can be dialed in to follow a desired path (“Need 50% decay? Ok, dial that in, and let’s chart out that path…”).

I still can’t grasp one thing.

  1. Why at 100% rate if 1 unit yields 1 unit [net 2 units] at the end of the year, then it should yield, x = .5 unit in six months? It should be the case of simple rate.

I’m asking this because you’ve said, instead of a year, if we count six months, 1 unit gives .5 unit- this is the core point because Your derivation of e starts from here as you’ve come to the decision that this .5 unit also yields some units for next six months. Then you divide the year into more & more sections and using same principle have come to e.

But when this is not the case of simple rate, how can you say it’s .5 after six months? As this is the case of compounding, Isn’t it more logical to think that after six months 1 unit would yield x units which together with 1 unit [net (1+x) units ] will yield 1 unit [net 1+ 1] units when compounded for next six months ?!

Hi Biplob, thanks for the comment. I’m not sure I understand the question exactly, but let me see if I can explain.

Simple interest is a type of artificial, man-made growth which looks something like this: You give me $100 on January 1st, and I promise 100% “simple interest” for the year. That means on Dec 31st I’ll give you $100 (original) + $100 (interest) = $200.

Ok. Now, what should you have in your account on June 30, halfway through the year? You’d think you should have $100 (original) + $50 (interest for half year). And then in the next six months, you earn the remaining $50 of interest.

This sounds nice – except it’s not fair! From Jan 1 - June 30, you had $100 which earned $50 in interest. From July 1 to Dec 31, you had $150 which earned $50 in interest. How could the larger amount earn the same interest as the smaller?

Well, banks love doing this (with simple interest they can pay you less), but nature doesn’t. Generally speaking, nature doesn’t have a “gap” between when interest is earned and when it’s used. Once the growth happens, its impact is accounted for immediately.

The goal was to show the difference between artificial, “staircase-like” simple interest (represented by (1 + r)^n) and the smooth accumulation of interest represented by e^rt.

you are earth, i mean i can dig you(blog) so i can get more resources.

very intuitive.