@skghosh: e^infinity is the same as any number to infinity (2^x), and will grow without bounds. e^-infinity will shrink to 0. -e is just -2.7818…
Thanks Simon, really appreciate it. I had that same euphoric feeling when it started to click as well.
where were you when i was struggling in math in high school…
not born yet…
shit !
no wonder i didn’t do so well…
you are so helping me make up for it now !!!
it was interesting.
Thank u for putting together such a thorough and clear explanation of e!!! I can’t believe teachers and textbooks just sorta throw it in without explaining it hardly at all, especially considering how important it is!!! Thank you! 
Misstatement (I think): “Of course, we can substitute 100% for any number (50%, 25%, 200%) and get the growth formula for that new rate.” I think you meant to say: Of course, we can substitute any number (50%, 25%, 200%) for 100% and get the growth formula for that new rate. Or did I miss something.
[…] one of my favorite numbers, can be defined like […]
You’re the man…now all that is left for me is figuring out why e^x is its own derivative
@hakizimana: We don’t really assign e a value, we discovered that all growth had a common internal rate, and called it “e” (which happens to be 2.718). This is like discovering that all light moves at the same speed, and calling that speed c (which happens to be 186,000 miles/sec).
Kalid: Thanks for the best explanation I`ve seen. I think I am starting to grasp some of it now.
How could I explain it to someone else by using the graphs of 2^x, e^x, 3^x and their derivatives, like this: http://geogebratube.org/student/m29170 ?
“- Hey, look here! The graph of the derivative of e^x is the same as that of e^x. If we imagine that the function is a plot of bacterial population, we see that the growth is proportional to current population-size.”
Would not 2^x and 3^x be that as well?
i don’t understand the reason why we always assign “e” with 2,718 yet you told us above that it’s not a number.
I found your website while searching for Fourier series tutorials.
I like your style and impressed by your selection of examples. I think example (especially from everyday life) is the best way to explain or create intuition in someone about the subject.
On a separate note, which statement is correct?
In article “An Intuitive Introduction to Limits" you declared “e” as your favourite number but in the article " An Intuitive Guide To Exponential Functions & e" you said you were always bothered by number “e”.
Regards,
Shahid Khilji
Glad you liked it! When I wrote the e article (2007), it had always bothered me, but I’ve since grown to enjoy it :).
Kalid,
This site has got to be one of the best of all time! How you have the patience and dedication astounds me. You even respond to all feedback!
@Doug: Thanks – glad you’re enjoying the diagrams, I find them sorely missing in most textbook explanations!
@Jim: Glad to hear it!
@Mike: Awesome, check out the rest of the articles when you have time. I have a book on Amazon (Math, Better Explained) but I guess I need to make the ads for it more prominent!
@David: Thanks, I really appreciate it. I just try to answer comments a little bit every day, and in batches :).
At least bank don’t continuously compound the loans we take FROM them either, right?
Khalid – this is truly amazing. First, your explanation of e is wonderful. Second, you take the time and energy to reply to everyone’s comments and follow-up. It is a testament to your dedication that more than 5 years after you wrote an article, it is still being commented on. Thank you!
@Niko: That’s right :). But, they’ll often advertise the better rate (i.e., the lower uncompounded APR when showing the interest you’ll have to pay, and the higher, compounded APY when you are earning interest).
@gautham: Whoops! I was probably a little sloppy with the 100% growth vs 200% of the previous value. I meant a doubling progression, so you gain your entire value (100%) and end up at 200% (2.0) of your original amount.
@Anonymous: Glad you liked it! Check out http://betterexplained.com/articles/developing-your-intuition-for-math/ for more about why d/dx e^x = e^x.
@Guarav: Thanks for the note! Really appreciate it :).