Now I have clue on e! Thanks! I want to be an actuary now!
Thanks Jim 
I just "e"njoyed my a-ha moment!
Awesome, very nice!
[…] e is a fundamental rate of growth, just like pi a fundamental ratio in every circle […]
WOW! Amazing how i was able to condense 16 years of learning into an hour long window! All those years spent into oblivion when teachers would come to class, scribble something in greek (literally)on the board, while students hoped helplessly that they’d someday understand all this! With your tutorial, the world has hope again! I wish teachers would focus on smaller syllabus, but more pertinent, practial, day-to-day stuff! Thank you! You are my official math messiah!
Hi Icegirl, thanks for the wonderful comment! I’m happy the article was able to help – yes, the goal is to focus on practical, “this makes sense” insights.
Kalid,
Thanks for the great explanation.
My comment is looking for a comment from you.
A new money system.
Every one gets 100 units a day for a year. At the end of the year there are 36,500 units of currency in circulation for each person. The 100 units have diminished as a proportion of the whole to the point of 1/365. On the first day of the second year the daily allotment is increased by 1/365. This keeps the proportion of the daily allotment constant at 1/365. Over year this approximates e, and if the growth is continuous, it is e. I call the type of money salmoney, and this particular version excalibrator.
I guess what I’m looking for is does this resonate with you as an apt or accurate use of e. Thanks for your time. My website is jaspersbox.com It is the attempt to create a new currency with a built in universal dividend.
Thanks for listening.
S. Clark
Hi Steven, thanks for writing, glad you liked the article. I’m not quite sure what you meant by “increasing by 1/365” – do you want to keep the proportion of money handed out the same? (Even in the first case, giving out 100 per day, the percent changes day after day).
You can use e to convert between simple and compound interest – I’m planning on a follow-up article on this topic, which may do a better job of answering your question than I’m doing now
This is a great explanation… It has been twenty years since I have studied this area of mathematics and then I just memorized it and went on about my business. I am now working on a wind turbine design and (to confess) having to look a few things that I have forgotten up. I came upon a speed gradient and surface roughness equation as applied to wind velocity that utilized the natural log of the height / surface roughness length. I did an internet search on the concept of the natural log to brush up on it and found this little jewel. This is one of the best explanations I’ve yet to come across.
Hi Todd, thanks for the wonderful comment! I’m really glad it was useful to you, and hope you figured out that turbine design
Hi Kalid, thanks a lot for the intuitive explanation. I’ve wondered a lot about “e”. I linked to this article from my lj above.
Keep your good work going!
Hi Deepak, thanks for the comment – glad you liked it!
Fantastic website. I love the explanation. Well definitely be back to read up some more.
Thanks Shaw, glad you like it!
Nice job, Kalid. Now I’m going to try to explain it to my wife using your approach.
Thanks Barry, hope she enjoys it
Hi Kalid, I have a question here. You have given an example to us as follows:
“I have $120 in an account bearing 5% interest. I keep it for 10 years. Assuming compound growth, I’d have 120 * e^(.05 * 10) = 197.85 after the 10 years.”
If I calculate it in the compound interest formula, the results are different:
$120 * (1 + 0.05)^10 = $195
Could you explain why? Thanks
Hi Denis, great question – clarifying the meaning between simple and continuous growth rates is something I’d like to write about more.
At a high level, “simple” interest is the final amount of growth, i.e. what you get at the end of the year. So 5% simple interest for 10 years would be (1.05)^10, as you wrote.
Continuous interest is the current rate of growth, i.e. how fast are we growing at this instant. In that case, you use e^(.05 * 10) to find the impact of 5% current growth for 10 time periods.
The reason continuous growth is more is because the “interest you earn, earns interest”. That is, after 1/2 a year you have earned 2.5%. After another half year, that 2.5% interest has earned 2.5% on itself. You can cut the time slice smaller and smaller (like we did for 1/3 of a year above) and see how small chunks of interest can earn more. With simple interest, you just say “I’ll give you 5% at the end of the year… and no compounding before then. If you’re lucky, I’ll spread out these payments each month.” Most banks list simple interest as their annual percentage rate, as it’s sometimes easier to think about.
However, continuous growth is more common in science and the real world (radioactive decay), as most things have a current growth rate. We don’t figure out the final amount and work backwards, as banks may. I’d like to write more about this, thanks for bringing it up.
Thank Kalid. So I guess it is more appropriate to use e for scientific calculation but not bank transaction calculation as bank’s compound growth rate is usually annual but not continuous.