This one page explained ‘e’ better than my 4 years in engineering school 
Love this site, Kalid!
I’m in year 11 and the typical textbook explanations just don’t cover it in enough depth for me, I like to build a deep intuition of the concepts, which is precisely why I love your site!
e is a fascinating number…
I was a little stuck at your explanation on how to use ‘e’ with different rates, but now I think I get it.
Basically, you can imagine a year of 50% growth as the same thing as half a year, with 100% growth per year. This is because per half year, the growth is 50%.
Also an intuitive explanation for why the derivative of e^x is e^x, is that e^x represents 100% growth, continuously. This means that at each instant, the gradient will be equal to the amount that there is currently.
But of course, I can’t explain it as well as you, Kalid, so I’ll leave it to those who do it best!
I googled “what does e to the power of x mean” and went across 3 sites before encountering your explanation. I can easily say Google needs changes in their search engine, as this site should result right at the top. Great work!!
I’m 49 years old and never understood e until now.
Of course I could have paid attention in class way back when…Better late than never though.
Thanks !
Thanks Heidi, Jasper and Chandra!
Kevin, really glad you’re enjoying it! You got it, from an instantaneous and continuously compounding perspective, 1 year of 50% growth = half a year of 100% growth.
I think of it like this: An individual “cell” intends to grow from 1 to 1.5 during the period in question. That’s the battle plan. Of course, with the compounding, the colony grows more (children get children) but each cell is unaware of that and just grows according to its own goal.
You got it, having your gradient equal to yourself every instant is 100% growth (more on this here: http://betterexplained.com/articles/developing-your-intuition-for-math/).
Btw, everyone has a different and useful take on a concept, please don’t feel like you can’t get your explanations out there! I write what clicked for me, but another analogy might resonate better for someone.
Great Job kalid, Where I got my education, after 10th grade of school, biology and math are mostly mutually exclusive subject. I could not have both. Therefore, I never ventured ahead of 10th grade math. Since you made this mystery of e quite clear to me, means your article is indeed very intuitive. However, one doubt is yet to be cleared, which may be because of my poor math understanding. Rate of 100% simply means double, if x is growing at 100%/year then after a year I get 2x, intuitively. I dont know how this compound addition can break the common sense meaning behind it. To me rate is always a change in amplitude(or variable) per unit event (or time). If 1 is becoming 2.718, intuitively, rate is not simply 100 %. Since 2.718 is an cumulative outcome after many events, the real rate should be the one that is applicable for individual even, not for a bunch of events. I shared this link with my brother, who again had to go through the same education system (not math after 10th grade school) and he also had same counter-intuitive feeling. I am not sure, May be there is a need to explain a difference between rate and rate constant. Or else if you can please add a couple of more sentences to existing article so as to make it more intuitive.
Thanks
Surinder
Hi Surinder, thanks for the note! Happy to clarify the doubt. Understanding the difference between a “nominal” rate (what is listed) and the “actual” rate (what you get) is the key to e.
Imagine a bank that promises 100% return at the end of the year (a magical bank, I know).
You deposit $1000 on Jan 1st – how much will you have on Dec 31st? $2000. After all, they promise 100% return.
Ok. But let’s say you’re not very patient, and want to peek at your account after 6 months (end of June). How much should you have?
You might say $1500 – after all, after halfway through you should have earned half the interest, right?
This sounds good, but there’s a problem. If the bank shows $1500 at the end of June, it means:
- From Jan 1st to June 30, you earned $500 on your $1000 initial balance
- From July 1st to Dec 31, you earned $500 on your $1500 initial balance
That isn’t good. Why should both periods return the same interest when you started with different amounts?
The only fair way to compute interest is to include all the effects of compounding.
Now, there is a concept of “doubling every year”. If you have an investment that goes $1000 (starting), $2000 (year 1), $4000 (year 2)… it is growing at 100% per year.
But, the continuous interest rate that causes this doubling is actually only 69.3%.
69.3% continuous growth = doubling at end of year (100% increase)
100% continuous growth = 2.718 at end of year (171.8% increase)
The terminology does get confusing though – we need to ask whether 100% means the final result “after all compounding effects are included” or whether 100% is the instantaneous rate of growth (and therefore becomes 2.718 after all compounding is included).
Try out the bank example and see what would make it “fair”. If you wanted to get 100% return at the end of the year, how much should each 6-month period pay? How about every 3 month period?
Thanks, I still can’t calculate e, but now I know how it is used with interest, which is a lot more than I figured out after reading Wikipedia.
I was playing around with (1 + 1/n)^n
and inputted large values of n in excel.
This may be an excel issue but when I input the following values for n
- 10^12 - result 2.718523496
- 10^13 - result 2.716110034 - why is it moving lower from 2.718?
- 10^14 - result 2.7161100
- 10^15 - result 3.0350352 - now it is greater than 2.718?
- 10^16 and greater- result 1.000? why 1.00?
I thought as it approaches infinity, the value would be 2.718.
Is this an excel computational bug issue?
Thank you. Finally I found a place that shows what is e. But can you give an example using the population growth dynamics in terms of survival rate?
First I want to thank you for this Better Explanation which surely put the physical concept of ‘e’ and ‘ln’ deeper into my brain.Now I have a query, can you please explain me the physical interpretation of (-e^x) and (-e^-x)? As you wonderfully explained it for e^x and e^-x.
For sure, I have to start maths afresh. I am doing an MSc. in signal processing and find this articles very useful. Do you have such notes in statistics and probability especially on stochastic processes, mutual information and Entropy and how they link up with “e”. I also think you are doing more that enough and I feel like donating to your site, so that you continue doing the great job. i will definitely be donating once I am through will my studies.
Actually dg – I tested this out on both Excel and Wolfram Alpha, and it is working fine. There is no issue with Excel – it is computing the values properly and it shows that the result starts to approach e.
Thank you very much. This post is very informative. I was looking for a simple explanation of e and I found it here.
In the above you compare 30% in one year vs 10 years of 3% growth by saying:
10 years of 3% growth means 30 changes of 1%. These changes happen over 10 years, so you are growing continuously at 3% per year.
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.
The same “30 changes of 1%” happen in each case. The faster your rate (30%) the less time you need to grow for the same effect (1 year).
Question:
How do you break it down to 30 changes of 1%? Isn’t there an infinite number of changes happening when considering continuous growth?
Unless you are saying that the number of continuous changes is 30?
Or am I missing something in your explanation?
I would think that the number of continuous changes over 10 year exceeds that over a 1 year period. I am having trouble understanding the merger of time and rate explanation - rate x time.
Wow. Just wow. Such beginner friendly explanations are hard to come by (espescially in maths 
). Got it on my first attempt.
Hi Kalid,
really great explanations. Wish all my math teachers were as ingenious and interesting as you.
That video was so perfect! Why weren’t you my math teacher? Many thanks!
Hi dg, that’s exactly it – Excel has computation limits on the precision. There are many floating-point issues which can happen, many scientists avoid using Excel for large computations.
Putting 1 + 1/(10^16) is likely to result in “1.0” (no remainder), so when you take it to the large power it just stays at 1. [This happens on Google’s Calculator too: https://www.google.com/?gws_rd=ssl#q=1+%2B+1%2F(10^16) ]
e was being discussed on the radio in the UK the other day (BBC Radio 4, In our time, 25 September), and they used the 100% interest example at the beginning of the show, but it was too quick for me and I spent the rest of the programme wishing I had grasped the vital explanation. Now I feel I have an understanding of e comparable to my understanding of pi. Thanks very much.