Sorry Kalid, I have already made a previous comment (641), I guess what got me confused is that after three years we are still left with material, when it says it decays 100% in a year.
Hi Kalid, Thankyou for the explanation and examples of e.
I just have a minor question about example 3 about radioactive decaying.
I didn’t understand what was happening with the paragraphs discussing the 5kg/year after few months and 2kg/year etc? doesn’t the whole 10kg of material decay completely after 1 year. what happens in the remaining 2 years? nothings left?
Thanks
Kelly
This is beautiful! With nothing more complicated than basic algebra you’ve explained something very important. (and no “lim” and hardly any ^!)
If we have two different exponential annual growth rate trend from two subsequent time period. Does it make mathematically sense to establish a trend change from the difference of the two rates. Or would that be wrong?
trend 1: 3,72% year 2005-2013
trend 2: 1,41% year 1990-2005
Trend change from first period to second: 2,31% ???
Btw amazing explanation, just love your website.
Kalid you are great in explaining! Though I have read this article in translation it excited my imagination.
thanks a lot now I understand why e=2.718268
Dude…you rock…you make Math interesting again…Kudos!
Best… explanation… ever… You’re explanation is so detailed, that i always will remember what e is and what is does. Thank you so much!
Great explanation! thank you so much. I had spend hours trying to find what ‘e’ really meant and all I got were definitions with more concepts I didn’t understand . This was very helpful.
Excellent explanation on the concepts of e. I read the article with my son who is in 10th grade.
dude, your explanations are like my own voice in my head, it is like you hit the right spots, when I read ur blog and see your intuitive explanations it gets better and better the longer i read your article, in the beginning when I started reading your blog I thought, OK I might find some stuff that will make sense to me and then a lot of obtuse stuff like in the other blogs so I had this attitude and expected that at any moment I will run into something you say that doesnt make sense but the more I read the more I changed my attitude or mind about running into “jagged” parts of the article now my reading has that effect of the natural constant e to the power of rt nowmy understanding flows like e into the graph, no jagged parts (doubts)
@Fation: Thanks, really appreciate it. One of my goals is to write how I wish someone explained things to me, guiding me along but also letting me work things out myself. Thanks for the comment.
I understood e for the first time even though I have been using it for such a long time. I can’t thank you enough. It really feels good to actually comprehend the things you use as tools.
To reiterate others, great to get a nice intuitive explanation for e.
This isn’t necessary but it gives another perspective on how e = (1=1/n)^n. A nice visual way of looking at it that lends itself to your blue and green dots graphs.
Dividing the period into two and applying the growth as done above you got the values which I’ve grouped and converted to fractions:
1 + 2*1/2 + (1/2)^2 which of course equals (1+1/2)^2
into 3 we get:
1 + 3(1/3) + 3(1/3)^2 + (1/3)^3 = (1+1/3)^3
I wonder if it would be better to display the decimals above (in the blue and green dots graphs) as fractions instead to make a clearer link to the formula.
Cheers,
Mike.
Hi,
Great explanation - I thought your table of (1+1/n)^n -> 2.7… must be wrong because:
(1+1/n) tends to 1 as n gets larger and 1^n = 1 for any (positive) n.
So I thought (1+1/n)^n must tend to 1 not 2.7…
Tried it on Excel 2010 using power(number,power) and it went to 2.7… [until something like 14 zeroes when it had a heart attack and jumped up to 3. something and then down to 1 so couldn’t test any larger n’s].
I do see that it tends to 2.7… but it still ‘feels’ as if it should go to1!
Regards, Don.
thank you for your article, it gave me a great insight on e. i am a beginner at math. if we have e^-t, it indicates decay. but what i dont get if we have e^-t/RC. which is the RC circuit from what i read. but i dont get what e^-t/RC indicates. the full formula is V(t)=V0e^-t/RC
thank you and sorry for my bad english
@Bader: Great question! In any equation, you can see it as
e^{rate * time}
In the case of Voltage = e^{-t/RC} we have:
t = time
negative sign = decay
1/RC = “rate”
What does this mean? If we have a higher resistance, our rate of decay will be reduced (i.e., we keep the voltage for longer). If we higher capacitance, our rate of decay will be reduced (we will again keep the voltage for longer).
The meaning is that we can slow our decay in voltage based on both “R” and “C” (and it turns out their interaction is multiplied, so doubling R and doubling C results in slowing decay by 4x).
Hi Kalid,
Happy to have found the site. Just posted this question above at the 'Ask a Question". Got some better clarification on the value of “e”. Still unclear on one thing. Why must the value of “e” be found using a 100% rate of growth than some other rate (like 50% or 300% to note your other instances).? If we settled on 50%, then “e” would = 1.64. . . . but that is only 1/2 of “e”. 300% would have “e” equal 20.085 . . . . but this is “e” raised by a power of 3.
Is it that 100% represents the ideal maximal growth rate? Then 300%, like 50% would be unsatisfactory then too because it doesn’t embody this ideal condition? Still unclear. 100% still seems arbitrary to me rather than rational or necessary in the computation. I see the 100% allows for the fraction formula using 1 / n, as 1 is the decimal equivalent for 100%. But then this just repeats the question: why not 0.5 or 3 for that numerator. Any explanations graciously welcome.
@William: Great question. I’m pasting a reply I had on reddit about why e is special:
I see e^x as the growth equivalent of the number “1”. We can say every number is a scaled version of 1. Of course, we could also say that every number is a scaled version of 1/2, but it adds extra complexity. 1 embodies the spirit of “being a number”.
e^x is the growth pattern that makes the fewest assumptions: you are always growing by the amount that you have. There is no delay in your interest payments, you are not growing at a fraction of what you have: it’s 100% growth, compounded continuously.
2^x isn’t so clean. It means you have some growth rate, that at the end of the full interval, results in doubling. Quick, you have a bacteria colony: what growth rate should it apply every second, so it ends up doubling at the end of the year?
Ack… hrm… that’d be ln(2) or 69.3%. That is, you need to tell the colony “grow at the instantaneous rate of 69.3%, and by the end of the year, your total compound growth will match 2^x”.
In nerdier terms, while f(x) = a^x always has a derivative f’(x) = c * f(x), proportional to the original growth, choosing a = 2.71828… lets that proportionality constant be 1. That is, we are growing by a rate exactly equal to our current amount.
This perfect symmetry makes e special (growing by 50% or 300% isn’t symmetric, your rate is different than your current amount).
Man, you really nailed it. Before reading this article, understanding e had always been very elusive. Math couldn’t be more interesting!! Keep up the good work. Nairobi, Kenya