Hi Denis, that’s right – banks usually express their interest rates in terms of simple interest over the year.
It seems trite to say, after so many comments of the same; however, I have to echo: this is explanation was fantastic!!! When I read about mathematic principles that are so clearly laid out, it reminds me of how much math is such a direct in-road to philosophy. Thanks for helping reveal the mystery. I’m looking forward to you writing more of the same. Very best regards, Mark
Hi Mark, thanks for the wonderful comment! It’s great to hear that explanations are working, it helps me tailor upcoming articles to do more of the same.
Yes, I think any math principle can be explained simply – it’s us humans who tend to complicate things :). The irony is that it takes a lot of thinking and effort to reveal the “mystery” which is simple after all. I’ll do my best to keep cranking out articles, thanks for dropping by.
I have one for you, How can you explain Euler’s formula re^j(theta) in a understandable way.
Read r multiplied by e to the power of imaginary unit j x angle theta
Euler’s formula is on the list – but first I need to do an explanations of complex numbers, possibly sine and cosine as well 
Very nice explanation indeed!
Thanks Garry!
Hi Kalid,
As others have already said, that’s a really nice explanation of the constant e.
I think you can explain Euler’s identity eiπ + 1 = 0 using not much more machinery than you’ve developed for the real-valued case. In fact, knowing the limit as n tends to infinity of (1 + x/n)n is ex is almost enough.
First, interpret the (1 + ix/n) term geometrically as a complex number with real part 1 and imaginary part x/n. For large n this is a number very close to 1 but rotated slightly towards towards the positive imaginary axis. The angle of that rotation will be approximately x/n.
Multiplying two complex numbers results in a new number with the sum of their angles and product of their moduli. So taking the nth power of (1 + ix/n) will result in a new complex number with angle approximately x for large n since the nth power is just n rotations by x/n.
Substituting x = π (a rotation of π radians) gets you to -1 and so the identity holds.
I realise that this is not really a proof, more a sketch really. Also, it takes advantage of a lot of facts about the polar representation of complex numbers which in turn depend on e. That said, in terms of giving me an intuition for Euler’s identity this is the most satisfying explanation I’ve heard so far.
Hmmm… it seems the sup HTML tags didn’t work in that last comment of mine. Also, “π” (pi) looks a lot like “n” (en) which is unfortunate.
So my second paragraph should read (in LaTeX) “… Euler’s identity $e^{i\pi} + 1 = 0 …” and “… of $(1 + x/n)^n$ is $e^x$ …”.
Hope that’s not too confusing.
Hi Mark, thanks for the wonderful comment! I really, really like that insight, I’ll have to use it when explaining Euler’s formula.
I’d first like to tackle complex numbers to give people (including myself) an intuition for it. Also, I’d like to cover the series expansion of e^x as well [1 + x + x^2/2 + …]. One of the great things about e is that it turns a “mind-bending” operation like imaginary exponent into a “understandable” operation like a series of multiplications/rotations.
Again, thanks for stopping by, that comment helped me a lot.
Very nice article. Explains the mystery of e
Thanks Siva :).
This is really beautiful. Congrats on an elegant explanation. There’s an alternative formula for e that might tickle you: e = lim as [delta x] -> 0 of (1 + [delta x])^(1/[delta x])
The simplest explanation I ever heard for the importance of exponential grwoth in phenomena like population growth is that with an exponetial function, the growth rate is proportional to the size. I.e., the bigger the population, the faster the growth. It works in reverse, viz., radioactive decay: the less radioisotpe remains, the slower the decay.
awesome i never thought of e as a base like that. When you compared it to the radius of a unit circle it just clicked. No one ever (or has yet to) said that to me in calc or any other of the engineering courses i’ve taken.
You didn’t understand what makes e so important.
It’s because it’s the ‘unity’ of diferentiation/integration
d(e^x)/dx = e^x
Any positive number different than 1 could be the e as you have been using in this article…
@mclaren: Thanks, that’s another way to think about it [It’s the same limit but restated]. Yes, the application of “growing as much as you currently have” is part of what makes e great.
@Anon1: Thanks, I’m glad to see it helped! Everyone has a different “a ha!” moment.
@Anon2: Being reversible is one great property of e^x and will be covered eventually, but I find it isn’t what helps people “get it” intuitively. A lot of the time you see e in formulas related to growth rates (heat transfer, radioactive decay), and you want to know why it’s there.
@Anon3: e is the unique number you get when you compound 100% (unit growth) continuously for a unit time period. Yes, any number can be the base of an exponent, but only 2.71828 (e) emerges from the unit rate and time period.
Hi Kalid,
I can follow your explanation of e but I can’t relate (1.33)^3 = e^x = 2.37037. If I take the natural log of 2.37037 it equals 0.863046. In other words for this particular case how do I determine what rate*time is?
Thanks, Richard
i feel smarter