Understanding $e$ with regard to growth and, in particular, the meaning of $e^i$ I understand that "$e$" refers to  Euler constant with a value $\approx 2.7182$.
a) However, I am having hard time understanding why we call it "continuous growth"? Shouldn't we call it "maximum growth", as I have seen the $\$1$ compounding example everywhere to explain it?
b) Is it right to say that any growth in real world can be expressed as $e$? Example if something is growing doubly, OR triply, then is that equivalent to $e^2$ or $e^3$, respectively? If yes, then how?
c) What is the intuition behind $e^i$ (ie, "$e$ raised to the power $i$"), where $i=\sqrt{-1}$ is the imaginary constant? What does it represent if I have to explain to a person who knows basic maths like plus, minus, multiply and divide? As $e$ represent "growth" and $i$ represents imaginary unit. So what does $e^i$ represent in layman terms? Does it have a geometrical representation?
Please clarify.
 A: A few things to unpack here.
First of all, even though the number $e$ is a constant, we dont call it "Euler's Constant".  We call it "Euler's Number".  The reason is because the term "Euler's Constant" is reserved for yet another constant, also known as the Euler-Mascheroni constant, denoted $\gamma$.
Secondly, we mathematicians dont call it anything to do with continuous growth.  An economist or accountant might, but he wouldnt be speaking as a mathematician.  Growth is an application, not something innate to the mathematics. $e$ has many properties and growth modelling is just one practical application.
In the case of the $1 bank account example, compounding is occurring continuously, so the account is continuously growing. New growth is based on current value, and that growth becomes a part of the value instantly. Every instant matters when compounding continuously.  An account that is not being compounded continuously is still continuously growing. That is to say, its always accruing interest. Its just that this interest doesnt go on to grow, itself, until the compounding period is over and the next one begins, which is when that growth becomes a part of the principle.  So "continuous growth" is a bit of a misnomer.  Continuous compounding is more appropriate. Every account continuously grows, in some sense, but only a continuously compounding account has a continuously growing principle.
And in fact, compounding and growth are very different.  You cant compound more than continuously; that is obviously the maximum. But that is not maximum growth.  The interest rate is still a variable in the equation; it isnt just the rate of compounding that matters.  An account with a 3% APR will grow faster than one with 1% APR... this is obviously true for normal compounding but its true for continuous compounding as well.
The growth equation should be expressed as $e^{rt}$, not merely as $e$.  There is no variable in just $e$. For continuous compounding, that is, where $r$ is the interest rate per unit of time, and $t$ is the time.
This kind of growth is quite artificial, despite the terms we use.  You see it in the continuous compounding interest models, but in reality no bank would offer such a thing, by pure fiat. In economics, there is too much fluctuation in every which way for it to ever be truly exponential. And in the natural world we dont see it either; logistic curves appear a lot like exponential growth when its small, but they are counterbalanced and will taper off at some point when it starts getting large.
And yes, if you want a continuously compounding account to double every year, you could just write $2^{t}$.  It could be expressed in terms of $e$ with the following expression: $e^{\ln(2) t}$, which indicates to you that the APR of the account is $\ln(2)\approx 69\%$. So indeed, an account with 69% interest will actually double (give back 100%) after one year, if its compounded continuously.
There is no suitable, intuitive explanation for $e^i$. There is no geometric representation, at least not one suitable to a layman.  This is the algebra of the complex numbers, and it isnt trivial to prove, either.  Complex graphs exist in four dimensions and is impossible to visualize for even the expert mathematician.  It is possible to take four dimensions and project them into 2 or 3, and its also possible to take a 3 dimensional slice of a 4 dimensional space. Doing so looses information and intuition. And additionally it makes little sense to do this with complex numbers because $\mathbb{C}^2$ and $\mathbb{R}^4$ are very different spaces with different properties.
And since Im here, I should warn you about people on this site who would rather spend their time mocking you for your question (for having the nerve to not already know), demanding that you show your work (when its clear to anyone that if your work bore fruit you wouldnt be here asking the question), accusing you of cheating (because all questions are homework questions, thats all math is), insist your question has already been asked (whether or not its been answered or is subtly different), belittle, otherwise imply that your question is beneath them, etc.  Even as they rush to point game off easier questions.
In the rare case that someone does give an answer its usually deliberately made complex and convoluted, using mathematics well beyond what is necessary, with no regard for the qualifications of the person asking... and more so that the person answering can tout their intellectualism and stroke their egos and brag about their mathematical prowess, rather than actually help you to learn the subject matter.  Then you have to also contend with the people who think embedding a link and referring you to other similar questions or resources constitutes an answer.  Its all very frustratingly futile to ask questions here.
My university professors warned the student body against ever coming to this site not because they feared us cheating, but because they feared we'd be turned off from mathematics because of the pretentious and bitter attitude of people who would call themselves mathematicians and would-be educators. The universities of this country have an increasingly negative view of the culture on this site and for good reason.  Ive called it out many times because I want to see it change for the better but I inevitably get banned for it instead. This very answer, despite it answering your question, could be censored any day, because no one here wants to hear the real criticisms.  They might call it a "rant" or "unfriendly" or "off topic" in order to help them feel good about censoring it.  I dont mind though; Im not here for the points, Im here because I believe in truth and truth should always be spoken. Thats what makes me a good mathematician and a good educator.
David G. Sarcasm in the comments could have given this answer to you, but that would have been too much work and too much writing and he wouldnt be able to whank to the malice of his own non-contribution afterwards.  You may just get better and more informed answers by going to layman sites where unqualified respondents actually want to learn and teach, like yahoo answers or quora.  This site is poison and I very infrequently pop in.
