How to interpret e^i in growth term? We can interpret e as e^1 i.e. starting with 1 we keep compounding continously at 100% rate  i.e. limit [1 + (1/t)]^t where t -> infinity..for e^2 we are compounding continuosly at 200% rate
We can apply euler's form and get cos(1) + isin(1) = 0.54 + 0.84 i but how do we interpret it, apart from jumping from one-dimension number line (e^1, e^2 etc.) to two-dimension complex plane (e^i)?
 A: Here’s a method I used for trying to get this idea across in the last course I taught. I wasn’t too successful with the students, but maybe I’ll be more successful now.
You know the explanation of $e$ as $\lim_{n\to\infty}(1+\frac1n)^n$. Perhaps you have also seen this:
$$
e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n
$$
If you haven’t, you can verify it by taking logarithms and doing a little algebraic magic. $(\star)$
Thus, if you want to, you can substitute $i$ for $x$ in the formula and look at $e^{\,i}$ as describing what you get by compounding for a year at an annualized rate of $i$, whatever that could mean.
But look rather at the slightly more general $e^{\,it}$ for $t$ real. When  you consider the complex number $1+\frac{it}n$, it’s just a little off the unit circle, and up almost exactly $t/n$ radians. You’ve probably seen something like this when you saw the proof of the relation $\lim_{h\to0}\frac{\sin h}h=1$. But make sure you draw the picture of $1+\frac{it}n$ in the Wessel-Argand-Gauss plane. Then think of what happens when you raise this to $n$-th power. The power is (almost) $t$ radians from the positive $x$-axis now, because of what happens when you multiply two complex numbers. And how far from the origin? Just a little more than a distance of $1$. And the two errors go to zero as you take $n\to\infty$.
You’ve just seen an argument of why $e^{\,it}=\cos t+i\sin t$. Of course you’d need to keep good control of the error terms in the finite power, and make sure
that in the limit they really go to zero. But it’s a sound argument.
$(\star)$ If you want to see the “algebraic magic”, I’ll add it in an edit.
