Understanding $e$ and $e$ to the power of imaginary number How did the value of $e$ come from compound interest equation. What does the value of $e$ really mean...
Capacitors and inductors charge and discharge exponentially, radioactive elements decay exponentially and even bacterial growth follows exponential i.e., $(2.71)^x$ ,why can't it be $2^x$ or something.
Also $e^2$ means $e*e$ ,$e^3$ means $e*e*e$
But what exactly $e^{ix}$ mean...
I want to know how to visualise $e^{i \pi} =-1 $in graphs... I knw how to get the value of such type of equations but Im not able to understand what they actually mean....
Plz help me...
 A: There are many properties that make the exponential function special; one that I find particularly instructive is:

There is exactly one function $f$ such that $f'(x)=f(x)$ for all $x$ and $f(0)=1$. This function is the exponential function, and it turns out that there is a particular real number $e$ such that $f(a)=e^a$ whenever $a\in \mathbb Q$. Therefore it makes sense to use the notation $e^x$ for $f(x)$ for all $x$.

The property $f'(x)=f(x)$ is what makes this particular exponential function useful for describing exponential growth and decay, because it makes it easy to relate the instantaneous rate of change to the current size of the thing that is growing or decaying.
In the complex plane it so happens that $f(ix)$ will be a point $x$ radians counterclockwise along the unit circle. This is forced by the relation $f'(x)=f(x)$, though it doesn't have any particular intuitive relation to repeated multiplication. One just has to get used to the fact that the unique function that obeys the nice rules we know from the real exponential happens to behave that way for complex arguments.
A: $e^{i\pi}$ is a point in the complex plane. You get this point if you "walk" $\pi$ radiants on the unit circle. This point turns out to be $(-1,0)$ or expressed in complex numbers $-1+0\cdot i$.
A: Adding to Henning's answer.  
If the exponential function $\exp(x) := e^x$ 
satisfies $\exp'(x) = \exp(x)$, then by the chain rule it also satisfies $\frac{d}{dx}\big(\exp(ax)\big) = a\exp(ax)$.
Check that the complex-valued function
$$
\phi(x) := \cos(x)+i\sin(x)
$$
satisfies $\phi'(x) = i\phi(x)$.  And $\phi(0)=1$.  Therefore $\phi(x) = \exp(i x)$.  That is:
$$
e^{ix} = \cos x + i \sin x
$$
A: 
How did the value of e come from compound interest equation.

Compound interest is when previously earned interest earns interest.
$$S=P\left(1+\frac{r}{n} \right)^{nt}$$
where

S = value after t periods
P = principal amount (initial investment)
r = annual nominal interest rate (not reflecting the compounding)
n = number of times the interest is compounded per year
t = number of years the money is borrowed for

Look at $n$. If $n=4$ you only get money 4 times year from the compound.
Let's say $n=12$ so you'd get your compound interest every month. That kind of makes sense if you also get your loan on a monthly basis.
Could there be a sweeter sound than money coming in at the beginning of every month? Actually, yes. What if you could get the compound money on a daily basis? It wouldn't be much, but you'd get money every day.
What you say? You want it every hour? Minute? OK! Fine, you know what? You can have it as a constant incoming stream of money. Are you happy now?
What would that mean for $n$? The number increases from 4, to 12 (month), 356 (day), 8544 (hour), ...up to $\infty$. That's exactly the definition of $e$: let $n$ go to infinity.
$$e^x=\lim_{n\to\infty}\left(1+\frac{x}{n} \right)^{n}$$
Which is exactly why the formula for continuous interest looks the way it does (I tried to stay conistent with variable names).
$$S=P\underbrace{\left(1+\frac{r}{n} \right)^{nt}}_{e^{rt}}=Pe^{rt}$$

What does the value of e really mean

For the interest example, $e$ means to do it continuously. Such values are often not immediately sound. But other statistical values aren't either, like the average for example:
The birthrate in the US (2012) is 1.88 So eeeer every woman gets almost 2 children? Or is ever second child missing an arm? This number makes no sense when thinking of it as a value like an amount of children. That's because it isn't an amount. It's an average amount.
The same goes for $e$. Given the different properties of interests, it can be hard to compare them. With $e$, there's a common ground. Instead of asking how much money you get how often per year, the question now boils down to "How much money do you get at the moment?".

Capacitors and inductors charge and discharge exponentially, radioactive elements decay exponentially and even bacterial growth follows exponential i.e., $2.71^x$ ,why can't it be $2^x$ or something.

It actually can be. But it's hard to compare exponential growths that have a different base. It's good to have a common base.
Now why is that base $e$?
It has to do with the physics. A lot of the principles of our world can be described with differential equations.
Some such equations can look something like
$$0=a\dddot y + b\ddot y +...$$
and without going into any detail, the solution for $y$ can look like
$$y(t) = C_1e^{\lambda_1t} + C_2e^{\lambda_2t}+...$$
This is because the derivative of $e^x$ is again $e^x$.
The derivative plays an important role when dealing with differential equations, which is why $e^x$ shows up in the general solution.
A: There is actually no particular reason why exponential growth must be
expressed as a power of $e$.
In the example of radioactive decay, for example, we often speak of
the half-life of a radioactive isotope, which is the amount of time
that it takes for half of the atoms in a sample to decay.
Thus if the half-life of an isotope is $H$ years, the portion of the sample
that will remain after $t$ years is $2^{t/H}$ —a power of $2$,
not expressed as a power of $e$.
Likewise, in finance there are things such as the "rule of $70$"
for estimating the amount of time it will take for an amount of money
to double at compound interest. If the result of the rule of $70$ is
$D$ years, and you leave $P$ dollars in an account at the same rate
of compound interest for $t$ years, the rule of $70$ says
you have $P \times 2^{t/D}$ dollars in the account at the end.
Again, a power of $2$, no $e$ in sight.
Of course you can easily convert any exponential growth or decay law
into a power of $e$ law: just introduce a constant factor in the exponent,
or change the constant that is already there.
For example, $2^x = e^{x \ln 2}$.
The particularly nice thing about $e$ that makes people want to use powers
of $e$ rather than powers of $2$ or $10$ in so many places is not
its exponential growth for large exponents, but the particularly
nice way it behaves for very small exponents.
In many places in mathematics and science where the exponent of something
is a variable, that exponent is not necessarily a whole number.
So yes, $2^3 = 2 \times 2 \times 2$ and $e^3 = e \times e \times e$,
but $2^{3/2} = 2 \sqrt 2$ (why?), $2^{1.4} = 2 \sqrt[5]{2^2}$,
and $2^{\sqrt2}$ … that's not so easy to write in other terms,
but it's a number between $2 \sqrt[5]{2^2}$ and $2 \sqrt 2$,
and it is well-defined as the limit of $2^x$ for real numbers $x$
as $x$ approaches $\sqrt2$.
By a small exponent I mean something like $0.001$, or better still
$0.000001$, or even $10^{-17}$.
The exponential function of any positive real number $R$ (where $R$ could
be  $2$, $10$, or $289$, not just $e$) has the nice feature that when
$x$ is small, you can write
$$R^x \approx 1 + kx,$$
where $k$ is a constant that depends on $R$ but not on $x$,
and this approximation is very good if $x$ is small enough.
The special thing about $e$ is that for small $x$,
$$e^x \approx 1 + x,$$
that is, the constant $k$ is not needed. (Or you could say $k=1$.)
This special feature of $e$ is intimately related to all kinds of other
special properties it has, such as the fact that $\frac{d}{dx}e^x = e^x$,
or that $\int \frac1x \;dx = \log_e x$, or the way $e$
helps in financial calculations.
This nice property is also a reason why $e^{ix}$, where $x$ is a real number,
can be viewed as a rotation around the center of the complex plane.
For very small $x$, $ix$ is also small in the way that makes this a
good approximation:
$$e^{ix} = 1 + ix.$$
This is a number close to $1$, but a little bit "off to one side" 
from the real-number axis. And the thing is that if we take any
point on the unit circle in the complex plane—the numbers of the
form $\cos\theta + i \sin\theta$—and multiply by $1 + ix$,
for small $x$, the result is approximately
$\cos(\theta + x) + i \sin(\theta + x)$, and if we multiply by $e^{ix}$,
that is the exact result: 
$$e^{ix}(\cos\theta + i \sin\theta) = \cos(\theta + x) + i \sin(\theta + x).$$
That is, multiplication by $e^{ix}$ moves a point $x$ radians around
the unit circle.
The unit circle is $2\pi$ radians from $1$ all the way around and back to $1$
again, so $e^{i2\pi} = 1$ Halfway around the circle is $\pi$ radians,
and the point opposite $1$ on the circle is $-1$, so $e^{i\pi} = -1$.
That's it. Multiplication by $e^{ix}$ rotates a complex number $x$ radians
around the center of the plane, and the number $e^{ix}$ itself is the 
point on the complex plane that $1$ gets to if you turn it $x$ radians
around the origin.
