I can still remember the "shock and awe" I felt when my math
teacher in high school wrote this formula, known as Euler's identity,
on the black board. He had been leading up to it through a series of
lectures on Taylor expansion and the theory of complex
numbers. In retrospect it reminds me of Andrew
Wiles' series of lectures at Cambridge in 1993 which ended dramatically
when he wrote down Fermat's Last Theorem on the board and said: "I
think I'll stop there.". Of course my math teacher had not
himself developed a proof of Euler's identity, and Wiles' proof of Fermat's
Last Theorem was vastly more difficult (and turned out to be flawed
in its initial form). The common aspect was the incredibly compact form
of the theorems despite the underlying complexity.
Euler's identity has been voted "the greatest equation ever"
in a poll of physicists conducted in 2004. The late great Richard Feynman
called it "our jewel" and "the most remarkable
formula in mathematics".
So what exactly is so remarkable about it?
Well, first of all it binds together the five most important numbers
in mathematics: 0, 1, i, π and e
together with the operators + and =. But even more
baffling, three of those numbers: i (the square root of minus
one), π (the ratio between the circumference and the diameter
of a circle), and e (the base of natural logarithms, closely
related to exponential growth) at first glance seem to have nothing
in common.
Some mathematicians consider that Euler's identity follows directly
from the definition of the expression e^{i},
so that Euler's identity amounts to nothing more than a tautology. At
the other end of the scale, we may quote a 19th century Harvard professor,
who said: "It is absolutely paradoxical; we cannot
understand it, and we don't know what it means, but we have proved it,
and therefore we know it must be the truth."
Most laymen would probably seize on the presence of i, a number
that offends many people due to its apparent absurdity. "There
is no such number as the square root of minus one!" It does
not help that mathematicians call it an "imaginary" number.
And while an expression such as e^{2}
or e^{5} may make sense, what
is to be made of e^{i}?

During my last year in school,
prior to university studies, it was customary for students in
Sweden to carry a funny hat during the final weeks between written
exams and the muchfeared oral exam conducted by a team of governmentappointed
"censors" or reviewers. The hat was decorated with pompoms
and bowknots corresponding to the grades received in the written
exams, and in the corresponding colors. In addition, we could
freely write slogans or mottos on the hat.  The text I chose
was "e^{iπ} +1 = 0", of course! (Unfortunately,
it does not show in the photo; perhaps it had not been added yet.) 
They would have a point. The definition of what it means to raise
a number to the i:th power, where i is the square of
minus one, is key to making sense of the equation. But the
accepted definition was not completely arbitrary; it was not pulled
out of a hat! And there is an underlying unity that transcends
the interpretation of the number i.
The unifying theme is that the value of the derivative of
the function e^{φ} is identical to the value of
the function itself at all points, while the derivatives of the trigonometric
functions sin(φ) and cos(φ) are, respectively,
cos(φ) and sin(φ), so that again the derivatives have
the same values as the functions themselves except for a phase
displacement. And the trigonometric functions are closely associated
with the properties of circles, which leads directly to the number
π.
This relationship becomes much clearer if we inspect the Taylor
expansions of these functions. This was how Euler arrived at his
celebrated formula e^{iφ} = cos(φ)
+ i*sin(φ). The special case φ = π gives
Euler's identity in the form e^{iπ} = 1. See also this
reference.
Euler's formula can be understood intuitively if we
interpret complex numbers as points in a twodimensional plane, with
real numbers along the xaxis and "imaginary numbers" (multiples
of i) along the yaxis. Each complex number will then have
a "real" and an "imaginary" component. By applying
the normal rules of vector algebra we can add, subtract and multiply
complex numbers with each other. It soon becomes clear that multiplication
by i corresponds to a counterclockwise rotation of 90 degrees
with unchanged magnitude (interpreted as the distance from the origin).
Complex numbers of the form e^{iφ} can be interpreted
as points on the unit circle (radius = 1) at an angle of φ above
the xaxis. When φ = π radians = 180 degrees, the circle intersects
the negative xaxis at x = 1, y = 0, consistent with e^{iπ}
= 1.