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 ei, 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 e2 or e5 may make sense, what is to be made of ei?
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 much-feared oral exam conducted by a team of government-appointed "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 "eiπ +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 eφ 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 eiφ = cos(φ) + i*sin(φ). The special case φ = π gives Euler's identity in the form eiπ = -1. See also this Wikipedia article.
Euler's formula can be understood intuitively if we interpret complex numbers as points in a two-dimensional plane, with real numbers along the x-axis and "imaginary numbers" (multiples of i) along the y-axis. 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 eiφ can be interpreted as points on the unit circle (radius = 1) at an angle of φ above the x-axis. When φ = π radians = 180 degrees, the circle intersects the negative x-axis at x = -1, y = 0, consistent with eiπ = -1.
