What a beautiful mathematical expression is Euler’s Formula:

where i is the square root of -1 (known as an imaginary number) and e is the natural exponential, 2.718281828… The formula above for x = 3.1415927… (pi), gives the amazing result: e^(i*pi) = -1 . Huh?

Geometric illustration of Euler's formula.

What do exponential functions and imaginary numbers have to do with trigonometric functions (sines and cosines) anyway? Exponential functions rise rapidly with increasing x, while trigonometric functions oscillate up and down with x. The short answer is that imaginary numbers are to blame. The long answer is that all functions can be expressed as a series of terms of increasing powers of x, using calculus.

I won’t bore you with the details, but Leonhard Euler did just this in 1748, and proved his formula using infinite series expansions and equating both sides. Why did he do this? Why not? Euler did not bother himself with the interpretation of his results.  As it turns out, many things are described by complex (real + imaginary) exponential as solutions to differential equations. In the theory of light, wave motion is described by the complex function e^(ik*x-wt).  There are more examples in nature for the use of imaginary numbers, but nobody really knows why.

Richard Feynman called Euler’s equation “the most remarkable formula in mathematics.” But what does it really mean? What is the fundamental nature of imaginary numbers that makes Euler’s formula so remarkable? I don’t have an answer  and perhaps somebody out there has some thoughts on this great mystery. As Shakespeare said, “There are more things in heaven and earth than are dreamed of in your philosophy.”

Benjamin Peirce (1809-1880, American mathematician, professor at Harvard) gave a lecture proving Euler’s equation, and may have said it best:

Gentlemen, that is surely true,
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.