The rich definition of e^x allows us not only to raise e to conventional powers to calculate its square, cube and so forth, but to do something unthinkable with ordinary exponents. e can be raised to an imaginary power.

Flashback: in my very first chemistry class I stood arm's length away from a beaker of mysterious aquamarine fluid into which a few colorless drops were slowly added. At first nothing happened; then suddenly the entire volume changed from blue to bright yellow, and gold flakes precipitated from nowhere onto the glass.

The next mathematical step is just as startling and dramatic. We combine the two most baroque values we've invented, e and i, to see what happens. The result is one of the most beautiful precipitates in mathematics.

Into the formula for e^x, substitute x = i:

The powers of i are familiar: the even ones are real (1 and − 1), the odd ones imaginary (i and −i). This causes the signs to alternate:

And by factoring out i it splits eⁱ into real and imaginary parts:

Each sum converges to a different value, and the result is the complex number:

And these values should be familiar, if entirely unexpected. They are, exactly, the sine and cosine of 1. The inescapable conclusion is:

A real number e, raised to an imaginary power, has created trigonometry.

The immediate implication of this result is that all the following can now be understood as identical to one another: