If you whirl a stone on a string, you can feel two opposing forces at work simultaneously that create the resulting circle. Inward pull and outward escape are exactly matched, changing in step with one another, but constantly opposed.

This sense appears in an exact way when the derivative of the sine and cosine functions are compared. It turns out that:

The two functions are precise duals of one another, up to a change of sign. It's a simple, beautiful relationship, each function changing at the rate specified by the other, mirroring the symmetry of the circle itself.

The trigonometric derivatives have another insight to yield. It is found by starting with the sine function, extracting its derivative, and repeating the process with each new result:

After exactly four repetitions, we're right back to sin x where we started — the same four-stage symmetry that emerged when i was raised to successively higher powers. It's another indication of the deep connection between the imaginary and trigonometric realms.

Negative and positive integers have the happy property of adding to give a single negative or positive result. The real and imaginary numbers are not so compatible. A complex number, whether written:

Or

remains an unresolved sum, expressed in two parts instead of a single unified value. Resolving this compound notation into a simpler one would be an elegant improvement — provided a way can be found to do so.

The dual behavior of the sine and cosine functions offers a starting point. It takes both functions to represent a complex number, just as their derivatives reflect one other. If we want to somehow combine these two functions into one and express a complex number more simply, we can ask what the derivative of that combined function would be. Intuitively at least, the answer is immediate: the function should be its own derivative.