In the world of electronics, i is taken - it is used for current. So electronics engineers use j. I like j. The descender makes it very conspicuous in a formula.

I have shown elsewhere on the Web that "Nature Will Not Imagine". Imaginary numbers were dreamt up by Rafael Bombelli, whose greatest supporter was Girolamo Cardano. They did not gain acceptance until Leonhard Euler produced his famous equation that links the antilogarithm of an imaginary number with the cosine and sine of the angle.

I showed that the Euler equation is a spiral in complex space:

There is another use for the term

**"Euler Spiral"**, so I had to specify that this is my Euler Spiral. On the back-plane, there is a red circle of radius 1. e

^{iX} causes the plotting-point to rotate in the i-Y complex plane whilst X moves smoothly along the real (X) axis. The projection of a smooth rotation onto a smooth linear translation produces a spiral.

This is useful for

**repetitive cyclic** functions, like a train of sine-waves in electronics. Here we use

**e**^{jwt}, where e is 2.718281828459045 approximately - the natural exponential constant, j is the imaginary prefix in electronics, w is omega - the angular frequency - and t is time. With repetitive functions, one cycle (for example) is much like another, so it does not matter whether complex numbers are natural or man-made. The important point is that they are useful.

The shadow on the real XY wall is a cosine (green) The shadow on the i-X complex plane is a sine. The orange dot on each shows that a sine and a cosine differ only in

**phase** - the value taken at zero degrees. Otherwise the shapes are the same.

So the electronics version of the Euler equation is

**e**^{jwt} = Cos(wt) + jSin(wt)

De Moivre showed that if this is so, when one multiplies two complex numbers together one must add the azimuth angles and multiply the distances from the origin (the ranges) by each other:

So one can consider the position on the plane not just in terms of the des Cartes (Descartes)

**Cartesian co-ordinates**, but also in terms of

**polar** co-ordinates, which up to now are easier to understand. For example, with Julia and Mandelbrot sets one simply doubles the angle and squares the range - because it is a complex squaring. Here we double the angle:

and here we square the range:

Whilst studying these things, and possibly using such graphical methods rather than arithmetic (mathematicians are always seeking the easiest way), Professor Gaston Julia added a tumble, along a fixed vector:

He then reiterated the process, and counted the number of times it was possible:

This tumble turned the crude de Moivre image:

into the

**first** fractals like this:

Without research into complex numbers, such things would never have arrived.

Charles