The e-book contains over four hundred pages and some thirty thousand words. There are also over five hundred line and surface graphs plus nearly ninety short videos, ten to twenty seconds in length, animating several of the functions for a total of almost six hundred illustrations. The videos are uploaded to *’youtube’* and will play automatically when cliicked.

Here are some of the many highlights:

### A) Euler’s Famous Formula Upgraded

Euler’s formula is, of course, stated in base exponential form as follow:

In section 6.0 his formula is upgraded for all bases, positive, negative, real, imaginary and complex.

### B) Complex Slope and Trajectories in Space

Consider an aircraft. In section 5.0 we will see that real slope is the aircraft’s climb or descent (violet line.) The black line is both zero real slope and zero imaginary slope. Imaginary slope is it’s heading (red line.) Complex slope is the sum of the two (blue line.) The algebra is a simple extension of the usual ‘slope intercept’ equation of a line.

Animation 28 ’Animate Complex Slope’

### C) Complex Branches to Polynomial Curves (Polynomial Bifurcation)

Using the cubic equation, the turning points of a typical cubic curve will show complex branches to the curve as *bifurcations* with an abrupt transition to the orthogonal plane.

Similar multi-branched graphs occur with a 4^{th} degree polynomial using the quartic equation and a 2^{nd} degree polynomial using the quadratic equation.

### D) Mordell and Elliptic Bifurcation

The interesting gaps of Mordell and elliptic curves will connect to each other at ‘bifurcation points’ where the curve exists on an orthogonal plane.