Conquest of the Plane

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Reading Notes for COTP 2011+, also at Zenodo 292257   

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Cover text

CONQUEST OF THE PLANE provides:

1. an integrated course for geometry and analysis
2. a didactic build-up that avoids traditional clutter
3. use of only the essentials for good understanding
4. proper place for vectors, complex numbers, linear algebra and trigonometry
5. an original and elegant development of trigonometry
6. an original and elegant foundation for calculus
7. examples from physics, economics and statistics
8. integration within the dynamic environment of Mathematica
9. considered didactic explanation in a 'meta' section.

For the professions that apply mathematics like physics, engineering, biology, economics and evidence based medicine, the book provides documentation to judge on the proposal to let the national parliaments look into mathematics education, as explained in the book Elegance with Substance by the same author.

Calculus can be developed with algebra and without the use of limits and infinitesimals. Define y / x as the outcome of division and y // x as the procedure of division. Using y // x with x possibly becoming zero will not be paradoxical when the paradoxical part has first been eliminated by algebraic simplification. The Weierstrasz epsilon and delta and its Cauchy shorthand with limits are paradoxical since those exclude the zero values that are precisely the values of interest at the point where the limit is taken. Much of calculus might well do without the limit idea and it could be advantageous to see calculus as part of algebra rather than a separate subject. This is not just a didactic observation but an essential refoundation of calculus.

Didactic issues in trigonometry concern the opaque names of sine and cosine and the cluttering of questions with  or 360 degrees whereas a simple 1 suffices. Basically the plane itself gives that unit of 1, and angles are mere sections. The solution is to use the unit turn or unit of measurement (meter) around (UMA) as the yardstick for angles. This gives the Xur and Yur functions, defined on the circle with unit circumference and generating the {X, Y} co-ordinates on the circle with unit radius. Finally students will understand what Cos and Sin are. The common term 'dimensionless' appears to confuse 'no unit of measurement specified' (with a metric, in planimetry and trigonometry) with 'no dimension' (a pure number, in number theory). The relevant mathematical constant is = 2(capital theta, reminiscent of a circle) rather than  and it comes into use much less when we use UMAs instead of radians. The sine and cosine remain relevant the derivative, and that can also be shown in an elegant manner.

You can benefit from this book also when you don't have the software. However, with the software, you will have an interactive environment in which you can test the propositions in this book and your own deductions. The software is included in The Economics Pack - by the same author - which is an application of Mathematica, a system for doing mathematics with the computer. The Pack has users in many countries in the world. The Pack is available for Windows XP, Macintosh and Unix platforms and requires Mathematica 8.0.1 or later. It can be freely downloaded, but you need a licence to run it.