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The Road to Reality Study Notes: Difference between revisions

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Finally, the Mandelbrot set is defined as the set of all points c in the complex plane so that repeated applications of the transformation mapping z to zĀ²+c, starting with z=0, do not escape to infinity.
Finally, the Mandelbrot set is defined as the set of all points c in the complex plane so that repeated applications of the transformation mapping z to zĀ²+c, starting with z=0, do not escape to infinity.
== Chapter 5 ==
This is a first pass of main topics in this chapter. This should be expanded.
=== 5.1 Geometry of complex algebra ===
* law of addition
* law of multiplication
* addition map
* multiplication map
** what does multiply by i do? rotate
=== 5.2 The idea of the complex logarithm ===
* Penrose shows similarities between addition and multiplication by talking about exponents
* $$b^{m+n} = b^m \times b^n$$
=== 5.3 Multiple valuedness, natural logarithms ===
* $$e^{i\theta}$$ is helpful notation for understanding rotating
* $$e^{i\theta} = cos \theta + i sin \theta$$
* (Worth looking into [https://en.wikipedia.org/wiki/Taylor_series Taylor Series], which is related.)


== Other Resources ==
== Other Resources ==