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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 == | ||