Difference between revisions of "The Road to Reality Study Notes"

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== Chapter 4 ==
== Chapter 4 ==


Penrose introduced the complex numbers, extending addition, subtraction, multiplication, and division of the reals to the system obtained by adjoining $i$, the square root of -1. Polynomial equations can be solved by complex numbers, this property is called ''algebraic closure'' and follows from the Fundamental Theorem of Algebra. Complex numbers can be visualized graphically as a plane, where the horizontal coordinate gives the real coordinate of the number and the vertical coordinate gives the imaginary part. This helps understand the behavior of power series; for example, the power series $1-x^2+x^4+\cdots$ converges to the function $1/(1+x²)$ only when $|x|<1$, despite the fact that the function doesn't seem to have "singular" behavior anywhere on the real line. This is explained by singularities at $x=i,-i$.  
Penrose introduced the complex numbers, extending addition, subtraction, multiplication, and division of the reals to the system obtained by adjoining i, the square root of -1. Polynomial equations can be solved by complex numbers, this property is called ''algebraic closure'' and follows from the Fundamental Theorem of Algebra. Complex numbers can be visualized graphically as a plane, where the horizontal coordinate gives the real coordinate of the number and the vertical coordinate gives the imaginary part. This helps understand the behavior of power series; for example, the power series  
$$1-x^2+x^4+\cdots$$
converges to the function $1/(1+x²)$ only when $|x|<1$, despite the fact that the function doesn't seem to have "singular" behavior anywhere on the real line. This is explained by singularities at $x=i,-i$.  


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 $z^2+c$, 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 $z^2+c$, starting with $z=0$, do not escape to infinity.

Revision as of 05:55, 16 March 2020

Each week The Road to Reality Book Club tackles a chapter of Sir Roger Penrose's Epic Tome. We use these meetings as an opportunity to write down the major points to be taken from our reading. Here we attempt to sum up what we believe Penrose was trying to convey and why. The hope is that these community-generated reading notes will benefit people in the future as they go on the same journey.


Chapter 1

  • Add a summary here

Chapter 2

  • summary

Chapter 3

  • and so on

Chapter 4

Penrose introduced the complex numbers, extending addition, subtraction, multiplication, and division of the reals to the system obtained by adjoining i, the square root of -1. Polynomial equations can be solved by complex numbers, this property is called algebraic closure and follows from the Fundamental Theorem of Algebra. Complex numbers can be visualized graphically as a plane, where the horizontal coordinate gives the real coordinate of the number and the vertical coordinate gives the imaginary part. This helps understand the behavior of power series; for example, the power series $$1-x^2+x^4+\cdots$$ converges to the function $1/(1+x²)$ only when $|x|<1$, despite the fact that the function doesn't seem to have "singular" behavior anywhere on the real line. This is explained by singularities at $x=i,-i$.

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 $z^2+c$, 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

What addition and multiplication look like geometrically on a complex plane.

  • 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

Relation between addition and multiplication when introducing exponents.

  • $$b^{m+n} = b^m \times b^n$$

5.3 Multiple valuedness, natural logarithms

Different values can arrive at the same value. Rotation brings you back to the same place repeatedly.

  • $$e^{i\theta}$$ is helpful notation for understanding rotating
  • $$e^{i\theta} = cos \theta + i sin \theta$$
  • (Worth looking into Taylor Series, which is related.)

Other Resources

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