The Road to Reality Study Notes: Difference between revisions
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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-x²+x<sup>4</sup>+… 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-x²+x<sup>4</sup>+… 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 z²+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 z²+c, starting with z=0, do not escape to infinity. | ||
Revision as of 09:45, 13 March 2020
Each week, as we go through each chapter in The Road to Reality Book Club, we could use that as an opportunity to write down what we think are the major points to be taken from what we've read. Let's sum up what we believe Penrose was trying to convey and why. These community-generated reading notes will very likely 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-x²+x4+… 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 z²+c, starting with z=0, do not escape to infinity.