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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. | ||