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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.  
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 ''[https://en.wikipedia.org/wiki/Algebraically_closed_field algebraic closure]'' and follows from the [https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra 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  
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 [https://en.wikipedia.org/wiki/Radius_of_convergence power series]; for example, the power series  
$$1-x^2+x^4+\cdots$$
$$1-x^2+x^4+\cdots$$
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 $$x=i,-i$$.  
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 [https://en.wikipedia.org/wiki/Mandelbrot_set 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 ==
== Chapter 5 ==
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