Editing The Road to Reality Study Notes (section)

== Chapter 4 Magical Complex Numbers ==

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 [https://en.wikipedia.org/wiki/Radius_of_convergence power series]; for example, the power series 
<math>1-x^2+x^4+\cdots</math>
converges to the function <math>1/(1+x²)</math> only when <math>|x|<1</math>, despite the fact that the function doesn't seem to have "singular" behavior anywhere on the real line. This can be explained by switching to the complex number system using <math>z=x+iy</math> whereby <math>1/(1+z²)</math> can be examined to have singularities at <math>x=i,-i</math>.  

With this, Penrose introduces us to the idea of the [https://mathworld.wolfram.com/RadiusofConvergence.html circle of convergence] as a circle in the complex plane centered at 0 with poles/singularities of <math>f(z)</math> defining the circle radius.  The series is convergent for any point z inside of this circle.

Finally, the [https://en.wikipedia.org/wiki/Mandelbrot_set Mandelbrot set] is defined as the set of all points <math>c</math> in the complex plane so that repeated applications of the transformation mapping <math>z</math> to <math>z^2+c</math>, starting with <math>z=0</math>, do not escape to infinity.