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

== 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+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.  

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.

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.