The Road to Reality Study Notes: Difference between revisions
→Chapter 4 Magical Complex Numbers
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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 | 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 can be explained by switching to the complex number system using $$z=x+iy$$ whereby $$1/(1+z²)$$ can be examined to have singularities at $$x=i,-i$$. 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 $$f(z)$$ defining the circle radius. The series is convergent for any point z inside of this circle. | 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 can be explained by switching to the complex number system using $$z=x+iy$$ whereby $$1/(1+z²)$$ can be examined to have singularities at $$x=i,-i$$. | ||
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 $$f(z)$$ 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 $$c$$ 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. | ||