angle.md

What is the external angle ?

External angle:

• is a real number -description by Robert P Munafo
• it is the angle of external ray which
  • goes through exterior point c of the Mandelbrot set
  • lands on the boundary point c of the Mandelbrot set
• it is the angle of point from the exterior of Mandelbrot set. It is not arg(c) but angle of the Boettcher coordinate of point = $arg_M(c)$
• angular part of complex potential

Names of the external angle

• external argument
• phase

How external angle is measured ?

Direction

The angle is measured in the counterclockwise or anticlockwise direction

Units and domain

• when measured in turns it is a real number between 0.0 and 1.0
• when measured in radians it is a real number between 0.0 and $2\Pi$
• when measured in degrees it is a real number between 0.0 and 360

base and expansion of numerical value

• nine-ways-to-display-a-floating-point-number
• [What Every Programmer Should Know About Floating-Point Arithmetic ](http://floating-point-gui.de/ What Every Programmer Should Know About Floating-Point Arithmetic )
• Stackoverflow : Why Are Floating Point Numbers Inaccurate?
• HOW TO WORK WITH ONE-DI MENSIONAL QUADRATIC MAPS G. Pastor , M. Romera, G. Álvarez, and F. Montoya
• tut
• home school math : The fascinating irrational numbers

decimal number (base = 10 )

• real number
  • ratio = fraction ( Finite Continued fraction ) = rational number ( if number can not be represented as a ratio then it is irrational number )
    • in lowest terms ( irreducible form ) : $\tfrac{1}{21}$
    • reducible form
      • in explicit normalized form ( only when denominator is odd ): $\tfrac{3}{63} = \tfrac{3}{2^{6}-1}$
  • irrational number( infinite continued fraction )
  • floating point number $0.\overline{047619}$
  • finite expansion
  • endless expansion
    • continue infinitely without repeating (in which case the number is called irrational = non-repeating non-terminating decimal numbers)
    • Recurring or repeating : (strictly) periodic ( preperiod = 0 , preiod > 0 ) or mixed = eventually periodic ( preperiod > 0 , period > 0 )

binary number ( base = 2

• rational number ( ratio) $\tfrac{1}{10101}$
• real number
  • floating point number ( scientific notation )
    • Raw binary ( raw IEEE format )
    • fixed point number ( notation)
    • with repeating sequences : $0.\overline{000011}$
    • with endless expansion $0.000011000011000011000011...$

technical notes

GitLab uses:

• the Redcarpet Ruby library for Markdown processing
• KaTeX to render math written with the LaTeX syntax, but only subset. Here is used version

git ( gitlab)

cd existing_folder
git init
git remote add origin git@gitlab.com:adammajewski/parameter_external_angle.git
git add .
git commit -m "Initial commit"
git push -u origin master