complex_analysis

What it is

The purpose of this repository is identifying what a complex number is, comprising a Real (Re) and Imaginary (Im) part, where Im can contain: a) some number part, and b) some iota and an exponent part.

Euclidean Plane

Represented some ordered pair, what some may call a Tuple, of two numbers within the Sets of Real Numbers (R).

We can work with this plane like so:

Drawing

[1] pry(main)> ab = EuclideanPlane.new(1)
[2] pry(main)> ab.draw

 |
- -
 |

=> nil
[3] pry(main)> ab = EuclideanPlane.new(2)
[4] pry(main)> ab.draw

  |
  |
-- --
  |
  |

=> nil
[5] pry(main)> ab = EuclideanPlane.new(3)
[6] pry(main)> ab.draw

   |
   |
   |
--- ---
   |
   |
   |

=> nil

Placing a Point

Please note, some offsetting is required, for now.

[1] pry(main)> ab = EuclideanPlane.new(2)
[2] pry(main)> ab.place_point(-1, 2)

 x |   
   |   
   |   
--- ---
   |   
   |   
   |

=> nil
[3] pry(main)> ab = EuclideanPlane.new(1)
[4] pry(main)> ab.place_point(1, 1)

  | x
  |  
-- --
  |  
  |

=> nil
[5] pry(main)> ab.place_point(-1, -1)

  |  
  |  
-- --
 x|  
  |

=> nil
[6] pry(main)> ab.place_point(1, -1)

  |  
  |  
-- --
  | x
  |

=> nil

Complex Plane

What makes the Complex Plane different, is the second item in the Tuple (on the y-axis) is within the Set of Complex Numbers (C), and thus the y-axis comprises a set of Imaginary (Im) Numbers.

Real (Re) numbers

Real numbers, represented as R, are all true Real numbers. In practice, we can represent an ordered pair on a graph through a Tuple of two Real Numbers. In strict Mathematics this is represented as some coordinate within R^2, i.e. a Euclidean Plane.

Imaginary (Im) numbers

Imaginary numbers represent a base, e.g. some number within all real numbers, and i (representing square root of -1) with some exponent, identifying evaluated result of i

Thus, this work can receive things like 3 + 3i^5 and appropriately:

1. convert: a) i^5 to i, and b) i^6 to -1
2. identify 3 as Re (real), and
3. also identify 3i as Im (imaginary)

These are accomplished thorugh through finding the mod of some exponent on a disk of imaginary numbers, as they are represented within the iota and exponent cycle.

How it Works

[1] pry(main)> ab = ComplexNumber.new("3 + 3i")
[2] pry(main)> ab.complex_conjugate.a
=> "3 - 3i"
[3] pry(main)> ab.re
=> "3"
[4] pry(main)> ab.im
=> "3i"

Analyzing Work

Given rspec is installed, calling rspec goes through a series of test cases on ComplexNumber, ComplexPlane, and EuclideanPlane.