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# A complex number can be represented as a paired real number with imaginary
# unit; a+bi.  Where a is real part, b is imaginary part and i is imaginary
# unit.  Real a equals complex a+0i mathematically.
#
# Complex object can be created as literal, and also by using Kernel#Complex,
# Complex::rect, Complex::polar or to_c method.
#
#     2+1i                 #=> (2+1i)
#     Complex(1)           #=> (1+0i)
#     Complex(2, 3)        #=> (2+3i)
#     Complex.polar(2, 3)  #=> (-1.9799849932008908+0.2822400161197344i)
#     3.to_c               #=> (3+0i)
#
# You can also create complex object from floating-point numbers or strings.
#
#     Complex(0.3)         #=> (0.3+0i)
#     Complex('0.3-0.5i')  #=> (0.3-0.5i)
#     Complex('2/3+3/4i')  #=> ((2/3)+(3/4)*i)
#     Complex('1@2')       #=> (-0.4161468365471424+0.9092974268256817i)
#
#     0.3.to_c             #=> (0.3+0i)
#     '0.3-0.5i'.to_c      #=> (0.3-0.5i)
#     '2/3+3/4i'.to_c      #=> ((2/3)+(3/4)*i)
#     '1@2'.to_c           #=> (-0.4161468365471424+0.9092974268256817i)
#
# A complex object is either an exact or an inexact number.
#
#     Complex(1, 1) / 2    #=> ((1/2)+(1/2)*i)
#     Complex(1, 1) / 2.0  #=> (0.5+0.5i)
#
class Complex < Numeric
  # Returns a complex object which denotes the given polar form.
  #
  #     Complex.polar(3, 0)            #=> (3.0+0.0i)
  #     Complex.polar(3, Math::PI/2)   #=> (1.836909530733566e-16+3.0i)
  #     Complex.polar(3, Math::PI)     #=> (-3.0+3.673819061467132e-16i)
  #     Complex.polar(3, -Math::PI/2)  #=> (1.836909530733566e-16-3.0i)
  #
  def self.polar: (Numeric, ?Numeric) -> Complex

  # Returns a complex object which denotes the given rectangular form.
  #
  #     Complex.rectangular(1, 2)  #=> (1+2i)
  #
  def self.rect: (Numeric, ?Numeric) -> Complex

  # Returns a complex object which denotes the given rectangular form.
  #
  #     Complex.rectangular(1, 2)  #=> (1+2i)
  #
  alias self.rectangular self.rect

  public

  # Performs multiplication.
  #
  #     Complex(2, 3)  * Complex(2, 3)   #=> (-5+12i)
  #     Complex(900)   * Complex(1)      #=> (900+0i)
  #     Complex(-2, 9) * Complex(-9, 2)  #=> (0-85i)
  #     Complex(9, 8)  * 4               #=> (36+32i)
  #     Complex(20, 9) * 9.8             #=> (196.0+88.2i)
  #
  def *: (Numeric) -> Complex

  # Performs exponentiation.
  #
  #     Complex('i') ** 2              #=> (-1+0i)
  #     Complex(-8) ** Rational(1, 3)  #=> (1.0000000000000002+1.7320508075688772i)
  #
  def **: (Numeric) -> Complex

  # Performs addition.
  #
  #     Complex(2, 3)  + Complex(2, 3)   #=> (4+6i)
  #     Complex(900)   + Complex(1)      #=> (901+0i)
  #     Complex(-2, 9) + Complex(-9, 2)  #=> (-11+11i)
  #     Complex(9, 8)  + 4               #=> (13+8i)
  #     Complex(20, 9) + 9.8             #=> (29.8+9i)
  #
  def +: (Numeric) -> Complex

  def +@: () -> Complex

  # Performs subtraction.
  #
  #     Complex(2, 3)  - Complex(2, 3)   #=> (0+0i)
  #     Complex(900)   - Complex(1)      #=> (899+0i)
  #     Complex(-2, 9) - Complex(-9, 2)  #=> (7+7i)
  #     Complex(9, 8)  - 4               #=> (5+8i)
  #     Complex(20, 9) - 9.8             #=> (10.2+9i)
  #
  def -: (Numeric) -> Complex

  # Returns negation of the value.
  #
  #     -Complex(1, 2)  #=> (-1-2i)
  #
  def -@: () -> Complex

  # Performs division.
  #
  #     Complex(2, 3)  / Complex(2, 3)   #=> ((1/1)+(0/1)*i)
  #     Complex(900)   / Complex(1)      #=> ((900/1)+(0/1)*i)
  #     Complex(-2, 9) / Complex(-9, 2)  #=> ((36/85)-(77/85)*i)
  #     Complex(9, 8)  / 4               #=> ((9/4)+(2/1)*i)
  #     Complex(20, 9) / 9.8             #=> (2.0408163265306123+0.9183673469387754i)
  #
  def /: (Numeric) -> Complex

  def <: (Numeric) -> bot

  def <=: (Numeric) -> bot

  # If `cmp`'s imaginary part is zero, and `object` is also a real number (or a
  # Complex number where the imaginary part is zero), compare the real part of
  # `cmp` to object.  Otherwise, return nil.
  #
  #     Complex(2, 3)  <=> Complex(2, 3)   #=> nil
  #     Complex(2, 3)  <=> 1               #=> nil
  #     Complex(2)     <=> 1               #=> 1
  #     Complex(2)     <=> 2               #=> 0
  #     Complex(2)     <=> 3               #=> -1
  #
  def <=>: (Numeric) -> Integer?

  # Returns true if cmp equals object numerically.
  #
  #     Complex(2, 3)  == Complex(2, 3)   #=> true
  #     Complex(5)     == 5               #=> true
  #     Complex(0)     == 0.0             #=> true
  #     Complex('1/3') == 0.33            #=> false
  #     Complex('1/2') == '1/2'           #=> false
  #
  def ==: (untyped) -> bool

  def >: (Numeric) -> bot

  def >=: (Numeric) -> bot

  # Returns the absolute part of its polar form.
  #
  #     Complex(-1).abs         #=> 1
  #     Complex(3.0, -4.0).abs  #=> 5.0
  #
  def abs: () -> Numeric

  # Returns square of the absolute value.
  #
  #     Complex(-1).abs2         #=> 1
  #     Complex(3.0, -4.0).abs2  #=> 25.0
  #
  def abs2: () -> Numeric

  # Returns the angle part of its polar form.
  #
  #     Complex.polar(3, Math::PI/2).arg  #=> 1.5707963267948966
  #
  def angle: () -> Float

  # Returns the angle part of its polar form.
  #
  #     Complex.polar(3, Math::PI/2).arg  #=> 1.5707963267948966
  #
  alias arg angle

  def ceil: (*untyped) -> bot

  def clone: (?freeze: bool) -> self

  def coerce: (Numeric) -> [ Complex, Complex ]

  # Returns the complex conjugate.
  #
  #     Complex(1, 2).conjugate  #=> (1-2i)
  #
  def conj: () -> Complex

  # Returns the complex conjugate.
  #
  #     Complex(1, 2).conjugate  #=> (1-2i)
  #
  def conjugate: () -> Complex

  # Returns the denominator (lcm of both denominator - real and imag).
  #
  # See numerator.
  #
  def denominator: () -> Integer

  def div: (Numeric) -> bot

  def divmod: (Numeric) -> bot

  def dup: () -> self

  def eql?: (untyped) -> bool

  # Performs division as each part is a float, never returns a float.
  #
  #     Complex(11, 22).fdiv(3)  #=> (3.6666666666666665+7.333333333333333i)
  #
  def fdiv: (Numeric) -> Complex

  # Returns `true` if `cmp`'s real and imaginary parts are both finite numbers,
  # otherwise returns `false`.
  #
  def finite?: () -> bool

  def floor: (?Integer) -> bot

  def hash: () -> Integer

  def i: () -> bot

  # Returns the imaginary part.
  #
  #     Complex(7).imaginary      #=> 0
  #     Complex(9, -4).imaginary  #=> -4
  #
  def imag: () -> Numeric

  # Returns the imaginary part.
  #
  #     Complex(7).imaginary      #=> 0
  #     Complex(9, -4).imaginary  #=> -4
  #
  def imaginary: () -> Numeric

  # Returns `1` if `cmp`'s real or imaginary part is an infinite number, otherwise
  # returns `nil`.
  #
  #     For example:
  #
  #        (1+1i).infinite?                   #=> nil
  #        (Float::INFINITY + 1i).infinite?   #=> 1
  #
  def infinite?: () -> Integer?

  # Returns the value as a string for inspection.
  #
  #     Complex(2).inspect                       #=> "(2+0i)"
  #     Complex('-8/6').inspect                  #=> "((-4/3)+0i)"
  #     Complex('1/2i').inspect                  #=> "(0+(1/2)*i)"
  #     Complex(0, Float::INFINITY).inspect      #=> "(0+Infinity*i)"
  #     Complex(Float::NAN, Float::NAN).inspect  #=> "(NaN+NaN*i)"
  #
  def inspect: () -> String

  def integer?: () -> bool

  # Returns the absolute part of its polar form.
  #
  #     Complex(-1).abs         #=> 1
  #     Complex(3.0, -4.0).abs  #=> 5.0
  #
  alias magnitude abs

  def modulo: (Numeric) -> bot

  def negative?: () -> bot

  def nonzero?: () -> self?

  # Returns the numerator.
  #
  #         1   2       3+4i  <-  numerator
  #         - + -i  ->  ----
  #         2   3        6    <-  denominator
  #
  #     c = Complex('1/2+2/3i')  #=> ((1/2)+(2/3)*i)
  #     n = c.numerator          #=> (3+4i)
  #     d = c.denominator        #=> 6
  #     n / d                    #=> ((1/2)+(2/3)*i)
  #     Complex(Rational(n.real, d), Rational(n.imag, d))
  #                              #=> ((1/2)+(2/3)*i)
  #
  # See denominator.
  #
  def numerator: () -> Complex

  # Returns the angle part of its polar form.
  #
  #     Complex.polar(3, Math::PI/2).arg  #=> 1.5707963267948966
  #
  alias phase angle

  # Returns an array; [cmp.abs, cmp.arg].
  #
  #     Complex(1, 2).polar  #=> [2.23606797749979, 1.1071487177940904]
  #
  def polar: () -> [Numeric, Float]

  def positive?: () -> bot

  # Performs division.
  #
  #     Complex(2, 3)  / Complex(2, 3)   #=> ((1/1)+(0/1)*i)
  #     Complex(900)   / Complex(1)      #=> ((900/1)+(0/1)*i)
  #     Complex(-2, 9) / Complex(-9, 2)  #=> ((36/85)-(77/85)*i)
  #     Complex(9, 8)  / 4               #=> ((9/4)+(2/1)*i)
  #     Complex(20, 9) / 9.8             #=> (2.0408163265306123+0.9183673469387754i)
  #
  def quo: (Numeric) -> Complex

  # Returns the value as a rational if possible (the imaginary part should be
  # exactly zero).
  #
  #     Complex(1.0/3, 0).rationalize  #=> (1/3)
  #     Complex(1, 0.0).rationalize    # RangeError
  #     Complex(1, 2).rationalize      # RangeError
  #
  # See to_r.
  #
  def rationalize: (?Numeric eps) -> Rational

  # Returns the real part.
  #
  #     Complex(7).real      #=> 7
  #     Complex(9, -4).real  #=> 9
  #
  def real: () -> Numeric

  # Returns false, even if the complex number has no imaginary part.
  #
  def real?: () -> false

  # Returns an array; [cmp.real, cmp.imag].
  #
  #     Complex(1, 2).rectangular  #=> [1, 2]
  #
  def rect: () -> [Numeric, Numeric]

  # Returns an array; [cmp.real, cmp.imag].
  #
  #     Complex(1, 2).rectangular  #=> [1, 2]
  #
  alias rectangular rect

  def reminder: (Numeric) -> bot

  def round: (*untyped) -> bot

  def step: (*untyped) ?{ (*untyped) -> untyped } -> bot

  # Returns self.
  #
  #     Complex(2).to_c      #=> (2+0i)
  #     Complex(-8, 6).to_c  #=> (-8+6i)
  #
  def to_c: () -> Complex

  # Returns the value as a float if possible (the imaginary part should be exactly
  # zero).
  #
  #     Complex(1, 0).to_f    #=> 1.0
  #     Complex(1, 0.0).to_f  # RangeError
  #     Complex(1, 2).to_f    # RangeError
  #
  def to_f: () -> Float

  # Returns the value as an integer if possible (the imaginary part should be
  # exactly zero).
  #
  #     Complex(1, 0).to_i    #=> 1
  #     Complex(1, 0.0).to_i  # RangeError
  #     Complex(1, 2).to_i    # RangeError
  #
  def to_i: () -> Integer

  alias to_int to_i

  # Returns the value as a rational if possible (the imaginary part should be
  # exactly zero).
  #
  #     Complex(1, 0).to_r    #=> (1/1)
  #     Complex(1, 0.0).to_r  # RangeError
  #     Complex(1, 2).to_r    # RangeError
  #
  # See rationalize.
  #
  def to_r: () -> Rational

  # Returns the value as a string.
  #
  #     Complex(2).to_s                       #=> "2+0i"
  #     Complex('-8/6').to_s                  #=> "-4/3+0i"
  #     Complex('1/2i').to_s                  #=> "0+1/2i"
  #     Complex(0, Float::INFINITY).to_s      #=> "0+Infinity*i"
  #     Complex(Float::NAN, Float::NAN).to_s  #=> "NaN+NaN*i"
  #
  def to_s: () -> String

  def truncate: (?Integer) -> bot

  def zero?: () -> bool
end

# The imaginary unit.
#
Complex::I: Complex