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# A rational number can be represented as a pair of integer numbers: a/b (b>0),
# where a is the numerator and b is the denominator. Integer a equals rational
# a/1 mathematically.
#
# In Ruby, you can create rational objects with the Kernel#Rational, to_r, or
# rationalize methods or by suffixing `r` to a literal. The return values will
# be irreducible fractions.
#
#     Rational(1)      #=> (1/1)
#     Rational(2, 3)   #=> (2/3)
#     Rational(4, -6)  #=> (-2/3)
#     3.to_r           #=> (3/1)
#     2/3r             #=> (2/3)
#
# You can also create rational objects from floating-point numbers or strings.
#
#     Rational(0.3)    #=> (5404319552844595/18014398509481984)
#     Rational('0.3')  #=> (3/10)
#     Rational('2/3')  #=> (2/3)
#
#     0.3.to_r         #=> (5404319552844595/18014398509481984)
#     '0.3'.to_r       #=> (3/10)
#     '2/3'.to_r       #=> (2/3)
#     0.3.rationalize  #=> (3/10)
#
# A rational object is an exact number, which helps you to write programs
# without any rounding errors.
#
#     10.times.inject(0) {|t| t + 0.1 }              #=> 0.9999999999999999
#     10.times.inject(0) {|t| t + Rational('0.1') }  #=> (1/1)
#
# However, when an expression includes an inexact component (numerical value or
# operation), it will produce an inexact result.
#
#     Rational(10) / 3   #=> (10/3)
#     Rational(10) / 3.0 #=> 3.3333333333333335
#
#     Rational(-8) ** Rational(1, 3)
#                        #=> (1.0000000000000002+1.7320508075688772i)
#
class Rational < Numeric
  public

  def %: (Integer) -> Rational
       | (Float) -> Float
       | (Rational) -> Rational
       | (Numeric) -> Numeric

  # Performs multiplication.
  #
  #     Rational(2, 3)  * Rational(2, 3)   #=> (4/9)
  #     Rational(900)   * Rational(1)      #=> (900/1)
  #     Rational(-2, 9) * Rational(-9, 2)  #=> (1/1)
  #     Rational(9, 8)  * 4                #=> (9/2)
  #     Rational(20, 9) * 9.8              #=> 21.77777777777778
  #
  def *: (Float) -> Float
       | (Complex) -> Complex
       | (Numeric) -> Numeric

  # Performs exponentiation.
  #
  #     Rational(2)    ** Rational(3)     #=> (8/1)
  #     Rational(10)   ** -2              #=> (1/100)
  #     Rational(10)   ** -2.0            #=> 0.01
  #     Rational(-4)   ** Rational(1, 2)  #=> (0.0+2.0i)
  #     Rational(1, 2) ** 0               #=> (1/1)
  #     Rational(1, 2) ** 0.0             #=> 1.0
  #
  def **: (Complex) -> Complex
        | (Numeric) -> Numeric

  # Performs addition.
  #
  #     Rational(2, 3)  + Rational(2, 3)   #=> (4/3)
  #     Rational(900)   + Rational(1)      #=> (901/1)
  #     Rational(-2, 9) + Rational(-9, 2)  #=> (-85/18)
  #     Rational(9, 8)  + 4                #=> (41/8)
  #     Rational(20, 9) + 9.8              #=> 12.022222222222222
  #
  def +: (Float) -> Float
       | (Complex) -> Complex
       | (Numeric) -> Rational

  def +@: () -> Rational

  # Performs subtraction.
  #
  #     Rational(2, 3)  - Rational(2, 3)   #=> (0/1)
  #     Rational(900)   - Rational(1)      #=> (899/1)
  #     Rational(-2, 9) - Rational(-9, 2)  #=> (77/18)
  #     Rational(9, 8)  - 4                #=> (-23/8)
  #     Rational(20, 9) - 9.8              #=> -7.577777777777778
  #
  def -: (Float) -> Float
       | (Complex) -> Complex
       | (Numeric) -> Rational

  # Negates `rat`.
  #
  def -@: () -> Rational

  # Performs division.
  #
  #     Rational(2, 3)  / Rational(2, 3)   #=> (1/1)
  #     Rational(900)   / Rational(1)      #=> (900/1)
  #     Rational(-2, 9) / Rational(-9, 2)  #=> (4/81)
  #     Rational(9, 8)  / 4                #=> (9/32)
  #     Rational(20, 9) / 9.8              #=> 0.22675736961451246
  #
  def /: (Float) -> Float
       | (Complex) -> Complex
       | (Numeric) -> Rational

  # Returns -1, 0, or +1 depending on whether `rational` is less than, equal to,
  # or greater than `numeric`.
  #
  # `nil` is returned if the two values are incomparable.
  #
  #     Rational(2, 3) <=> Rational(2, 3)  #=> 0
  #     Rational(5)    <=> 5               #=> 0
  #     Rational(2, 3) <=> Rational(1, 3)  #=> 1
  #     Rational(1, 3) <=> 1               #=> -1
  #     Rational(1, 3) <=> 0.3             #=> 1
  #
  #     Rational(1, 3) <=> "0.3"           #=> nil
  #
  def <=>: (Numeric) -> Integer?

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

  # Returns the absolute value of `rat`.
  #
  #     (1/2r).abs    #=> (1/2)
  #     (-1/2r).abs   #=> (1/2)
  #
  # Rational#magnitude is an alias for Rational#abs.
  #
  def abs: () -> Rational

  def abs2: () -> Rational

  def angle: () -> (Integer | Float)

  alias arg angle

  # Returns the smallest number greater than or equal to `rat` with a precision of
  # `ndigits` decimal digits (default: 0).
  #
  # When the precision is negative, the returned value is an integer with at least
  # `ndigits.abs` trailing zeros.
  #
  # Returns a rational when `ndigits` is positive, otherwise returns an integer.
  #
  #     Rational(3).ceil      #=> 3
  #     Rational(2, 3).ceil   #=> 1
  #     Rational(-3, 2).ceil  #=> -1
  #
  #       #    decimal      -  1  2  3 . 4  5  6
  #       #                   ^  ^  ^  ^   ^  ^
  #       #   precision      -3 -2 -1  0  +1 +2
  #
  #     Rational('-123.456').ceil(+1).to_f  #=> -123.4
  #     Rational('-123.456').ceil(-1)       #=> -120
  #
  def ceil: () -> Integer
          | (Integer digits) -> (Integer | Rational)

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

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

  def conj: () -> Rational

  def conjugate: () -> Rational

  # Returns the denominator (always positive).
  #
  #     Rational(7).denominator             #=> 1
  #     Rational(7, 1).denominator          #=> 1
  #     Rational(9, -4).denominator         #=> 4
  #     Rational(-2, -10).denominator       #=> 5
  #
  def denominator: () -> Integer

  def div: (Numeric) -> Integer

  def divmod: (Numeric) -> [Numeric, Numeric]

  def dup: () -> self

  def eql?: (untyped) -> bool

  # Performs division and returns the value as a Float.
  #
  #     Rational(2, 3).fdiv(1)       #=> 0.6666666666666666
  #     Rational(2, 3).fdiv(0.5)     #=> 1.3333333333333333
  #     Rational(2).fdiv(3)          #=> 0.6666666666666666
  #
  def fdiv: (Numeric) -> Float

  def finite?: () -> bool

  # Returns the largest number less than or equal to `rat` with a precision of
  # `ndigits` decimal digits (default: 0).
  #
  # When the precision is negative, the returned value is an integer with at least
  # `ndigits.abs` trailing zeros.
  #
  # Returns a rational when `ndigits` is positive, otherwise returns an integer.
  #
  #     Rational(3).floor      #=> 3
  #     Rational(2, 3).floor   #=> 0
  #     Rational(-3, 2).floor  #=> -2
  #
  #       #    decimal      -  1  2  3 . 4  5  6
  #       #                   ^  ^  ^  ^   ^  ^
  #       #   precision      -3 -2 -1  0  +1 +2
  #
  #     Rational('-123.456').floor(+1).to_f  #=> -123.5
  #     Rational('-123.456').floor(-1)       #=> -130
  #
  def floor: () -> Integer
           | (Integer digits) -> (Integer | Rational)

  def hash: () -> Integer

  def i: () -> Complex

  def imag: () -> Integer

  def imaginary: () -> Integer

  def infinite?: () -> Integer?

  # Returns the value as a string for inspection.
  #
  #     Rational(2).inspect      #=> "(2/1)"
  #     Rational(-8, 6).inspect  #=> "(-4/3)"
  #     Rational('1/2').inspect  #=> "(1/2)"
  #
  def inspect: () -> String

  def integer?: () -> bool

  # Returns the absolute value of `rat`.
  #
  #     (1/2r).abs    #=> (1/2)
  #     (-1/2r).abs   #=> (1/2)
  #
  # Rational#magnitude is an alias for Rational#abs.
  #
  alias magnitude abs

  def modulo: (Float) -> Float
            | (Numeric) -> Rational

  # Returns `true` if `rat` is less than 0.
  #
  def negative?: () -> bool

  def nonzero?: () -> self?

  # Returns the numerator.
  #
  #     Rational(7).numerator        #=> 7
  #     Rational(7, 1).numerator     #=> 7
  #     Rational(9, -4).numerator    #=> -9
  #     Rational(-2, -10).numerator  #=> 1
  #
  def numerator: () -> Integer

  alias phase angle

  def polar: () -> [ Rational, Integer | Float ]

  # Returns `true` if `rat` is greater than 0.
  #
  def positive?: () -> bool

  # Performs division.
  #
  #     Rational(2, 3)  / Rational(2, 3)   #=> (1/1)
  #     Rational(900)   / Rational(1)      #=> (900/1)
  #     Rational(-2, 9) / Rational(-9, 2)  #=> (4/81)
  #     Rational(9, 8)  / 4                #=> (9/32)
  #     Rational(20, 9) / 9.8              #=> 0.22675736961451246
  #
  def quo: (Float) -> Float
         | (Complex) -> Complex
         | (Numeric) -> Rational

  # Returns a simpler approximation of the value if the optional argument `eps` is
  # given (rat-|eps| <= result <= rat+|eps|), self otherwise.
  #
  #     r = Rational(5033165, 16777216)
  #     r.rationalize                    #=> (5033165/16777216)
  #     r.rationalize(Rational('0.01'))  #=> (3/10)
  #     r.rationalize(Rational('0.1'))   #=> (1/3)
  #
  def rationalize: (?Numeric eps) -> Rational

  def real: () -> Rational

  def real?: () -> true

  def rect: () -> [ Rational, Numeric ]

  alias rectangular rect

  def remainder: (Float) -> Float
               | (Numeric) -> Rational

  # Returns `rat` rounded to the nearest value with a precision of `ndigits`
  # decimal digits (default: 0).
  #
  # When the precision is negative, the returned value is an integer with at least
  # `ndigits.abs` trailing zeros.
  #
  # Returns a rational when `ndigits` is positive, otherwise returns an integer.
  #
  #     Rational(3).round      #=> 3
  #     Rational(2, 3).round   #=> 1
  #     Rational(-3, 2).round  #=> -2
  #
  #       #    decimal      -  1  2  3 . 4  5  6
  #       #                   ^  ^  ^  ^   ^  ^
  #       #   precision      -3 -2 -1  0  +1 +2
  #
  #     Rational('-123.456').round(+1).to_f  #=> -123.5
  #     Rational('-123.456').round(-1)       #=> -120
  #
  # The optional `half` keyword argument is available similar to Float#round.
  #
  #     Rational(25, 100).round(1, half: :up)    #=> (3/10)
  #     Rational(25, 100).round(1, half: :down)  #=> (1/5)
  #     Rational(25, 100).round(1, half: :even)  #=> (1/5)
  #     Rational(35, 100).round(1, half: :up)    #=> (2/5)
  #     Rational(35, 100).round(1, half: :down)  #=> (3/10)
  #     Rational(35, 100).round(1, half: :even)  #=> (2/5)
  #     Rational(-25, 100).round(1, half: :up)   #=> (-3/10)
  #     Rational(-25, 100).round(1, half: :down) #=> (-1/5)
  #     Rational(-25, 100).round(1, half: :even) #=> (-1/5)
  #
  def round: (?half: :up | :down | :even) -> Integer
           | (Integer digits, ?half: :up | :down | :even) -> (Integer | Rational)

  def step: (?Numeric limit, ?Numeric step) { (Rational) -> void } -> self
          | (?Numeric limit, ?Numeric step) -> Enumerator[Rational, self]
          | (?by: Numeric, ?to: Numeric) { (Rational) -> void } -> self
          | (?by: Numeric, ?to: Numeric) -> Enumerator[Rational, self]

  def to_c: () -> Complex

  # Returns the value as a Float.
  #
  #     Rational(2).to_f      #=> 2.0
  #     Rational(9, 4).to_f   #=> 2.25
  #     Rational(-3, 4).to_f  #=> -0.75
  #     Rational(20, 3).to_f  #=> 6.666666666666667
  #
  def to_f: () -> Float

  # Returns the truncated value as an integer.
  #
  # Equivalent to Rational#truncate.
  #
  #     Rational(2, 3).to_i    #=> 0
  #     Rational(3).to_i       #=> 3
  #     Rational(300.6).to_i   #=> 300
  #     Rational(98, 71).to_i  #=> 1
  #     Rational(-31, 2).to_i  #=> -15
  #
  def to_i: () -> Integer

  alias to_int to_i

  # Returns self.
  #
  #     Rational(2).to_r      #=> (2/1)
  #     Rational(-8, 6).to_r  #=> (-4/3)
  #
  def to_r: () -> Rational

  # Returns the value as a string.
  #
  #     Rational(2).to_s      #=> "2/1"
  #     Rational(-8, 6).to_s  #=> "-4/3"
  #     Rational('1/2').to_s  #=> "1/2"
  #
  def to_s: () -> String

  # Returns `rat` truncated (toward zero) to a precision of `ndigits` decimal
  # digits (default: 0).
  #
  # When the precision is negative, the returned value is an integer with at least
  # `ndigits.abs` trailing zeros.
  #
  # Returns a rational when `ndigits` is positive, otherwise returns an integer.
  #
  #     Rational(3).truncate      #=> 3
  #     Rational(2, 3).truncate   #=> 0
  #     Rational(-3, 2).truncate  #=> -1
  #
  #       #    decimal      -  1  2  3 . 4  5  6
  #       #                   ^  ^  ^  ^   ^  ^
  #       #   precision      -3 -2 -1  0  +1 +2
  #
  #     Rational('-123.456').truncate(+1).to_f  #=> -123.4
  #     Rational('-123.456').truncate(-1)       #=> -120
  #
  def truncate: () -> Integer
              | (Integer ndigits) -> (Integer | Rational)

  def zero?: () -> bool
end