numeric.rbs

# <!-- rdoc-file=numeric.c -->
# Numeric is the class from which all higher-level numeric classes should
# inherit.
#
# Numeric allows instantiation of heap-allocated objects. Other core numeric
# classes such as Integer are implemented as immediates, which means that each
# Integer is a single immutable object which is always passed by value.
#
#     a = 1
#     1.object_id == a.object_id   #=> true
#
# There can only ever be one instance of the integer `1`, for example. Ruby
# ensures this by preventing instantiation. If duplication is attempted, the
# same instance is returned.
#
#     Integer.new(1)                   #=> NoMethodError: undefined method `new' for Integer:Class
#     1.dup                            #=> 1
#     1.object_id == 1.dup.object_id   #=> true
#
# For this reason, Numeric should be used when defining other numeric classes.
#
# Classes which inherit from Numeric must implement `coerce`, which returns a
# two-member Array containing an object that has been coerced into an instance
# of the new class and `self` (see #coerce).
#
# Inheriting classes should also implement arithmetic operator methods (`+`,
# `-`, `*` and `/`) and the `<=>` operator (see Comparable). These methods may
# rely on `coerce` to ensure interoperability with instances of other numeric
# classes.
#
#     class Tally < Numeric
#       def initialize(string)
#         @string = string
#       end
#
#       def to_s
#         @string
#       end
#
#       def to_i
#         @string.size
#       end
#
#       def coerce(other)
#         [self.class.new('|' * other.to_i), self]
#       end
#
#       def <=>(other)
#         to_i <=> other.to_i
#       end
#
#       def +(other)
#         self.class.new('|' * (to_i + other.to_i))
#       end
#
#       def -(other)
#         self.class.new('|' * (to_i - other.to_i))
#       end
#
#       def *(other)
#         self.class.new('|' * (to_i * other.to_i))
#       end
#
#       def /(other)
#         self.class.new('|' * (to_i / other.to_i))
#       end
#     end
#
#     tally = Tally.new('||')
#     puts tally * 2            #=> "||||"
#     puts tally > 1            #=> true
#
# ## What's Here
#
# First, what's elsewhere. Class Numeric:
#
# *   Inherits from [class Object](rdoc-ref:Object@What-27s+Here).
# *   Includes [module Comparable](rdoc-ref:Comparable@What-27s+Here).
#
# Here, class Numeric provides methods for:
#
# *   [Querying](rdoc-ref:Numeric@Querying)
# *   [Comparing](rdoc-ref:Numeric@Comparing)
# *   [Converting](rdoc-ref:Numeric@Converting)
# *   [Other](rdoc-ref:Numeric@Other)
#
# ### Querying
#
# *   #finite?: Returns true unless `self` is infinite or not a number.
# *   #infinite?: Returns -1, `nil` or +1, depending on whether `self` is
#     `-Infinity<tt>, finite, or <tt>+Infinity`.
# *   #integer?: Returns whether `self` is an integer.
# *   #negative?: Returns whether `self` is negative.
# *   #nonzero?: Returns whether `self` is not zero.
# *   #positive?: Returns whether `self` is positive.
# *   #real?: Returns whether `self` is a real value.
# *   #zero?: Returns whether `self` is zero.
#
# ### Comparing
#
# *   #<=>: Returns:
#
#     *   -1 if  `self` is less than the given value.
#     *   0 if `self` is equal to the given value.
#     *   1 if `self` is greater than the given value.
#     *   `nil` if `self` and the given value are not comparable.
#
# *   #eql?: Returns whether `self` and the given value have the same value and
#     type.
#
# ### Converting
#
# *   #% (aliased as #modulo): Returns the remainder of `self` divided by the
#     given value.
# *   #-@: Returns the value of `self`, negated.
# *   #abs (aliased as #magnitude): Returns the absolute value of `self`.
# *   #abs2: Returns the square of `self`.
# *   #angle (aliased as #arg and #phase): Returns 0 if `self` is positive,
#     Math::PI otherwise.
# *   #ceil: Returns the smallest number greater than or equal to `self`, to a
#     given precision.
# *   #coerce: Returns array `[coerced_self, coerced_other]` for the given other
#     value.
# *   #conj (aliased as #conjugate): Returns the complex conjugate of `self`.
# *   #denominator: Returns the denominator (always positive) of the Rational
#     representation of `self`.
# *   #div: Returns the value of `self` divided by the given value and converted
#     to an integer.
# *   #divmod: Returns array `[quotient, modulus]` resulting from dividing
#     `self` the given divisor.
# *   #fdiv: Returns the Float result of dividing `self` by the given divisor.
# *   #floor: Returns the largest number less than or equal to `self`, to a
#     given precision.
# *   #i: Returns the Complex object `Complex(0, self)`. the given value.
# *   #imaginary (aliased as #imag): Returns the imaginary part of the `self`.
# *   #numerator: Returns the numerator of the Rational representation of
#     `self`; has the same sign as `self`.
# *   #polar: Returns the array `[self.abs, self.arg]`.
# *   #quo: Returns the value of `self` divided by the given value.
# *   #real: Returns the real part of `self`.
# *   #rect (aliased as #rectangular): Returns the array `[self, 0]`.
# *   #remainder: Returns `self-arg*(self/arg).truncate` for the given `arg`.
# *   #round: Returns the value of `self` rounded to the nearest value for the
#     given a precision.
# *   #to_c: Returns the Complex representation of `self`.
# *   #to_int: Returns the Integer representation of `self`, truncating if
#     necessary.
# *   #truncate: Returns `self` truncated (toward zero) to a given precision.
#
# ### Other
#
# *   #clone: Returns `self`; does not allow freezing.
# *   #dup (aliased as #+@): Returns `self`.
# *   #step: Invokes the given block with the sequence of specified numbers.
#
class Numeric
  include Comparable

  # <!--
  #   rdoc-file=numeric.c
  #   - self % other -> real_numeric
  # -->
  # Returns `self` modulo `other` as a real number.
  #
  # Of the Core and Standard Library classes, only Rational uses this
  # implementation.
  #
  # For Rational `r` and real number `n`, these expressions are equivalent:
  #
  #     r % n
  #     r-n*(r/n).floor
  #     r.divmod(n)[1]
  #
  # See Numeric#divmod.
  #
  # Examples:
  #
  #     r = Rational(1, 2)    # => (1/2)
  #     r2 = Rational(2, 3)   # => (2/3)
  #     r % r2                # => (1/2)
  #     r % 2                 # => (1/2)
  #     r % 2.0               # => 0.5
  #
  #     r = Rational(301,100) # => (301/100)
  #     r2 = Rational(7,5)    # => (7/5)
  #     r % r2                # => (21/100)
  #     r % -r2               # => (-119/100)
  #     (-r) % r2             # => (119/100)
  #     (-r) %-r2             # => (-21/100)
  #
  def %: (Numeric) -> Numeric

  # Performs addition: the class of the resulting object depends on the class of
  # `numeric`.
  #
  def +: (Numeric) -> Numeric

  # <!--
  #   rdoc-file=numeric.rb
  #   - +self -> self
  # -->
  # Returns `self`.
  #
  def +@: () -> self

  # Performs subtraction: the class of the resulting object depends on the class
  # of `numeric`.
  #
  def -: (Numeric) -> Numeric

  # <!--
  #   rdoc-file=numeric.c
  #   - -self -> numeric
  # -->
  # Unary Minus---Returns the receiver, negated.
  #
  def -@: () -> self

  # <!--
  #   rdoc-file=numeric.c
  #   - self <=> other -> zero or nil
  # -->
  # Returns zero if `self` is the same as `other`, `nil` otherwise.
  #
  # No subclass in the Ruby Core or Standard Library uses this implementation.
  #
  def <=>: (Numeric other) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - abs -> numeric
  # -->
  # Returns the absolute value of `self`.
  #
  #     12.abs        #=> 12
  #     (-34.56).abs  #=> 34.56
  #     -34.56.abs    #=> 34.56
  #
  def abs: () -> Numeric

  # <!--
  #   rdoc-file=complex.c
  #   - abs2 -> real
  # -->
  # Returns the square of `self`.
  #
  def abs2: () -> self

  # <!-- rdoc-file=complex.c -->
  # Returns zero if `self` is positive, Math::PI otherwise.
  #
  def angle: () -> (0 | Float)

  # <!--
  #   rdoc-file=complex.c
  #   - arg -> 0 or Math::PI
  # -->
  # Returns zero if `self` is positive, Math::PI otherwise.
  #
  alias arg angle

  # <!--
  #   rdoc-file=numeric.c
  #   - ceil(ndigits = 0) -> float or integer
  # -->
  # Returns the smallest float or integer that is greater than or equal to `self`,
  # as specified by the given `ndigits`, which must be an [integer-convertible
  # object](rdoc-ref:implicit_conversion.rdoc@Integer-Convertible+Objects).
  #
  # Equivalent to `self.to_f.ceil(ndigits)`.
  #
  # Related: #floor, Float#ceil.
  #
  def ceil: () -> Integer
          | (Integer digits) -> (Integer | Numeric)

  # <!--
  #   rdoc-file=numeric.c
  #   - coerce(other) -> array
  # -->
  # Returns a 2-element array containing two numeric elements, formed from the two
  # operands `self` and `other`, of a common compatible type.
  #
  # Of the Core and Standard Library classes, Integer, Rational, and Complex use
  # this implementation.
  #
  # Examples:
  #
  #     i = 2                    # => 2
  #     i.coerce(3)              # => [3, 2]
  #     i.coerce(3.0)            # => [3.0, 2.0]
  #     i.coerce(Rational(1, 2)) # => [0.5, 2.0]
  #     i.coerce(Complex(3, 4))  # Raises RangeError.
  #
  #     r = Rational(5, 2)       # => (5/2)
  #     r.coerce(2)              # => [(2/1), (5/2)]
  #     r.coerce(2.0)            # => [2.0, 2.5]
  #     r.coerce(Rational(2, 3)) # => [(2/3), (5/2)]
  #     r.coerce(Complex(3, 4))  # => [(3+4i), ((5/2)+0i)]
  #
  #     c = Complex(2, 3)        # => (2+3i)
  #     c.coerce(2)              # => [(2+0i), (2+3i)]
  #     c.coerce(2.0)            # => [(2.0+0i), (2+3i)]
  #     c.coerce(Rational(1, 2)) # => [((1/2)+0i), (2+3i)]
  #     c.coerce(Complex(3, 4))  # => [(3+4i), (2+3i)]
  #
  # Raises an exception if any type conversion fails.
  #
  def coerce: (Numeric) -> [ Numeric, Numeric ]

  # <!--
  #   rdoc-file=numeric.rb
  #   - conj -> self
  # -->
  # Returns `self`.
  #
  def conjugate: () -> self

  # <!--
  #   rdoc-file=numeric.rb
  #   - conj()
  # -->
  #
  alias conj conjugate

  # <!--
  #   rdoc-file=rational.c
  #   - num.denominator  ->  integer
  # -->
  # Returns the denominator (always positive).
  #
  def denominator: () -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - div(other) -> integer
  # -->
  # Returns the quotient `self/other` as an integer (via `floor`), using method
  # `/` in the derived class of `self`. (Numeric itself does not define method
  # `/`.)
  #
  # Of the Core and Standard Library classes, Only Float and Rational use this
  # implementation.
  #
  def div: (Numeric) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - divmod(other) -> array
  # -->
  # Returns a 2-element array `[q, r]`, where
  #
  #     q = (self/other).floor                  # Quotient
  #     r = self % other                        # Remainder
  #
  # Of the Core and Standard Library classes, only Rational uses this
  # implementation.
  #
  # Examples:
  #
  #     Rational(11, 1).divmod(4)               # => [2, (3/1)]
  #     Rational(11, 1).divmod(-4)              # => [-3, (-1/1)]
  #     Rational(-11, 1).divmod(4)              # => [-3, (1/1)]
  #     Rational(-11, 1).divmod(-4)             # => [2, (-3/1)]
  #
  #     Rational(12, 1).divmod(4)               # => [3, (0/1)]
  #     Rational(12, 1).divmod(-4)              # => [-3, (0/1)]
  #     Rational(-12, 1).divmod(4)              # => [-3, (0/1)]
  #     Rational(-12, 1).divmod(-4)             # => [3, (0/1)]
  #
  #     Rational(13, 1).divmod(4.0)             # => [3, 1.0]
  #     Rational(13, 1).divmod(Rational(4, 11)) # => [35, (3/11)]
  #
  def divmod: (Numeric) -> [ Numeric, Numeric ]

  # <!--
  #   rdoc-file=numeric.c
  #   - eql?(other) -> true or false
  # -->
  # Returns `true` if `self` and `other` are the same type and have equal values.
  #
  # Of the Core and Standard Library classes, only Integer, Rational, and Complex
  # use this implementation.
  #
  # Examples:
  #
  #     1.eql?(1)              # => true
  #     1.eql?(1.0)            # => false
  #     1.eql?(Rational(1, 1)) # => false
  #     1.eql?(Complex(1, 0))  # => false
  #
  # Method `eql?` is different from `==` in that `eql?` requires matching types,
  # while `==` does not.
  #
  def eql?: (untyped) -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - fdiv(other) -> float
  # -->
  # Returns the quotient `self/other` as a float, using method `/` in the derived
  # class of `self`. (Numeric itself does not define method `/`.)
  #
  # Of the Core and Standard Library classes, only BigDecimal uses this
  # implementation.
  #
  def fdiv: (Numeric) -> Numeric

  # <!--
  #   rdoc-file=numeric.rb
  #   - finite? -> true or false
  # -->
  # Returns `true` if `self` is a finite number, `false` otherwise.
  #
  def finite?: () -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - floor(ndigits = 0) -> float or integer
  # -->
  # Returns the largest float or integer that is less than or equal to `self`, as
  # specified by the given `ndigits`, which must be an [integer-convertible
  # object](rdoc-ref:implicit_conversion.rdoc@Integer-Convertible+Objects).
  #
  # Equivalent to `self.to_f.floor(ndigits)`.
  #
  # Related: #ceil, Float#floor.
  #
  def floor: () -> Integer
           | (Integer digits) -> Numeric

  # <!--
  #   rdoc-file=numeric.c
  #   - i -> complex
  # -->
  # Returns `Complex(0, self)`:
  #
  #     2.i              # => (0+2i)
  #     -2.i             # => (0-2i)
  #     2.0.i            # => (0+2.0i)
  #     Rational(1, 2).i # => (0+(1/2)*i)
  #     Complex(3, 4).i  # Raises NoMethodError.
  #
  def i: () -> Complex

  # <!--
  #   rdoc-file=numeric.rb
  #   - imag -> 0
  # -->
  # Returns zero.
  #
  def imaginary: () -> 0

  # <!--
  #   rdoc-file=numeric.rb
  #   - imag()
  # -->
  #
  alias imag imaginary

  # <!--
  #   rdoc-file=numeric.rb
  #   - infinite? -> -1, 1, or nil
  # -->
  # Returns `nil`, -1, or 1 depending on whether `self` is finite, `-Infinity`, or
  # `+Infinity`.
  #
  def infinite?: () -> Integer?

  # <!--
  #   rdoc-file=numeric.rb
  #   - integer? -> true or false
  # -->
  # Returns `true` if `self` is an Integer.
  #
  #     1.0.integer? # => false
  #     1.integer?   # => true
  #
  def integer?: () -> bool

  # <!-- rdoc-file=numeric.c -->
  # Returns the absolute value of `self`.
  #
  #     12.abs        #=> 12
  #     (-34.56).abs  #=> 34.56
  #     -34.56.abs    #=> 34.56
  #
  alias magnitude abs

  # <!-- rdoc-file=numeric.c -->
  # Returns `self` modulo `other` as a real number.
  #
  # Of the Core and Standard Library classes, only Rational uses this
  # implementation.
  #
  # For Rational `r` and real number `n`, these expressions are equivalent:
  #
  #     r % n
  #     r-n*(r/n).floor
  #     r.divmod(n)[1]
  #
  # See Numeric#divmod.
  #
  # Examples:
  #
  #     r = Rational(1, 2)    # => (1/2)
  #     r2 = Rational(2, 3)   # => (2/3)
  #     r % r2                # => (1/2)
  #     r % 2                 # => (1/2)
  #     r % 2.0               # => 0.5
  #
  #     r = Rational(301,100) # => (301/100)
  #     r2 = Rational(7,5)    # => (7/5)
  #     r % r2                # => (21/100)
  #     r % -r2               # => (-119/100)
  #     (-r) % r2             # => (119/100)
  #     (-r) %-r2             # => (-21/100)
  #
  def modulo: (Numeric) -> Numeric

  # <!--
  #   rdoc-file=numeric.c
  #   - negative? -> true or false
  # -->
  # Returns `true` if `self` is less than 0, `false` otherwise.
  #
  def negative?: () -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - nonzero?  ->  self or nil
  # -->
  # Returns +self+ if +self+ is not a zero value, +nil+ otherwise;
  #     uses method <tt>zero?</tt> for the evaluation.
  #
  #     The returned +self+ allows the method to be chained:
  #
  #       a = %w[z Bb bB bb BB a aA Aa AA A]
  #       a.sort {|a, b| (a.downcase <=> b.downcase).nonzero? || a <=> b }
  #       # => ["A", "a", "AA", "Aa", "aA", "BB", "Bb", "bB", "bb", "z"]
  #
  #     Of the Core and Standard Library classes,
  #     Integer, Float, Rational, and Complex use this implementation.
  #
  # Related: #zero?
  #
  def nonzero?: () -> self?

  # <!--
  #   rdoc-file=rational.c
  #   - num.numerator  ->  integer
  # -->
  # Returns the numerator.
  #
  def numerator: () -> Numeric

  # <!-- rdoc-file=complex.c -->
  # Returns zero if `self` is positive, Math::PI otherwise.
  #
  alias phase angle

  # <!--
  #   rdoc-file=complex.c
  #   - polar -> array
  # -->
  # Returns array `[self.abs, self.arg]`.
  #
  def polar: () -> [ Numeric, Numeric ]

  # <!--
  #   rdoc-file=numeric.c
  #   - positive? -> true or false
  # -->
  # Returns `true` if `self` is greater than 0, `false` otherwise.
  #
  def positive?: () -> bool

  # <!--
  #   rdoc-file=rational.c
  #   - num.quo(int_or_rat)   ->  rat
  #   - num.quo(flo)          ->  flo
  # -->
  # Returns the most exact division (rational for integers, float for floats).
  #
  def quo: (Numeric) -> Numeric

  # <!--
  #   rdoc-file=numeric.rb
  #   - real -> self
  # -->
  # Returns `self`.
  #
  def real: () -> self

  # <!--
  #   rdoc-file=numeric.rb
  #   - real? -> true or false
  # -->
  # Returns `true` if `self` is a real number (i.e. not Complex).
  #
  def real?: () -> true

  # <!-- rdoc-file=complex.c -->
  # Returns array `[self, 0]`.
  #
  def rect: () -> [ Numeric, Numeric ]

  # <!--
  #   rdoc-file=complex.c
  #   - rect -> array
  # -->
  # Returns array `[self, 0]`.
  #
  alias rectangular rect

  # <!--
  #   rdoc-file=numeric.c
  #   - remainder(other) -> real_number
  # -->
  # Returns the remainder after dividing `self` by `other`.
  #
  # Of the Core and Standard Library classes, only Float and Rational use this
  # implementation.
  #
  # Examples:
  #
  #     11.0.remainder(4)              # => 3.0
  #     11.0.remainder(-4)             # => 3.0
  #     -11.0.remainder(4)             # => -3.0
  #     -11.0.remainder(-4)            # => -3.0
  #
  #     12.0.remainder(4)              # => 0.0
  #     12.0.remainder(-4)             # => 0.0
  #     -12.0.remainder(4)             # => -0.0
  #     -12.0.remainder(-4)            # => -0.0
  #
  #     13.0.remainder(4.0)            # => 1.0
  #     13.0.remainder(Rational(4, 1)) # => 1.0
  #
  #     Rational(13, 1).remainder(4)   # => (1/1)
  #     Rational(13, 1).remainder(-4)  # => (1/1)
  #     Rational(-13, 1).remainder(4)  # => (-1/1)
  #     Rational(-13, 1).remainder(-4) # => (-1/1)
  #
  def remainder: (Numeric) -> Numeric

  # <!--
  #   rdoc-file=numeric.c
  #   - round(digits = 0) -> integer or float
  # -->
  # Returns `self` rounded to the nearest value with a precision of `digits`
  # decimal digits.
  #
  # Numeric implements this by converting `self` to a Float and invoking
  # Float#round.
  #
  def round: () -> Integer
           | (Integer digits) -> Numeric

  # <!--
  #   rdoc-file=numeric.c
  #   - step(to = nil, by = 1) {|n| ... } ->  self
  #   - step(to = nil, by = 1)            ->  enumerator
  #   - step(to = nil, by: 1) {|n| ... }  ->  self
  #   - step(to = nil, by: 1)             ->  enumerator
  #   - step(by: 1, to: ) {|n| ... }      ->  self
  #   - step(by: 1, to: )                 ->  enumerator
  #   - step(by: , to: nil) {|n| ... }    ->  self
  #   - step(by: , to: nil)               ->  enumerator
  # -->
  # Generates a sequence of numbers; with a block given, traverses the sequence.
  #
  # Of the Core and Standard Library classes, Integer, Float, and Rational use
  # this implementation.
  #
  # A quick example:
  #
  #     squares = []
  #     1.step(by: 2, to: 10) {|i| squares.push(i*i) }
  #     squares # => [1, 9, 25, 49, 81]
  #
  # The generated sequence:
  #
  # *   Begins with `self`.
  # *   Continues at intervals of `by` (which may not be zero).
  # *   Ends with the last number that is within or equal to `to`; that is, less
  #     than or equal to `to` if `by` is positive, greater than or equal to `to`
  #     if `by` is negative. If `to` is `nil`, the sequence is of infinite length.
  #
  # If a block is given, calls the block with each number in the sequence; returns
  # `self`. If no block is given, returns an Enumerator::ArithmeticSequence.
  #
  # **Keyword Arguments**
  #
  # With keyword arguments `by` and `to`, their values (or defaults) determine the
  # step and limit:
  #
  #     # Both keywords given.
  #     squares = []
  #     4.step(by: 2, to: 10) {|i| squares.push(i*i) }    # => 4
  #     squares # => [16, 36, 64, 100]
  #     cubes = []
  #     3.step(by: -1.5, to: -3) {|i| cubes.push(i*i*i) } # => 3
  #     cubes   # => [27.0, 3.375, 0.0, -3.375, -27.0]
  #     squares = []
  #     1.2.step(by: 0.2, to: 2.0) {|f| squares.push(f*f) }
  #     squares # => [1.44, 1.9599999999999997, 2.5600000000000005, 3.24, 4.0]
  #
  #     squares = []
  #     Rational(6/5).step(by: 0.2, to: 2.0) {|r| squares.push(r*r) }
  #     squares # => [1.0, 1.44, 1.9599999999999997, 2.5600000000000005, 3.24, 4.0]
  #
  #     # Only keyword to given.
  #     squares = []
  #     4.step(to: 10) {|i| squares.push(i*i) }           # => 4
  #     squares # => [16, 25, 36, 49, 64, 81, 100]
  #     # Only by given.
  #
  #     # Only keyword by given
  #     squares = []
  #     4.step(by:2) {|i| squares.push(i*i); break if i > 10 }
  #     squares # => [16, 36, 64, 100, 144]
  #
  #     # No block given.
  #     e = 3.step(by: -1.5, to: -3) # => (3.step(by: -1.5, to: -3))
  #     e.class                      # => Enumerator::ArithmeticSequence
  #
  # **Positional Arguments**
  #
  # With optional positional arguments `to` and `by`, their values (or defaults)
  # determine the step and limit:
  #
  #     squares = []
  #     4.step(10, 2) {|i| squares.push(i*i) }    # => 4
  #     squares # => [16, 36, 64, 100]
  #     squares = []
  #     4.step(10) {|i| squares.push(i*i) }
  #     squares # => [16, 25, 36, 49, 64, 81, 100]
  #     squares = []
  #     4.step {|i| squares.push(i*i); break if i > 10 }  # => nil
  #     squares # => [16, 25, 36, 49, 64, 81, 100, 121]
  #
  # **Implementation Notes**
  #
  # If all the arguments are integers, the loop operates using an integer counter.
  #
  # If any of the arguments are floating point numbers, all are converted to
  # floats, and the loop is executed *floor(n + n*Float::EPSILON) + 1* times,
  # where *n = (limit - self)/step*.
  #
  def step: (?Numeric limit, ?Numeric step) { (Numeric) -> void } -> self
          | (?Numeric limit, ?Numeric step) -> Enumerator::ArithmeticSequence
          | (?by: Numeric, ?to: Numeric) { (Numeric) -> void } -> self
          | (?by: Numeric, ?to: Numeric) -> Enumerator::ArithmeticSequence

  # <!--
  #   rdoc-file=complex.c
  #   - to_c -> complex
  # -->
  # Returns `self` as a Complex object.
  #
  def to_c: () -> Complex

  # <!--
  #   rdoc-file=numeric.c
  #   - to_int -> integer
  # -->
  # Returns `self` as an integer; converts using method `to_i` in the derived
  # class.
  #
  # Of the Core and Standard Library classes, only Rational and Complex use this
  # implementation.
  #
  # Examples:
  #
  #     Rational(1, 2).to_int # => 0
  #     Rational(2, 1).to_int # => 2
  #     Complex(2, 0).to_int  # => 2
  #     Complex(2, 1).to_int  # Raises RangeError (non-zero imaginary part)
  #
  def to_int: () -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - truncate(digits = 0) -> integer or float
  # -->
  # Returns `self` truncated (toward zero) to a precision of `digits` decimal
  # digits.
  #
  # Numeric implements this by converting `self` to a Float and invoking
  # Float#truncate.
  #
  def truncate: () -> Integer
              | (Integer ndigits) -> (Integer | Numeric)

  # <!--
  #   rdoc-file=numeric.c
  #   - zero? -> true or false
  # -->
  # Returns `true` if `zero` has a zero value, `false` otherwise.
  #
  # Of the Core and Standard Library classes, only Rational and Complex use this
  # implementation.
  #
  def zero?: () -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - clone(freeze: true) -> self
  # -->
  # Returns `self`.
  #
  # Raises an exception if the value for `freeze` is neither `true` nor `nil`.
  #
  # Related: Numeric#dup.
  #
  def clone: (?freeze: true?) -> self
end