# <!-- rdoc-file=numeric.c -->
# An Integer object represents an integer value.
#
# You can create an Integer object explicitly with:
#
# *   An [integer literal](rdoc-ref:syntax/literals.rdoc@Integer+Literals).
#
# You can convert certain objects to Integers with:
#
# *   Method #Integer.
#
# An attempt to add a singleton method to an instance of this class causes an
# exception to be raised.
#
# ## What's Here
#
# First, what's elsewhere. Class Integer:
#
# *   Inherits from [class Numeric](rdoc-ref:Numeric@What-27s+Here) and [class
#     Object](rdoc-ref:Object@What-27s+Here).
# *   Includes [module Comparable](rdoc-ref:Comparable@What-27s+Here).
#
# Here, class Integer provides methods for:
#
# *   [Querying](rdoc-ref:Integer@Querying)
# *   [Comparing](rdoc-ref:Integer@Comparing)
# *   [Converting](rdoc-ref:Integer@Converting)
# *   [Other](rdoc-ref:Integer@Other)
#
# ### Querying
#
# *   #allbits?: Returns whether all bits in `self` are set.
# *   #anybits?: Returns whether any bits in `self` are set.
# *   #nobits?: Returns whether no bits in `self` are set.
#
# ### Comparing
#
# *   #<: Returns whether `self` is less than the given value.
# *   #<=: Returns whether `self` is less than or equal to the given value.
# *   #<=>: Returns a number indicating whether `self` is less than, equal to,
#     or greater than the given value.
# *   #== (aliased as #===): Returns whether `self` is equal to the given
#         value.
#
# *   #>: Returns whether `self` is greater than the given value.
# *   #>=: Returns whether `self` is greater than or equal to the given value.
#
# ### Converting
#
# *   ::sqrt: Returns the integer square root of the given value.
# *   ::try_convert: Returns the given value converted to an Integer.
# *   #% (aliased as #modulo): Returns `self` modulo the given value.
# *   #&: Returns the bitwise AND of `self` and the given value.
# *   #*: Returns the product of `self` and the given value.
# *   #**: Returns the value of `self` raised to the power of the given value.
# *   #+: Returns the sum of `self` and the given value.
# *   #-: Returns the difference of `self` and the given value.
# *   #/: Returns the quotient of `self` and the given value.
# *   #<<: Returns the value of `self` after a leftward bit-shift.
# *   #>>: Returns the value of `self` after a rightward bit-shift.
# *   #[]: Returns a slice of bits from `self`.
# *   #^: Returns the bitwise EXCLUSIVE OR of `self` and the given value.
# *   #|: Returns the bitwise OR of `self` and the given value.
# *   #ceil: Returns the smallest number greater than or equal to `self`.
# *   #chr: Returns a 1-character string containing the character represented by
#     the value of `self`.
# *   #digits: Returns an array of integers representing the base-radix digits
#     of `self`.
# *   #div: Returns the integer result of dividing `self` by the given value.
# *   #divmod: Returns a 2-element array containing the quotient and remainder
#     results of dividing `self` by the given value.
# *   #fdiv: Returns the Float result of dividing `self` by the given value.
# *   #floor: Returns the greatest number smaller than or equal to `self`.
# *   #pow: Returns the modular exponentiation of `self`.
# *   #pred: Returns the integer predecessor of `self`.
# *   #remainder: Returns the remainder after dividing `self` by the given
#     value.
# *   #round: Returns `self` rounded to the nearest value with the given
#     precision.
# *   #succ (aliased as #next): Returns the integer successor of `self`.
# *   #to_f: Returns `self` converted to a Float.
# *   #to_s (aliased as #inspect): Returns a string containing the place-value
#     representation of `self` in the given radix.
# *   #truncate: Returns `self` truncated to the given precision.
#
# ### Other
#
# *   #downto: Calls the given block with each integer value from `self` down to
#     the given value.
# *   #times: Calls the given block `self` times with each integer in
#     `(0..self-1)`.
# *   #upto: Calls the given block with each integer value from `self` up to the
#     given value.
#
class Integer < Numeric
  # <!--
  #   rdoc-file=numeric.c
  #   - Integer.sqrt(numeric) -> integer
  # -->
  # Returns the integer square root of the non-negative integer `n`, which is the
  # largest non-negative integer less than or equal to the square root of
  # `numeric`.
  #
  #     Integer.sqrt(0)       # => 0
  #     Integer.sqrt(1)       # => 1
  #     Integer.sqrt(24)      # => 4
  #     Integer.sqrt(25)      # => 5
  #     Integer.sqrt(10**400) # => 10**200
  #
  # If `numeric` is not an Integer, it is converted to an Integer:
  #
  #     Integer.sqrt(Complex(4, 0))  # => 2
  #     Integer.sqrt(Rational(4, 1)) # => 2
  #     Integer.sqrt(4.0)            # => 2
  #     Integer.sqrt(3.14159)        # => 1
  #
  # This method is equivalent to `Math.sqrt(numeric).floor`, except that the
  # result of the latter code may differ from the true value due to the limited
  # precision of floating point arithmetic.
  #
  #     Integer.sqrt(10**46)    # => 100000000000000000000000
  #     Math.sqrt(10**46).floor # => 99999999999999991611392
  #
  # Raises an exception if `numeric` is negative.
  #
  def self.sqrt: (int n) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - Integer.try_convert(object) -> object, integer, or nil
  # -->
  # If `object` is an Integer object, returns `object`.
  #     Integer.try_convert(1) # => 1
  #
  # Otherwise if `object` responds to `:to_int`, calls `object.to_int` and returns
  # the result.
  #     Integer.try_convert(1.25) # => 1
  #
  # Returns `nil` if `object` does not respond to `:to_int`
  #     Integer.try_convert([]) # => nil
  #
  # Raises an exception unless `object.to_int` returns an Integer object.
  #
  def self.try_convert: (int) -> Integer
                      | (untyped) -> Integer?

  # <!--
  #   rdoc-file=numeric.c
  #   - self % other -> real_number
  # -->
  # Returns `self` modulo `other` as a real number.
  #
  # For integer `n` and real number `r`, these expressions are equivalent:
  #
  #     n % r
  #     n-r*(n/r).floor
  #     n.divmod(r)[1]
  #
  # See Numeric#divmod.
  #
  # Examples:
  #
  #     10 % 2              # => 0
  #     10 % 3              # => 1
  #     10 % 4              # => 2
  #
  #     10 % -2             # => 0
  #     10 % -3             # => -2
  #     10 % -4             # => -2
  #
  #     10 % 3.0            # => 1.0
  #     10 % Rational(3, 1) # => (1/1)
  #
  def %: (Float) -> Float
       | (Rational) -> Rational
       | (Integer) -> Integer
       | (Numeric) -> Numeric

  # <!--
  #   rdoc-file=numeric.c
  #   - self & other ->  integer
  # -->
  # Bitwise AND; each bit in the result is 1 if both corresponding bits in `self`
  # and `other` are 1, 0 otherwise:
  #
  #     "%04b" % (0b0101 & 0b0110) # => "0100"
  #
  # Raises an exception if `other` is not an Integer.
  #
  # Related: Integer#| (bitwise OR), Integer#^ (bitwise EXCLUSIVE OR).
  #
  def &: (Integer) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - self * numeric -> numeric_result
  # -->
  # Performs multiplication:
  #
  #     4 * 2              # => 8
  #     4 * -2             # => -8
  #     -4 * 2             # => -8
  #     4 * 2.0            # => 8.0
  #     4 * Rational(1, 3) # => (4/3)
  #     4 * Complex(2, 0)  # => (8+0i)
  #
  def *: (Float) -> Float
       | (Rational) -> Rational
       | (Complex) -> Complex
       | (Integer) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - self ** numeric -> numeric_result
  # -->
  # Raises `self` to the power of `numeric`:
  #
  #     2 ** 3              # => 8
  #     2 ** -3             # => (1/8)
  #     -2 ** 3             # => -8
  #     -2 ** -3            # => (-1/8)
  #     2 ** 3.3            # => 9.849155306759329
  #     2 ** Rational(3, 1) # => (8/1)
  #     2 ** Complex(3, 0)  # => (8+0i)
  #
  def **: (Integer) -> Numeric
        | (Float) -> Numeric
        | (Rational) -> Numeric
        | (Complex) -> Complex

  # <!--
  #   rdoc-file=numeric.c
  #   - self + numeric -> numeric_result
  # -->
  # Performs addition:
  #
  #     2 + 2              # => 4
  #     -2 + 2             # => 0
  #     -2 + -2            # => -4
  #     2 + 2.0            # => 4.0
  #     2 + Rational(2, 1) # => (4/1)
  #     2 + Complex(2, 0)  # => (4+0i)
  #
  def +: (Integer) -> Integer
       | (Float) -> Float
       | (Rational) -> Rational
       | (Complex) -> Complex

  def +@: () -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - self - numeric -> numeric_result
  # -->
  # Performs subtraction:
  #
  #     4 - 2              # => 2
  #     -4 - 2             # => -6
  #     -4 - -2            # => -2
  #     4 - 2.0            # => 2.0
  #     4 - Rational(2, 1) # => (2/1)
  #     4 - Complex(2, 0)  # => (2+0i)
  #
  def -: (Integer) -> Integer
       | (Float) -> Float
       | (Rational) -> Rational
       | (Complex) -> Complex

  # <!--
  #   rdoc-file=numeric.rb
  #   - -int -> integer
  # -->
  # Returns `self`, negated.
  #
  def -@: () -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - self / numeric -> numeric_result
  # -->
  # Performs division; for integer `numeric`, truncates the result to an integer:
  #
  #      4 / 3              # => 1
  #      4 / -3             # => -2
  #      -4 / 3             # => -2
  #      -4 / -3            # => 1
  #
  #     For other +numeric+, returns non-integer result:
  #
  #      4 / 3.0            # => 1.3333333333333333
  #      4 / Rational(3, 1) # => (4/3)
  #      4 / Complex(3, 0)  # => ((4/3)+0i)
  #
  def /: (Integer) -> Integer
       | (Float) -> Float
       | (Rational) -> Rational
       | (Complex) -> Complex

  # <!--
  #   rdoc-file=numeric.c
  #   - self < other -> true or false
  # -->
  # Returns `true` if the value of `self` is less than that of `other`:
  #
  #       1 < 0              # => false
  #       1 < 1              # => false
  #       1 < 2              # => true
  #       1 < 0.5            # => false
  #       1 < Rational(1, 2) # => false
  #
  #     Raises an exception if the comparison cannot be made.
  #
  def <: (Numeric) -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - self << count -> integer
  # -->
  # Returns `self` with bits shifted `count` positions to the left, or to the
  # right if `count` is negative:
  #
  #     n = 0b11110000
  #     "%08b" % (n << 1)  # => "111100000"
  #     "%08b" % (n << 3)  # => "11110000000"
  #     "%08b" % (n << -1) # => "01111000"
  #     "%08b" % (n << -3) # => "00011110"
  #
  # Related: Integer#>>.
  #
  def <<: (int) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - self <= real -> true or false
  # -->
  # Returns `true` if the value of `self` is less than or equal to that of
  # `other`:
  #
  #     1 <= 0              # => false
  #     1 <= 1              # => true
  #     1 <= 2              # => true
  #     1 <= 0.5            # => false
  #     1 <= Rational(1, 2) # => false
  #
  # Raises an exception if the comparison cannot be made.
  #
  def <=: (Numeric) -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - self <=> other  ->  -1, 0, +1, or nil
  # -->
  # Returns:
  #
  # *   -1, if `self` is less than `other`.
  # *   0, if `self` is equal to `other`.
  # *   1, if `self` is greater then `other`.
  # *   `nil`, if `self` and `other` are incomparable.
  #
  # Examples:
  #
  #     1 <=> 2              # => -1
  #     1 <=> 1              # => 0
  #     1 <=> 0              # => 1
  #     1 <=> 'foo'          # => nil
  #
  #     1 <=> 1.0            # => 0
  #     1 <=> Rational(1, 1) # => 0
  #     1 <=> Complex(1, 0)  # => 0
  #
  # This method is the basis for comparisons in module Comparable.
  #
  def <=>: (Integer | Rational) -> Integer
         | (untyped) -> Integer?

  # <!-- rdoc-file=numeric.c -->
  # Returns `true` if `self` is numerically equal to `other`; `false` otherwise.
  #
  #     1 == 2     #=> false
  #     1 == 1.0   #=> true
  #
  # Related: Integer#eql? (requires `other` to be an Integer).
  #
  def ==: (untyped) -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - self == other -> true or false
  # -->
  # Returns `true` if `self` is numerically equal to `other`; `false` otherwise.
  #
  #     1 == 2     #=> false
  #     1 == 1.0   #=> true
  #
  # Related: Integer#eql? (requires `other` to be an Integer).
  #
  def ===: (untyped) -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - self > other -> true or false
  # -->
  # Returns `true` if the value of `self` is greater than that of `other`:
  #
  #       1 > 0              # => true
  #       1 > 1              # => false
  #       1 > 2              # => false
  #       1 > 0.5            # => true
  #       1 > Rational(1, 2) # => true
  #
  #     Raises an exception if the comparison cannot be made.
  #
  def >: (Numeric) -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - self >= real -> true or false
  # -->
  # Returns `true` if the value of `self` is greater than or equal to that of
  # `other`:
  #
  #     1 >= 0              # => true
  #     1 >= 1              # => true
  #     1 >= 2              # => false
  #     1 >= 0.5            # => true
  #     1 >= Rational(1, 2) # => true
  #
  # Raises an exception if the comparison cannot be made.
  #
  def >=: (Numeric) -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - self >> count -> integer
  # -->
  # Returns `self` with bits shifted `count` positions to the right, or to the
  # left if `count` is negative:
  #
  #     n = 0b11110000
  #     "%08b" % (n >> 1)  # => "01111000"
  #     "%08b" % (n >> 3)  # => "00011110"
  #     "%08b" % (n >> -1) # => "111100000"
  #     "%08b" % (n >> -3) # => "11110000000"
  #
  # Related: Integer#<<.
  #
  def >>: (int) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - self[offset]    -> 0 or 1
  #   - self[offset, size] -> integer
  #   - self[range] -> integer
  # -->
  # Returns a slice of bits from `self`.
  #
  # With argument `offset`, returns the bit at the given offset, where offset 0
  # refers to the least significant bit:
  #
  #     n = 0b10 # => 2
  #     n[0]     # => 0
  #     n[1]     # => 1
  #     n[2]     # => 0
  #     n[3]     # => 0
  #
  # In principle, `n[i]` is equivalent to `(n >> i) & 1`. Thus, negative index
  # always returns zero:
  #
  #     255[-1] # => 0
  #
  # With arguments `offset` and `size`, returns `size` bits from `self`, beginning
  # at `offset` and including bits of greater significance:
  #
  #     n = 0b111000       # => 56
  #     "%010b" % n[0, 10] # => "0000111000"
  #     "%010b" % n[4, 10] # => "0000000011"
  #
  # With argument `range`, returns `range.size` bits from `self`, beginning at
  # `range.begin` and including bits of greater significance:
  #
  #     n = 0b111000      # => 56
  #     "%010b" % n[0..9] # => "0000111000"
  #     "%010b" % n[4..9] # => "0000000011"
  #
  # Raises an exception if the slice cannot be constructed.
  #
  def []: (int) -> Integer
        | (int i, int len) -> Integer
        | (Range[int]) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - self ^ other -> integer
  # -->
  # Bitwise EXCLUSIVE OR; each bit in the result is 1 if the corresponding bits in
  # `self` and `other` are different, 0 otherwise:
  #
  #     "%04b" % (0b0101 ^ 0b0110) # => "0011"
  #
  # Raises an exception if `other` is not an Integer.
  #
  # Related: Integer#& (bitwise AND), Integer#| (bitwise OR).
  #
  def ^: (Integer) -> Integer

  # <!--
  #   rdoc-file=numeric.rb
  #   - abs -> integer
  # -->
  # Returns the absolute value of `self`.
  #
  #     (-12345).abs # => 12345
  #     -12345.abs   # => 12345
  #     12345.abs    # => 12345
  #
  def abs: () -> Integer

  def abs2: () -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - allbits?(mask) -> true or false
  # -->
  # Returns `true` if all bits that are set (=1) in `mask` are also set in `self`;
  # returns `false` otherwise.
  #
  # Example values:
  #
  #     0b1010101  self
  #     0b1010100  mask
  #     0b1010100  self & mask
  #          true  self.allbits?(mask)
  #
  #     0b1010100  self
  #     0b1010101  mask
  #     0b1010100  self & mask
  #         false  self.allbits?(mask)
  #
  # Related: Integer#anybits?, Integer#nobits?.
  #
  def allbits?: (int mask) -> bool

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

  # <!--
  #   rdoc-file=numeric.c
  #   - anybits?(mask) -> true or false
  # -->
  # Returns `true` if any bit that is set (=1) in `mask` is also set in `self`;
  # returns `false` otherwise.
  #
  # Example values:
  #
  #     0b10000010  self
  #     0b11111111  mask
  #     0b10000010  self & mask
  #           true  self.anybits?(mask)
  #
  #     0b00000000  self
  #     0b11111111  mask
  #     0b00000000  self & mask
  #          false  self.anybits?(mask)
  #
  # Related: Integer#allbits?, Integer#nobits?.
  #
  def anybits?: (int mask) -> bool

  alias arg angle

  # <!--
  #   rdoc-file=numeric.rb
  #   - bit_length -> integer
  # -->
  # Returns the number of bits of the value of `self`, which is the bit position
  # of the highest-order bit that is different from the sign bit (where the least
  # significant bit has bit position 1). If there is no such bit (zero or minus
  # one), returns zero.
  #
  # This method returns `ceil(log2(self < 0 ? -self : self + 1))`>.
  #
  #     (-2**1000-1).bit_length   # => 1001
  #     (-2**1000).bit_length     # => 1000
  #     (-2**1000+1).bit_length   # => 1000
  #     (-2**12-1).bit_length     # => 13
  #     (-2**12).bit_length       # => 12
  #     (-2**12+1).bit_length     # => 12
  #     -0x101.bit_length         # => 9
  #     -0x100.bit_length         # => 8
  #     -0xff.bit_length          # => 8
  #     -2.bit_length             # => 1
  #     -1.bit_length             # => 0
  #     0.bit_length              # => 0
  #     1.bit_length              # => 1
  #     0xff.bit_length           # => 8
  #     0x100.bit_length          # => 9
  #     (2**12-1).bit_length      # => 12
  #     (2**12).bit_length        # => 13
  #     (2**12+1).bit_length      # => 13
  #     (2**1000-1).bit_length    # => 1000
  #     (2**1000).bit_length      # => 1001
  #     (2**1000+1).bit_length    # => 1001
  #
  # For Integer *n*, this method can be used to detect overflow in Array#pack:
  #
  #     if n.bit_length < 32
  #       [n].pack('l') # No overflow.
  #     else
  #       raise 'Overflow'
  #     end
  #
  def bit_length: () -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - ceil(ndigits = 0) -> integer
  # -->
  # Returns an integer that is a "ceiling" value for `self`,
  # as specified by the given `ndigits`,
  # which must be an
  # [integer-convertible
  # object](rdoc-ref:implicit_conversion.rdoc@Integer-Convertible+Objects).
  # *   When `self` is zero, returns zero (regardless of the value of `ndigits`):
  #         0.ceil(2)  # => 0
  # 0.ceil(-2) # => 0
  #
  # *   When `self` is non-zero and `ndigits` is non-negative, returns `self`:
  #         555.ceil     # => 555
  # 555.ceil(50) # => 555
  #
  # *   When `self` is non-zero and `ndigits` is negative,
  #      returns a value based on a computed granularity:
  #     *   The granularity is `10 ** ndigits.abs`.
  #     *   The returned value is the smallest multiple of the granularity
  #          that is greater than or equal to `self`.
  #     Examples with positive `self`:
  # ndigits|Granularity|1234.ceil(ndigits)
  # -------|-----------|------------------
  #      -1|         10|              1240
  #      -2|        100|              1300
  #      -3|       1000|              2000
  #      -4|      10000|             10000
  #      -5|     100000|            100000
  #     Examples with negative `self`:
  # ndigits|Granularity|-1234.ceil(ndigits)
  # -------|-----------|-------------------
  #      -1|         10|              -1230
  #      -2|        100|              -1200
  #      -3|       1000|              -1000
  #      -4|      10000|                  0
  #      -5|     100000|                  0
  # Related: Integer#floor.
  #
  def ceil: () -> Integer
          | (int digits) -> (Integer | Float)

  # <!--
  #   rdoc-file=numeric.rb
  #   - ceildiv(numeric) -> integer
  # -->
  # Returns the result of division `self` by `numeric`. rounded up to the nearest
  # integer.
  #
  #     3.ceildiv(3)   # => 1
  #     4.ceildiv(3)   # => 2
  #
  #     4.ceildiv(-3)  # => -1
  #     -4.ceildiv(3)  # => -1
  #     -4.ceildiv(-3) # => 2
  #
  #     3.ceildiv(1.2) # => 3
  #
  def ceildiv: (Numeric other) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - chr           -> string
  #   - chr(encoding) -> string
  # -->
  # Returns a 1-character string containing the character represented by the value
  # of `self`, according to the given `encoding`.
  #
  #     65.chr                   # => "A"
  #     0.chr                    # => "\x00"
  #     255.chr                  # => "\xFF"
  #     string = 255.chr(Encoding::UTF_8)
  #     string.encoding          # => Encoding::UTF_8
  #
  # Raises an exception if `self` is negative.
  #
  # Related: Integer#ord.
  #
  def chr: (?encoding) -> String

  # <!--
  #   rdoc-file=bignum.c
  #   - int.coerce(numeric)  ->  array
  # -->
  # Returns an array with both a `numeric` and a `int` represented as Integer
  # objects or Float objects.
  #
  # This is achieved by converting `numeric` to an Integer or a Float.
  #
  # A TypeError is raised if the `numeric` is not an Integer or a Float type.
  #
  #     (0x3FFFFFFFFFFFFFFF+1).coerce(42)   #=> [42, 4611686018427387904]
  #
  def coerce: (Numeric) -> [ Numeric, Numeric ]

  def conj: () -> Integer

  def conjugate: () -> Integer

  # <!--
  #   rdoc-file=numeric.rb
  #   - denominator -> 1
  # -->
  # Returns `1`.
  #
  def denominator: () -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - digits(base = 10) -> array_of_integers
  # -->
  # Returns an array of integers representing the `base`-radix digits of `self`;
  # the first element of the array represents the least significant digit:
  #
  #     12345.digits      # => [5, 4, 3, 2, 1]
  #     12345.digits(7)   # => [4, 6, 6, 0, 5]
  #     12345.digits(100) # => [45, 23, 1]
  #
  # Raises an exception if `self` is negative or `base` is less than 2.
  #
  def digits: (?int base) -> ::Array[Integer]

  # <!--
  #   rdoc-file=numeric.c
  #   - div(numeric)  -> integer
  # -->
  # Performs integer division; returns the integer result of dividing `self` by
  # `numeric`:
  #
  #     4.div(3)              # => 1
  #     4.div(-3)             # => -2
  #     -4.div(3)             # => -2
  #     -4.div(-3)            # => 1
  #     4.div(3.0)            # => 1
  #     4.div(Rational(3, 1)) # => 1
  #
  # Raises an exception if `numeric` does not have method `div`.
  #
  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
  #
  # Examples:
  #
  #     11.divmod(4)              # => [2, 3]
  #     11.divmod(-4)             # => [-3, -1]
  #     -11.divmod(4)             # => [-3, 1]
  #     -11.divmod(-4)            # => [2, -3]
  #
  #     12.divmod(4)              # => [3, 0]
  #     12.divmod(-4)             # => [-3, 0]
  #     -12.divmod(4)             # => [-3, 0]
  #     -12.divmod(-4)            # => [3, 0]
  #
  #     13.divmod(4.0)            # => [3, 1.0]
  #     13.divmod(Rational(4, 1)) # => [3, (1/1)]
  #
  def divmod: (Integer) -> [ Integer, Integer ]
            | (Float) -> [ Integer, Float ]
            | (Rational) -> [ Integer, Rational ]
            | (Numeric) -> [ Numeric, Numeric ]

  # <!--
  #   rdoc-file=numeric.c
  #   - downto(limit) {|i| ... } -> self
  #   - downto(limit)            ->  enumerator
  # -->
  # Calls the given block with each integer value from `self` down to `limit`;
  # returns `self`:
  #
  #     a = []
  #     10.downto(5) {|i| a << i }              # => 10
  #     a                                       # => [10, 9, 8, 7, 6, 5]
  #     a = []
  #     0.downto(-5) {|i| a << i }              # => 0
  #     a                                       # => [0, -1, -2, -3, -4, -5]
  #     4.downto(5) {|i| fail 'Cannot happen' } # => 4
  #
  # With no block given, returns an Enumerator.
  #
  def downto: (Numeric limit) { (Integer) -> void } -> Integer
            | (Numeric limit) -> ::Enumerator[Integer, self]

  def dup: () -> self

  def eql?: (untyped) -> bool

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

  # <!--
  #   rdoc-file=numeric.c
  #   - fdiv(numeric) -> float
  # -->
  # Returns the Float result of dividing `self` by `numeric`:
  #
  #     4.fdiv(2)      # => 2.0
  #     4.fdiv(-2)      # => -2.0
  #     -4.fdiv(2)      # => -2.0
  #     4.fdiv(2.0)      # => 2.0
  #     4.fdiv(Rational(3, 4))      # => 5.333333333333333
  #
  # Raises an exception if `numeric` cannot be converted to a Float.
  #
  def fdiv: (Numeric) -> Float

  def finite?: () -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - floor(ndigits = 0) -> integer
  # -->
  # Returns an integer that is a "floor" value for `self`,
  # as specified by the given `ndigits`,
  # which must be an
  # [integer-convertible
  # object](rdoc-ref:implicit_conversion.rdoc@Integer-Convertible+Objects).
  # *   When `self` is zero, returns zero (regardless of the value of `ndigits`):
  #         0.floor(2)  # => 0
  # 0.floor(-2) # => 0
  #
  # *   When `self` is non-zero and `ndigits` is non-negative, returns `self`:
  #         555.floor     # => 555
  # 555.floor(50) # => 555
  #
  # *   When `self` is non-zero and `ndigits` is negative,
  #      returns a value based on a computed granularity:
  #     *   The granularity is `10 ** ndigits.abs`.
  #     *   The returned value is the largest multiple of the granularity
  #          that is less than or equal to `self`.
  #     Examples with positive `self`:
  # ndigits|Granularity|1234.floor(ndigits)
  # -------|-----------|-------------------
  #      -1|         10|               1230
  #      -2|        100|               1200
  #      -3|       1000|               1000
  #      -4|      10000|                  0
  #      -5|     100000|                  0
  #     Examples with negative `self`:
  # ndigits|Granularity|-1234.floor(ndigits)
  # -------|-----------|--------------------
  #      -1|         10|               -1240
  #      -2|        100|               -1300
  #      -3|       1000|               -2000
  #      -4|      10000|              -10000
  #      -5|     100000|             -100000
  # Related: Integer#ceil.
  #
  def floor: (?int digits) -> Integer

  # <!--
  #   rdoc-file=rational.c
  #   - int.gcd(other_int)  ->  integer
  # -->
  # Returns the greatest common divisor of the two integers. The result is always
  # positive. 0.gcd(x) and x.gcd(0) return x.abs.
  #
  #     36.gcd(60)                  #=> 12
  #     2.gcd(2)                    #=> 2
  #     3.gcd(-7)                   #=> 1
  #     ((1<<31)-1).gcd((1<<61)-1)  #=> 1
  #
  def gcd: (Integer) -> Integer

  # <!--
  #   rdoc-file=rational.c
  #   - int.gcdlcm(other_int)  ->  array
  # -->
  # Returns an array with the greatest common divisor and the least common
  # multiple of the two integers, [gcd, lcm].
  #
  #     36.gcdlcm(60)                  #=> [12, 180]
  #     2.gcdlcm(2)                    #=> [2, 2]
  #     3.gcdlcm(-7)                   #=> [1, 21]
  #     ((1<<31)-1).gcdlcm((1<<61)-1)  #=> [1, 4951760154835678088235319297]
  #
  def gcdlcm: (Integer) -> [ Integer, Integer ]

  def i: () -> Complex

  def imag: () -> Integer

  def imaginary: () -> Integer

  def infinite?: () -> Integer?

  # <!-- rdoc-file=numeric.c -->
  # Returns a string containing the place-value representation of `self` in radix
  # `base` (in 2..36).
  #
  #     12345.to_s               # => "12345"
  #     12345.to_s(2)            # => "11000000111001"
  #     12345.to_s(8)            # => "30071"
  #     12345.to_s(10)           # => "12345"
  #     12345.to_s(16)           # => "3039"
  #     12345.to_s(36)           # => "9ix"
  #     78546939656932.to_s(36)  # => "rubyrules"
  #
  # Raises an exception if `base` is out of range.
  #
  alias inspect to_s

  # <!--
  #   rdoc-file=numeric.rb
  #   - integer? -> true
  # -->
  # Since `self` is already an Integer, always returns `true`.
  #
  def integer?: () -> true

  # <!--
  #   rdoc-file=rational.c
  #   - int.lcm(other_int)  ->  integer
  # -->
  # Returns the least common multiple of the two integers. The result is always
  # positive. 0.lcm(x) and x.lcm(0) return zero.
  #
  #     36.lcm(60)                  #=> 180
  #     2.lcm(2)                    #=> 2
  #     3.lcm(-7)                   #=> 21
  #     ((1<<31)-1).lcm((1<<61)-1)  #=> 4951760154835678088235319297
  #
  def lcm: (Integer) -> Integer

  # <!--
  #   rdoc-file=numeric.rb
  #   - magnitude()
  # -->
  #
  def magnitude: () -> Integer

  # <!-- rdoc-file=numeric.c -->
  # Returns `self` modulo `other` as a real number.
  #
  # For integer `n` and real number `r`, these expressions are equivalent:
  #
  #     n % r
  #     n-r*(n/r).floor
  #     n.divmod(r)[1]
  #
  # See Numeric#divmod.
  #
  # Examples:
  #
  #     10 % 2              # => 0
  #     10 % 3              # => 1
  #     10 % 4              # => 2
  #
  #     10 % -2             # => 0
  #     10 % -3             # => -2
  #     10 % -4             # => -2
  #
  #     10 % 3.0            # => 1.0
  #     10 % Rational(3, 1) # => (1/1)
  #
  alias modulo %

  def negative?: () -> bool

  # <!-- rdoc-file=numeric.c -->
  # Returns the successor integer of `self` (equivalent to `self + 1`):
  #
  #     1.succ  #=> 2
  #     -1.succ #=> 0
  #
  # Related: Integer#pred (predecessor value).
  #
  def next: () -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - nobits?(mask) -> true or false
  # -->
  # Returns `true` if no bit that is set (=1) in `mask` is also set in `self`;
  # returns `false` otherwise.
  #
  # Example values:
  #
  #     0b11110000  self
  #     0b00001111  mask
  #     0b00000000  self & mask
  #           true  self.nobits?(mask)
  #
  #     0b00000001  self
  #     0b11111111  mask
  #     0b00000001  self & mask
  #          false  self.nobits?(mask)
  #
  # Related: Integer#allbits?, Integer#anybits?.
  #
  def nobits?: (int mask) -> bool

  def nonzero?: () -> self?

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

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

  # <!--
  #   rdoc-file=numeric.rb
  #   - ord -> self
  # -->
  # Returns `self`; intended for compatibility to character literals in Ruby 1.9.
  #
  def ord: () -> Integer

  alias phase angle

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

  def positive?: () -> bool

  # <!--
  #   rdoc-file=numeric.c
  #   - integer.pow(numeric)           ->  numeric
  #   - integer.pow(integer, integer)  ->  integer
  # -->
  # Returns (modular) exponentiation as:
  #
  #     a.pow(b)     #=> same as a**b
  #     a.pow(b, m)  #=> same as (a**b) % m, but avoids huge temporary values
  #
  def pow: (Integer other) -> (Integer | Rational)
         | (Integer other, Integer modulo) -> Integer
         | (Float) -> (Float | Complex)
         | (Rational) -> (Float | Rational | Complex)
         | (Complex) -> Complex

  # <!--
  #   rdoc-file=numeric.c
  #   - pred -> next_integer
  # -->
  # Returns the predecessor of `self` (equivalent to `self - 1`):
  #
  #     1.pred  #=> 0
  #     -1.pred #=> -2
  #
  # Related: Integer#succ (successor value).
  #
  def pred: () -> Integer

  def quo: (Integer) -> Rational
         | (Float) -> Float
         | (Rational) -> Rational
         | (Complex) -> Complex
         | (Numeric) -> Numeric

  # <!--
  #   rdoc-file=rational.c
  #   - int.rationalize([eps])  ->  rational
  # -->
  # Returns the value as a rational.  The optional argument `eps` is always
  # ignored.
  #
  def rationalize: (?Numeric eps) -> Rational

  def real: () -> self

  def real?: () -> true

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

  alias rectangular rect

  # <!--
  #   rdoc-file=numeric.c
  #   - remainder(other) -> real_number
  # -->
  # Returns the remainder after dividing `self` by `other`.
  #
  # Examples:
  #
  #     11.remainder(4)              # => 3
  #     11.remainder(-4)             # => 3
  #     -11.remainder(4)             # => -3
  #     -11.remainder(-4)            # => -3
  #
  #     12.remainder(4)              # => 0
  #     12.remainder(-4)             # => 0
  #     -12.remainder(4)             # => 0
  #     -12.remainder(-4)            # => 0
  #
  #     13.remainder(4.0)            # => 1.0
  #     13.remainder(Rational(4, 1)) # => (1/1)
  #
  def remainder: (Integer) -> Integer
               | (Float) -> Float
               | (Rational) -> Rational
               | (Numeric) -> Numeric

  # <!--
  #   rdoc-file=numeric.c
  #   - round(ndigits= 0, half: :up) -> integer
  # -->
  # Returns `self` rounded to the nearest value with a precision of `ndigits`
  # decimal digits.
  #
  # When `ndigits` is negative, the returned value has at least `ndigits.abs`
  # trailing zeros:
  #
  #     555.round(-1)      # => 560
  #     555.round(-2)      # => 600
  #     555.round(-3)      # => 1000
  #     -555.round(-2)     # => -600
  #     555.round(-4)      # => 0
  #
  # Returns `self` when `ndigits` is zero or positive.
  #
  #     555.round     # => 555
  #     555.round(1)  # => 555
  #     555.round(50) # => 555
  #
  # If keyword argument `half` is given, and `self` is equidistant from the two
  # candidate  values, the rounding is according to the given `half` value:
  #
  # *   `:up` or `nil`: round away from zero:
  #
  #         25.round(-1, half: :up)      # => 30
  #         (-25).round(-1, half: :up)   # => -30
  #
  # *   `:down`: round toward zero:
  #
  #         25.round(-1, half: :down)    # => 20
  #         (-25).round(-1, half: :down) # => -20
  #
  # *   `:even`: round toward the candidate whose last nonzero digit is even:
  #
  #         25.round(-1, half: :even)    # => 20
  #         15.round(-1, half: :even)    # => 20
  #         (-25).round(-1, half: :even) # => -20
  #
  # Raises and exception if the value for `half` is invalid.
  #
  # Related: Integer#truncate.
  #
  def round: (?half: :up | :down | :even) -> Integer
           | (int digits, ?half: :up | :down | :even) -> (Integer | Float)

  # <!--
  #   rdoc-file=numeric.rb
  #   - size -> integer
  # -->
  # Returns the number of bytes in the machine representation of `self`; the value
  # is system-dependent:
  #
  #     1.size             # => 8
  #     -1.size            # => 8
  #     2147483647.size    # => 8
  #     (256**10 - 1).size # => 10
  #     (256**20 - 1).size # => 20
  #     (256**40 - 1).size # => 40
  #
  def size: () -> Integer

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

  # <!--
  #   rdoc-file=numeric.c
  #   - succ -> next_integer
  # -->
  # Returns the successor integer of `self` (equivalent to `self + 1`):
  #
  #     1.succ  #=> 2
  #     -1.succ #=> 0
  #
  # Related: Integer#pred (predecessor value).
  #
  def succ: () -> Integer

  # <!--
  #   rdoc-file=numeric.rb
  #   - times {|i| ... } -> self
  #   - times            -> enumerator
  # -->
  # Calls the given block `self` times with each integer in `(0..self-1)`:
  #
  #     a = []
  #     5.times {|i| a.push(i) } # => 5
  #     a                        # => [0, 1, 2, 3, 4]
  #
  # With no block given, returns an Enumerator.
  #
  def times: () { (Integer) -> void } -> self
           | () -> ::Enumerator[Integer, self]

  def to_c: () -> Complex

  # <!--
  #   rdoc-file=numeric.c
  #   - to_f -> float
  # -->
  # Converts `self` to a Float:
  #
  #     1.to_f  # => 1.0
  #     -1.to_f # => -1.0
  #
  # If the value of `self` does not fit in a Float, the result is infinity:
  #
  #     (10**400).to_f  # => Infinity
  #     (-10**400).to_f # => -Infinity
  #
  def to_f: () -> Float

  # <!--
  #   rdoc-file=numeric.rb
  #   - to_i -> self
  # -->
  # Returns `self` (which is already an Integer).
  #
  def to_i: () -> Integer

  # <!--
  #   rdoc-file=numeric.rb
  #   - to_int -> self
  # -->
  # Returns `self` (which is already an Integer).
  #
  alias to_int to_i

  # <!--
  #   rdoc-file=rational.c
  #   - int.to_r  ->  rational
  # -->
  # Returns the value as a rational.
  #
  #     1.to_r        #=> (1/1)
  #     (1<<64).to_r  #=> (18446744073709551616/1)
  #
  def to_r: () -> Rational

  # <!--
  #   rdoc-file=numeric.c
  #   - to_s(base = 10)  ->  string
  # -->
  # Returns a string containing the place-value representation of `self` in radix
  # `base` (in 2..36).
  #
  #     12345.to_s               # => "12345"
  #     12345.to_s(2)            # => "11000000111001"
  #     12345.to_s(8)            # => "30071"
  #     12345.to_s(10)           # => "12345"
  #     12345.to_s(16)           # => "3039"
  #     12345.to_s(36)           # => "9ix"
  #     78546939656932.to_s(36)  # => "rubyrules"
  #
  # Raises an exception if `base` is out of range.
  #
  def to_s: () -> String
          | (2) -> String
          | (3) -> String
          | (4) -> String
          | (5) -> String
          | (6) -> String
          | (7) -> String
          | (8) -> String
          | (9) -> String
          | (10) -> String
          | (11) -> String
          | (12) -> String
          | (13) -> String
          | (14) -> String
          | (15) -> String
          | (16) -> String
          | (17) -> String
          | (18) -> String
          | (19) -> String
          | (20) -> String
          | (21) -> String
          | (22) -> String
          | (23) -> String
          | (24) -> String
          | (25) -> String
          | (26) -> String
          | (27) -> String
          | (28) -> String
          | (29) -> String
          | (30) -> String
          | (31) -> String
          | (32) -> String
          | (33) -> String
          | (34) -> String
          | (35) -> String
          | (36) -> String
          | (int base) -> String

  # <!--
  #   rdoc-file=numeric.c
  #   - truncate(ndigits = 0) -> integer
  # -->
  # Returns `self` truncated (toward zero) to a precision of `ndigits` decimal
  # digits.
  #
  # When `ndigits` is negative, the returned value has at least `ndigits.abs`
  # trailing zeros:
  #
  #     555.truncate(-1)  # => 550
  #     555.truncate(-2)  # => 500
  #     -555.truncate(-2) # => -500
  #
  # Returns `self` when `ndigits` is zero or positive.
  #
  #     555.truncate     # => 555
  #     555.truncate(50) # => 555
  #
  # Related: Integer#round.
  #
  def truncate: () -> Integer
              | (int ndigits) -> Integer

  # <!--
  #   rdoc-file=numeric.c
  #   - upto(limit) {|i| ... } -> self
  #   - upto(limit)            ->  enumerator
  # -->
  # Calls the given block with each integer value from `self` up to `limit`;
  # returns `self`:
  #
  #     a = []
  #     5.upto(10) {|i| a << i }              # => 5
  #     a                                     # => [5, 6, 7, 8, 9, 10]
  #     a = []
  #     -5.upto(0) {|i| a << i }              # => -5
  #     a                                     # => [-5, -4, -3, -2, -1, 0]
  #     5.upto(4) {|i| fail 'Cannot happen' } # => 5
  #
  # With no block given, returns an Enumerator.
  #
  def upto: (Numeric limit) { (Integer) -> void } -> Integer
          | (Numeric limit) -> ::Enumerator[Integer, self]

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

  # <!--
  #   rdoc-file=numeric.c
  #   - self | other -> integer
  # -->
  # Bitwise OR; each bit in the result is 1 if either corresponding bit in `self`
  # or `other` is 1, 0 otherwise:
  #
  #     "%04b" % (0b0101 | 0b0110) # => "0111"
  #
  # Raises an exception if `other` is not an Integer.
  #
  # Related: Integer#& (bitwise AND), Integer#^ (bitwise EXCLUSIVE OR).
  #
  def |: (Integer) -> Integer

  # <!--
  #   rdoc-file=numeric.rb
  #   - ~int -> integer
  # -->
  # One's complement: returns the value of `self` with each bit inverted.
  #
  # Because an integer value is conceptually of infinite length, the result acts
  # as if it had an infinite number of one bits to the left. In hex
  # representations, this is displayed as two periods to the left of the digits:
  #
  #     sprintf("%X", ~0x1122334455)    # => "..FEEDDCCBBAA"
  #
  def ~: () -> Integer
end
