diff --git a/core/integer.rbs b/core/integer.rbs
index fd2ef0ae4..c52c8ecb2 100644
--- a/core/integer.rbs
+++ b/core/integer.rbs
@@ -93,6 +93,8 @@
 #     given value.
 #
 class Integer < Numeric
+  # TODO: `undef self.new`
+
   # <!--
   #   rdoc-file=numeric.c
   #   - Integer.sqrt(numeric) -> integer
@@ -421,7 +423,7 @@ class Integer < Numeric
   #
   # Related: Integer#eql? (requires `other` to be an Integer).
   #
-  def ==: (untyped) -> bool
+  def ==: (untyped other) -> bool
 
   # <!--
   #   rdoc-file=numeric.c
@@ -434,7 +436,7 @@ class Integer < Numeric
   #
   # Related: Integer#eql? (requires `other` to be an Integer).
   #
-  def ===: (untyped) -> bool
+  alias === ==
 
   # <!--
   #   rdoc-file=numeric.c
@@ -687,8 +689,7 @@ class Integer < Numeric
   #      -5|     100000|                  0
   # Related: Integer#floor.
   #
-  def ceil: () -> Integer
-          | (int digits) -> (Integer | Float)
+  def ceil: (?int digits) -> Integer
 
   # <!--
   #   rdoc-file=numeric.rb
@@ -726,7 +727,7 @@ class Integer < Numeric
   #
   # Related: Integer#ord.
   #
-  def chr: (?encoding) -> String
+  def chr: (?encoding encoding) -> String
 
   # <!--
   #   rdoc-file=bignum.c
@@ -749,7 +750,7 @@ class Integer < Numeric
   # -->
   # Returns `1`.
   #
-  def denominator: () -> Integer
+  def denominator: () -> 1
 
   # <!--
   #   rdoc-file=numeric.c
@@ -764,7 +765,7 @@ class Integer < Numeric
   #
   # Raises an exception if `self` is negative or `base` is less than 2.
   #
-  def digits: (?int base) -> ::Array[Integer]
+  def digits: (?int base) -> Array[Integer]
 
   # <!--
   #   rdoc-file=numeric.c
@@ -912,7 +913,7 @@ class Integer < Numeric
   #     3.gcd(-7)                   #=> 1
   #     ((1<<31)-1).gcd((1<<61)-1)  #=> 1
   #
-  def gcd: (Integer) -> Integer
+  def gcd: (Integer other_int) -> Integer
 
   # <!--
   #   rdoc-file=rational.c
@@ -926,7 +927,7 @@ class Integer < Numeric
   #     3.gcdlcm(-7)                   #=> [1, 21]
   #     ((1<<31)-1).gcdlcm((1<<61)-1)  #=> [1, 4951760154835678088235319297]
   #
-  def gcdlcm: (Integer) -> [ Integer, Integer ]
+  def gcdlcm: (Integer other_int) -> [ Integer, Integer ]
 
   # <!-- rdoc-file=numeric.c -->
   # Returns a string containing the place-value representation of `self` in radix
@@ -964,14 +965,14 @@ class Integer < Numeric
   #     3.lcm(-7)                   #=> 21
   #     ((1<<31)-1).lcm((1<<61)-1)  #=> 4951760154835678088235319297
   #
-  def lcm: (Integer) -> Integer
+  def lcm: (Integer other_int) -> Integer
 
   # <!--
   #   rdoc-file=numeric.rb
   #   - magnitude()
   # -->
   #
-  def magnitude: () -> Integer
+  alias magnitude abs
 
   # <!-- rdoc-file=numeric.c -->
   # Returns `self` modulo `other` as a real numeric (Integer, Float, or Rational).
@@ -1009,7 +1010,7 @@ class Integer < Numeric
   #
   # Related: Integer#pred (predecessor value).
   #
-  def next: () -> Integer
+  alias next succ
 
   # <!--
   #   rdoc-file=numeric.c
@@ -1040,7 +1041,7 @@ class Integer < Numeric
   # -->
   # Returns `self`.
   #
-  def numerator: () -> Integer
+  def numerator: () -> self
 
   # <!--
   #   rdoc-file=numeric.rb
@@ -1056,7 +1057,7 @@ class Integer < Numeric
   # -->
   # Returns `self`; intended for compatibility to character literals in Ruby 1.9.
   #
-  def ord: () -> Integer
+  def ord: () -> self
 
   def polar: () -> [ Integer, Integer | Float ]
 
@@ -1180,8 +1181,7 @@ class Integer < Numeric
   #
   # Related: Integer#truncate.
   #
-  def round: (?half: :up | :down | :even) -> Integer
-           | (int digits, ?half: :up | :down | :even) -> (Integer | Float)
+  def round: (?int digits, ?half: Numeric::round_mode) -> Integer
 
   # <!--
   #   rdoc-file=numeric.rb
@@ -1226,8 +1226,8 @@ class Integer < Numeric
   #
   # With no block given, returns an Enumerator.
   #
-  def times: () { (Integer) -> void } -> self
-           | () -> ::Enumerator[Integer, self]
+  def times: () { (Integer i) -> void } -> self
+           | () -> Enumerator[Integer, self]
 
   # <!--
   #   rdoc-file=numeric.c
@@ -1251,7 +1251,7 @@ class Integer < Numeric
   # -->
   # Returns `self` (which is already an Integer).
   #
-  def to_i: () -> Integer
+  def to_i: () -> self
 
   # <!--
   #   rdoc-file=numeric.rb
@@ -1289,43 +1289,7 @@ class Integer < Numeric
   #
   # 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
+  def to_s: (?int base) -> String
 
   # <!--
   #   rdoc-file=numeric.c
@@ -1348,8 +1312,7 @@ class Integer < Numeric
   #
   # Related: Integer#round.
   #
-  def truncate: () -> Integer
-              | (int ndigits) -> Integer
+  def truncate: (?int ndigits) -> Integer
 
   # <!--
   #   rdoc-file=numeric.c
diff --git a/core/numeric.rbs b/core/numeric.rbs
index fe6f800cd..f81b4c79d 100644
--- a/core/numeric.rbs
+++ b/core/numeric.rbs
@@ -158,6 +158,9 @@
 class Numeric
   include Comparable
 
+  # The direction you can round
+  type round_mode = :up | :down | :even | 'up' | 'down' | 'even' | string | nil
+
   # <!--
   #   rdoc-file=numeric.c
   #   - self % other -> real_numeric
diff --git a/test/stdlib/Integer_test.rb b/test/stdlib/Integer_test.rb
index 79d4d1c7f..db26b2e1f 100644
--- a/test/stdlib/Integer_test.rb
+++ b/test/stdlib/Integer_test.rb
@@ -1,385 +1,407 @@
-require_relative "test_helper"
+require_relative 'test_helper'
 
-class IntegerTest < StdlibTest
-  target Integer
+class IntegerSingletonTest < Test::Unit::TestCase
+  include TestHelper
+
+  testing 'singleton(Integer)'
 
   def test_sqrt
-    Integer.sqrt(4)
-    Integer.sqrt(4.0)
-    Integer.sqrt(4/1r)
-    Integer.sqrt(ToInt.new)
+    with_int 38 do |int|
+      assert_send_type  '(int) -> Integer',
+                        Integer, :sqrt, int
+    end
   end
 
-  def test_modulo
-    3 % 1
-    3 % 1.1
-    3 % 1.5r
-  end
-
-  def test_bitwise_ops
-    3 & 1
-    1 ^ 3
-  end
-
-  def test_calc
-    3 * 1
-    3 * 1.0
-    3 * (1r/3)
-
-    3 - 1
-    3 - 0.1
-    3 - 1r
-    3 - 1.to_c
-
-    1 ** 1
-    2 ** 2.1
-    3 ** 3r
-    3 ** 10.to_c
-
-    3 + 1
-    3 + 1.0
-    3 + (1r/3)
-    3 + 10.to_c
-
-    3 / 1
-    3 / 1.0
-    3 / (1r/3)
-    30 / 10.to_c
-  end
-
-  def test_compare
-    3 < 1
-    3 < 1.0
-    3 < 1r
-
-    1 > 1
-    1 > 1.0
-    1 > 1r
-
-    1 <= 3
-    1 <= 1.3
-    1 <= 3r
-
-    1 >= 3
-    1 >= 1.3
-    1 >= 3r
-
-    1 <=> 1
-    1 <=> 1.0
-    1 <=> 3r
-
-    3 === 3.0
-    3 === ""
-  end
-
-  def test_shift
-    1 << 30
-    1 << 30.to_f
-    1 << ToInt.new
-
-    1 >> 30
-    1 >> 30.to_f
-    1 >> ToInt.new
-  end
-
-  def test_aref
-    3[0]
-    3[0.3]
-    3[ToInt.new]
-
-    3[1,2]
-    3[1...3]
-  end
-
-  def test_to_s
-    1.to_s
-    1.to_s(2)
-    1.to_s(3)
-    1.to_s(4)
-    1.to_s(5)
-    1.to_s(6)
-    1.to_s(7)
-    1.to_s(8)
-    1.to_s(9)
-    1.to_s(10)
-    1.to_s(11)
-    1.to_s(12)
-    1.to_s(13)
-    1.to_s(14)
-    1.to_s(15)
-    1.to_s(16)
-    1.to_s(17)
-    1.to_s(18)
-    1.to_s(19)
-    1.to_s(20)
-    1.to_s(21)
-    1.to_s(22)
-    1.to_s(23)
-    1.to_s(24)
-    1.to_s(25)
-    1.to_s(26)
-    1.to_s(27)
-    1.to_s(28)
-    1.to_s(29)
-    1.to_s(30)
-    1.to_s(31)
-    1.to_s(32)
-    1.to_s(33)
-    1.to_s(34)
-    1.to_s(35)
-    1.to_s(36)
-    30.to_s(ToInt.new)
-  end
-
-  def test_abs_abs2
-    3.abs
-    3.abs2
+  def test_try_convert
+    with_int 1 do |int|
+      assert_send_type  '(int) -> Integer',
+                        Integer, :try_convert, int
+    end
+
+    with_untyped do |untyped|
+      assert_send_type '(untyped) -> Integer?',
+                       Integer, :try_convert, untyped
+    end
   end
+end
 
-  def test_allbits?
-    1.allbits?(1)
-    2.allbits?(1)
-    3.allbits?(ToInt.new)
+class IntegerInstanceTest < Test::Unit::TestCase
+  include TestHelper
+
+  testing 'Integer'
+
+  def test_op_mod(method: :%)
+    # omit 'todo'
   end
 
-  def test_angle
-    3.angle()
+  def test_op_and
+    # omit 'todo'
   end
 
-  def test_anybits?
-    0xf0.anybits?(0xf)
-    0xf1.anybits?(0xf)
-    0xf1.anybits?(ToInt.new)
+  def test_op_mul
+    # omit 'todo'
+  end
+
+  def test_op_pow
+    # omit 'todo'
+  end
+
+  def test_op_add
+    # omit 'todo'
+  end
+
+  def test_op_sub
+    # omit 'todo'
+  end
+
+  def test_op_uneg
+    assert_send_type  '() -> Integer',
+                      38, :-@
+  end
+
+  def test_op_div
+    # omit 'todo'
+  end
+
+  def test_op_lt
+    # omit 'todo'
+  end
+
+  def test_op_lsh
+    # omit 'todo'
+  end
+
+  def test_op_leq
+    # omit 'todo'
+  end
+
+  def test_op_cmp
+    # omit 'todo'
+  end
+
+  def test_op_eq(method: :==)
+    with_untyped.and 38 do |other|
+      next unless defined? other.== # even for the `===` case heh
+      assert_send_type  '(untyped) -> bool',
+                        38, method, other
+    end
+  end
+
+  def test_op_eqq
+    test_op_eq(method: :===)
+  end
+
+  def test_op_gt
+    # omit 'todo'
+  end
+
+  def test_op_geq
+    # omit 'todo'
+  end
+
+  def test_op_rsh
+    # omit 'todo'
+  end
+
+  def test_op_aref
+    # omit 'todo'
+  end
+
+  def test_op_xor
+    # omit 'todo'
   end
 
-  def test_arg
-    3.arg
+  def test_abs(method: :magnitude)
+    assert_send_type  '() -> Integer',
+                      38, method
+    assert_send_type  '() -> Integer',
+                      -38, method
+  end
+
+  def test_allbits?
+    with_int 1 do |mask|
+      assert_send_type  '(int) -> bool',
+                        0, :allbits?, mask
+      assert_send_type  '(int) -> bool',
+                        1, :allbits?, mask
+    end
+  end
+
+  def test_anybits?
+    with_int 1 do |mask|
+      assert_send_type  '(int) -> bool',
+                        0, :anybits?, mask
+      assert_send_type  '(int) -> bool',
+                        1, :anybits?, mask
+    end
   end
 
   def test_bit_length
-    3.bit_length
+    assert_send_type  '() -> Integer',
+                      38, :bit_length
+    assert_send_type  '() -> Integer',
+                      3838, :bit_length
   end
 
   def test_ceil
-    3.ceil
-    3.ceil(10)
-    3.ceil(ToInt.new)
+    assert_send_type  '() -> Integer',
+                      38, :ceil
+    with_int(-1) do |digits|
+      assert_send_type  '(int) -> Integer',
+                        38, :ceil, digits
+    end
   end
 
   def test_ceildiv
-    3.ceildiv(10)
-    3.ceildiv(1.3)
+    # omit 'todo'
   end
 
   def test_chr
-    3.chr
-    3.chr(Encoding::UTF_8)
-    3.chr("UTF-7")
-    3.chr(ToStr.new("ASCII-8BIT"))
+    assert_send_type  '() -> String',
+                      38, :chr
+
+    with_encoding do |encoding|
+      assert_send_type  '(encoding) -> String',
+                        38, :chr, encoding
+    end
   end
 
-  def test_conj
-    3.conj
+  def test_coerce
+    # omit 'todo'
   end
 
   def test_denominator
-    3.denominator
+    assert_send_type  '() -> 1',
+                      38, :denominator
   end
 
   def test_digits
-    3.digits
-    3.digits(3)
-    3.digits(3.0)
-    30.digits(ToInt.new)
+    assert_send_type  '() -> Array[Integer]',
+                      38, :digits
+
+    with_int 5 do |base|
+      assert_send_type  '(int) -> Array[Integer]',
+                        38, :digits, base
+    end
   end
 
   def test_div
-    30.div(10)
+    # omit 'todo'
   end
 
-  def test_div_mod
-    3.divmod(3)
-    40.divmod(1.0)
-    30.divmod(30r)
+  def test_divmod
+    # omit 'todo'
   end
 
-  def test_down_to
-    30.downto(1) {}
-    30.downto(31)
-    30.downto(4.2)
+  def test_downto
+    # omit 'todo'
   end
 
   def test_even?
-    30.even?
+    assert_send_type  '() -> bool',
+                      38, :even?
+    assert_send_type  '() -> bool',
+                      39, :even?
   end
 
   def test_fdiv
-    30.fdiv(30)
-    30.fdiv(3r)
-    30.fdiv(3.1)
-  end
-
-  def test_finite?
-    30.finite?
+    # omit 'todo'
   end
 
   def test_floor
-    30.floor
-    30.floor(3)
-    30.floor(ToInt.new)
+    assert_send_type  '() -> Integer',
+                      38, :floor
+    with_int(-1) do |digits|
+      assert_send_type  '(int) -> Integer',
+                        38, :floor, digits
+    end
   end
 
   def test_gcd
-    30.gcd(1)
+    assert_send_type  '(Integer) -> Integer',
+                      38, :gcd, 2
+    refute_send_type  '(_ToInt) -> Integer',
+                      39, :gcd, ToInt.new(2) # explicitly only supports ints
   end
 
   def test_gcdlcm
-    30.gcdlcm(31)
+    assert_send_type  '(Integer) -> [Integer, Integer]',
+                      38, :gcdlcm, 2
+    refute_send_type  '(_ToInt) -> [Integer, Integer]',
+                      39, :gcdlcm, ToInt.new(2) # explicitly only supports ints
+  end
+
+  def test_inspect
+    test_to_s(method: :inspect)
+  end
+
+  def test_integer?
+    assert_send_type  '() -> true',
+                      38, :integer?
   end
 
   def test_lcm
-    30.lcm(50)
+    assert_send_type  '(Integer) -> Integer',
+                      38, :lcm, 2
+    refute_send_type  '(_ToInt) -> Integer',
+                      39, :lcm, ToInt.new(2) # explicitly only supports ints
   end
 
   def test_magnitude
-    30.magnitude
+    test_abs(method: :magnitude)
   end
 
-  def test_modulo_
-    30.modulo(30)
-    30.modulo(3.1)
-    30.modulo(3r/5)
+  def test_modulo
+    test_op_mod(method: :modulo)
   end
 
   def test_next
-    30.next
+    test_succ(method: :next)
   end
 
   def test_nobits?
-    0xf0.nobits?(0xf)
-    0xf1.nobits?(0xf)
-    30.nobits?(ToInt.new)
+    with_int 1 do |mask|
+      assert_send_type  '(int) -> bool',
+                        0, :nobits?, mask
+      assert_send_type  '(int) -> bool',
+                        1, :nobits?, mask
+    end
   end
 
-  def test_nonzero?
-    30.nonzero?
-    0.nonzero?
+  def test_numerator
+    assert_send_type  '() -> 38',
+                      38, :numerator
   end
 
-  def test_numerator
-    30.numerator
+  def test_odd?
+    assert_send_type  '() -> bool',
+                      38, :odd?
+    assert_send_type  '() -> bool',
+                      39, :odd?
+  end
+
+  def test_ord
+    assert_send_type  '() -> 38',
+                      38, :ord
   end
 
   def test_pow
-    1.pow(30)
-    1.pow(2.0)
-    1.pow(30.to_c)
-    3.pow(3, 5)
+    # omit 'todo'
   end
 
-  def test_quo
-    3.quo(1)
-    3.quo(2.1)
-    3.quo(4r/5)
-    3.quo(10.to_c)
+  def test_pred
+    assert_send_type  '() -> Integer',
+                      1, :pred
   end
 
   def test_rationalize
-    3.rationalize
-    3.rationalize(30)
+    assert_send_type  '() -> Rational',
+                      38, :rationalize
+
+    with Numeric.new do |eps|
+      assert_send_type  '(Numeric) -> Rational',
+                        38, :rationalize, eps
+    end
   end
 
   def test_remainder
-    3.remainder(1)
-    3.remainder(1.3)
-    3.remainder(1r/3)
+    # omit 'todo'
   end
 
   def test_round
-    13.round()
-    13.round(half: :up)
-    14.round(-1, half: :down)
-    15.round(ToInt.new)
+    assert_send_type  '() -> Integer',
+                      38, :round
+
+    with_round_mode do |mode|
+      assert_send_type  '(half: Numeric::round_mode) -> Integer',
+                        38, :round, half: mode
+    end
+
+    with_int(-1) do |digits|
+      assert_send_type  '(int) -> Integer',
+                        38, :round, digits
+
+      with_round_mode do |mode|
+        assert_send_type  '(int, half: Numeric::round_mode) -> Integer',
+                          38, :round, digits, half: mode
+      end
+    end
   end
 
-  def test_step
-    3.step { break }
-    3.step
-    3.step(10, 2) {}
-    3.step(10, 2)
-    3.step(10, 1.1) {}
-    3.step(10, 1.1)
+  def test_size
+    assert_send_type  '() -> Integer',
+                      38, :size
+    assert_send_type  '() -> Integer',
+                      3838, :size
+  end
 
-    3.step(to: 30) { break }
-    3.step(to: 30)
-    3.step(to: 30, by: 100) {}
-    3.step(to: 30, by: 100)
-    3.step(to: 30, by: 10.0) {}
-    3.step(to: 30, by: 10.0)
+  def test_succ(method: :succ)
+    assert_send_type  '() -> Integer',
+                      1, method
   end
 
   def test_times
-    3.times {}
-    3.times
+    assert_send_type  '() { (Integer) -> void } -> 38',
+                      38, :times do end
+    assert_send_type  '() -> Enumerator[Integer, 38]',
+                      38, :times
   end
 
-  def test_truncate
-    100.truncate
-    100.truncate(10)
-    100.truncate(ToInt.new(-2))
+  def test_to_f
+    assert_send_type  '() -> Float',
+                      38, :to_f
   end
 
-  def test_upto
-    5.upto(10) {}
-    5.upto(10.1) {}
+  def test_to_i(method: :to_i)
+    assert_send_type  '() -> 38',
+                      38, method
   end
-end
 
+  def test_to_int
+    test_to_i(method: :to_int)
+  end
 
-class IntegerSingletonTest < Test::Unit::TestCase
-  include TestHelper
+  def test_to_r
+    assert_send_type  '() -> Rational',
+                      38, :to_r
+  end
 
-  testing "singleton(::Integer)"
+  def test_to_s(method: :to_s)
+    assert_send_type  '() -> String',
+                      38, method
 
-  def test_try_convert
-    assert_send_type(
-      "(Integer) -> Integer",
-      Integer, :try_convert, 10
-    )
-    assert_send_type(
-      "(ToInt) -> Integer",
-      Integer, :try_convert, ToInt.new(10)
-    )
-    assert_send_type(
-      "(String) -> nil",
-      Integer, :try_convert, "10"
-    )
+    with_int 8 do |base|
+      assert_send_type  '(int) -> String',
+                        38, method, base
+    end
   end
-end
 
-class IntegerInstanceTest < Test::Unit::TestCase
-  include TestHelper
+  def test_truncate
+    assert_send_type  '() -> Integer',
+                      38, :truncate
+    with_int(-1) do |digits|
+      assert_send_type  '(int) -> Integer',
+                        38, :truncate, digits
+    end
+  end
 
-  testing "::Integer"
+  def test_upto
+    # omit 'todo'
+  end
 
-  def test_pow
-    assert_send_type "(Integer) -> Integer",
-                     1, :pow, 2
-    assert_send_type "(Integer) -> Rational",
-                     -2, :pow, -1
-    assert_send_type "(Integer, Integer) -> Integer",
-                     1, :pow, 2, 10
-    assert_send_type "(Float) -> Float",
-                     1, :pow, 1.0
-    assert_send_type "(Float) -> Complex",
-                     -9, :pow, 0.1
-    assert_send_type "(Rational) -> Float",
-                     2, :pow, 1/2r
-    assert_send_type "(Rational) -> Rational",
-                     1, :pow, 1r
-    assert_send_type "(Rational) -> Complex",
-                     -3, :pow, -4/3r
-    assert_send_type "(Complex) -> Complex",
-                     1, :pow, 1i
+  def test_zero?
+    assert_send_type  '() -> bool',
+                      0, :zero?
+    assert_send_type  '() -> bool',
+                      38, :zero?
+  end
+
+  def test_op_or
+    # omit 'todo'
+  end
+
+  def test_op_not
+    assert_send_type  '() -> Integer',
+                      38, :~
   end
 end
diff --git a/test/stdlib/test_helper.rb b/test/stdlib/test_helper.rb
index b11939079..e9848e69c 100644
--- a/test/stdlib/test_helper.rb
+++ b/test/stdlib/test_helper.rb
@@ -41,6 +41,15 @@ def if_ruby(range, skip: true)
 end
 
 module WithStdlibAliases
+  def with_round_mode(&block)
+    block.call(nil)
+
+    %i[up down even].each do |mode|
+      block.call(mode)
+      with_string(mode.to_s, &block)
+    end
+  end
+
   def with_timeout(seconds: 1, nanoseconds: 0)
     unless block_given?
       return RBS::UnitTest::WithAliases::WithEnum.new(