diff --git a/stdlib/source/library/lux/compiler/language/lux/phase/extension/analysis/jvm.lux b/stdlib/source/library/lux/compiler/language/lux/phase/extension/analysis/jvm.lux
index 51d14f0299..687271c598 100644
--- a/stdlib/source/library/lux/compiler/language/lux/phase/extension/analysis/jvm.lux
+++ b/stdlib/source/library/lux/compiler/language/lux/phase/extension/analysis/jvm.lux
@@ -35,8 +35,8 @@
    [math
     [number
      ["n" natural]
-     [integer
-      ["i[0]" /32]]]]
+     ["i[0]" /32
+      ["[1]" integer]]]]
    [macro
     ["^" pattern]
     ["[0]" template]]
diff --git a/stdlib/source/library/lux/compiler/language/lux/phase/extension/declaration/jvm.lux b/stdlib/source/library/lux/compiler/language/lux/phase/extension/declaration/jvm.lux
index 86a213787a..f04a4f3293 100644
--- a/stdlib/source/library/lux/compiler/language/lux/phase/extension/declaration/jvm.lux
+++ b/stdlib/source/library/lux/compiler/language/lux/phase/extension/declaration/jvm.lux
@@ -30,8 +30,8 @@
    [math
     [number
      ["n" natural]
-     [integer
-      ["i[0]" /32]]]]
+     ["i[0]" /32
+      ["[1]" integer]]]]
    [macro
     ["[0]" template]]
    [type
diff --git a/stdlib/source/library/lux/compiler/language/lux/phase/extension/translation/jvm/common.lux b/stdlib/source/library/lux/compiler/language/lux/phase/extension/translation/jvm/common.lux
index 5cb7ff5a23..343812e436 100644
--- a/stdlib/source/library/lux/compiler/language/lux/phase/extension/translation/jvm/common.lux
+++ b/stdlib/source/library/lux/compiler/language/lux/phase/extension/translation/jvm/common.lux
@@ -20,8 +20,8 @@
      ["[0]" dictionary]]]
    [math
     [number
-     [integer
-      ["[0]" /32]]]]
+     ["[0]" /32
+      ["[1]" integer]]]]
    [macro
     ["[0]" template]]
    [target
diff --git a/stdlib/source/library/lux/compiler/language/lux/phase/extension/translation/jvm/host.lux b/stdlib/source/library/lux/compiler/language/lux/phase/extension/translation/jvm/host.lux
index 0a4ff75886..e990492f57 100644
--- a/stdlib/source/library/lux/compiler/language/lux/phase/extension/translation/jvm/host.lux
+++ b/stdlib/source/library/lux/compiler/language/lux/phase/extension/translation/jvm/host.lux
@@ -29,8 +29,8 @@
    [math
     [number
      ["n" natural]
-     [integer
-      ["i[0]" /32]]]]
+     ["i[0]" /32
+      ["[1]" integer]]]]
    [macro
     ["^" pattern]
     ["[0]" template]
diff --git a/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/complex.lux b/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/complex.lux
index 6980d4b871..9f14a0f82f 100644
--- a/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/complex.lux
+++ b/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/complex.lux
@@ -14,8 +14,8 @@
      ["[0]" list]]]
    [math
     [number
-     [integer
-      ["i[0]" /32]]]]
+     ["i[0]" /32
+      ["[1]" integer]]]]
    [target
     [jvm
      ["_" bytecode (.only Bytecode)]
diff --git a/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/function/method/of.lux b/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/function/method/of.lux
index 4d48e11dc9..9de78aedd0 100644
--- a/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/function/method/of.lux
+++ b/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/function/method/of.lux
@@ -13,8 +13,8 @@
     [number
      ["n" natural]
      ["i" integer]
-     [integer
-      ["[0]" /32]]]]
+     ["[0]" /32
+      ["[1]" integer]]]]
    [target
     [jvm
      ["_" bytecode (.only Label Bytecode) (.use "[1]#[0]" monad)]
diff --git a/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/runtime.lux b/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/runtime.lux
index 271ff99e3f..3e71f3f18e 100644
--- a/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/runtime.lux
+++ b/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/runtime.lux
@@ -23,8 +23,8 @@
     [number
      ["n" natural]
      ["[0]" i64]
-     [integer
-      ["i[0]" /32]]]]
+     ["i[0]" /32
+      ["[1]" integer]]]]
    [macro
     ["[0]" template]]
    [target
diff --git a/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/when.lux b/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/when.lux
index 74f02d3f47..847079895f 100644
--- a/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/when.lux
+++ b/stdlib/source/library/lux/compiler/language/lux/phase/translation/jvm/when.lux
@@ -14,8 +14,8 @@
    [math
     [number
      ["n" natural]
-     [integer
-      ["i[0]" /32]]]]
+     ["i[0]" /32
+      ["[1]" integer]]]]
    [macro
     ["[0]" template]]
    [target
diff --git a/stdlib/source/library/lux/data/bit.lux b/stdlib/source/library/lux/data/bit.lux
index 30365e7725..c8f7b0aa3f 100644
--- a/stdlib/source/library/lux/data/bit.lux
+++ b/stdlib/source/library/lux/data/bit.lux
@@ -84,3 +84,10 @@
 
        _
        {try.#Failure ..not_valid}))))
+
+(the .public (xor left right)
+  (-> Bit Bit
+      Bit)
+  (if left
+      (not right)
+      right))
diff --git a/stdlib/source/library/lux/math/number/integer/08.lux b/stdlib/source/library/lux/math/number/08/integer.lux
similarity index 97%
rename from stdlib/source/library/lux/math/number/integer/08.lux
rename to stdlib/source/library/lux/math/number/08/integer.lux
index e579bf74a2..7f53f09331 100644
--- a/stdlib/source/library/lux/math/number/integer/08.lux
+++ b/stdlib/source/library/lux/math/number/08/integer.lux
@@ -19,10 +19,10 @@
     ["[0]" nominal]]
    [macro
     ["[0]" template]]]]
- ["[0]" // (.only)
-  [//
-   ["[0]" natural]
-   ["[0]" i64]]])
+ [///
+  ["//" integer]
+  ["[0]" natural]
+  ["[0]" i64]])
 
 (the .public for
   (template.macro (_ ,size ,injection)
diff --git a/stdlib/source/library/lux/math/number/natural/08.lux b/stdlib/source/library/lux/math/number/08/natural.lux
similarity index 98%
rename from stdlib/source/library/lux/math/number/natural/08.lux
rename to stdlib/source/library/lux/math/number/08/natural.lux
index 762d23213f..a0942dcf25 100644
--- a/stdlib/source/library/lux/math/number/natural/08.lux
+++ b/stdlib/source/library/lux/math/number/08/natural.lux
@@ -19,9 +19,9 @@
     ["[0]" nominal]]
    [macro
     ["[0]" template]]]]
- ["[0]" // (.only)
-  [//
-   ["[0]" i64]]])
+ [///
+  ["//" natural]
+  ["[0]" i64]])
 
 (the .public for
   (template.macro (_ ,size ,injection)
diff --git a/stdlib/source/library/lux/math/number/integer/16.lux b/stdlib/source/library/lux/math/number/16/integer.lux
similarity index 86%
rename from stdlib/source/library/lux/math/number/integer/16.lux
rename to stdlib/source/library/lux/math/number/16/integer.lux
index b32cd24c6a..4b0b96b620 100644
--- a/stdlib/source/library/lux/math/number/integer/16.lux
+++ b/stdlib/source/library/lux/math/number/16/integer.lux
@@ -7,10 +7,10 @@
    [data
     ["[0]" binary
      ["[1]" \\injection]]]]]
- [//
-  ["[0]" /08]
-  [//
-   ["[0]" natural]]])
+ [///
+  ["[0]" /08
+   ["[1]" integer]]
+  ["[0]" natural]])
 
 (/08.for (natural.+ /08.size /08.size)
          binary.bits_16)
diff --git a/stdlib/source/library/lux/math/number/natural/16.lux b/stdlib/source/library/lux/math/number/16/natural.lux
similarity index 86%
rename from stdlib/source/library/lux/math/number/natural/16.lux
rename to stdlib/source/library/lux/math/number/16/natural.lux
index 2663b8cb01..f0329c8af7 100644
--- a/stdlib/source/library/lux/math/number/natural/16.lux
+++ b/stdlib/source/library/lux/math/number/16/natural.lux
@@ -7,8 +7,10 @@
    [data
     ["[0]" binary
      ["[1]" \\injection]]]]]
- ["[0]" // (.only)
-  ["[0]" /08]])
+ [///
+  ["//" natural]
+  ["[0]" /08
+   ["[1]" natural]]])
 
 (/08.for (//.+ /08.size /08.size)
          binary.bits_16)
diff --git a/stdlib/source/library/lux/math/number/integer/32.lux b/stdlib/source/library/lux/math/number/32/integer.lux
similarity index 80%
rename from stdlib/source/library/lux/math/number/integer/32.lux
rename to stdlib/source/library/lux/math/number/32/integer.lux
index 7ead899cc3..2fcfc9feee 100644
--- a/stdlib/source/library/lux/math/number/integer/32.lux
+++ b/stdlib/source/library/lux/math/number/32/integer.lux
@@ -7,11 +7,12 @@
    [data
     ["[0]" binary
      ["[1]" \\injection]]]]]
- [//
-  ["[0]" /08]
-  ["[0]" /16]
-  [//
-   ["[0]" natural]]])
+ [///
+  ["[0]" /08
+   ["[1]" integer]]
+  ["[0]" /16
+   ["[1]" integer]]
+  ["[0]" natural]])
 
 (/08.for (natural.+ /16.size /16.size)
          binary.bits_32)
diff --git a/stdlib/source/library/lux/math/number/natural/32.lux b/stdlib/source/library/lux/math/number/32/natural.lux
similarity index 80%
rename from stdlib/source/library/lux/math/number/natural/32.lux
rename to stdlib/source/library/lux/math/number/32/natural.lux
index b3c558c3ea..e0ac2faf39 100644
--- a/stdlib/source/library/lux/math/number/natural/32.lux
+++ b/stdlib/source/library/lux/math/number/32/natural.lux
@@ -7,9 +7,12 @@
    [data
     ["[0]" binary
      ["[1]" \\injection]]]]]
- ["[0]" // (.only)
-  ["[0]" /08]
-  ["[0]" /16]])
+ [///
+  ["//" natural]
+  ["[0]" /08
+   ["[1]" natural]]
+  ["[0]" /16
+   ["[1]" natural]]])
 
 (/08.for (//.+ /16.size /16.size)
          binary.bits_32)
diff --git a/stdlib/source/library/lux/math/number/big/integer.lux b/stdlib/source/library/lux/math/number/big/integer.lux
index 319ed9f16e..7cbf483784 100644
--- a/stdlib/source/library/lux/math/number/big/integer.lux
+++ b/stdlib/source/library/lux/math/number/big/integer.lux
@@ -25,22 +25,28 @@
   [//
    ["//_64" natural]
    ["/_64" integer]
+   ["[0]" signum]
    [//
     [arithmetic (.only Arithmetic)]]]])
 
 (the with_template (.in_module# .prelude .with_template))
 (the macro (.in_module# .prelude .template#macro))
 
-(every .public Signum
-  Bit)
+(with_template [,value ,character ,signum]
+  [(the ,value
+     (its signum.#value ,signum))
 
-(the .public positive 0b)
-(the .public negative 1b)
+   (the ,character
+     (its signum.#character ,signum))]
+
+  [positive positive_signum signum.positive]
+  [negative negative_signum signum.negative]
+  )
 
 (every .public Number
   (Record
    [... https://en.wikipedia.org/wiki/Sign_function
-    #signum Signum
+    #signum signum.Value
     ... https://en.wikipedia.org/wiki/Absolute_value
     #absolute //.Number]))
 
@@ -217,7 +223,7 @@
      (when [origin it]
        [[signum_of_origin origin]
         [signum_of_it it]]
-       [#signum (bit.= signum_of_origin signum_of_it)
+       [#signum (signum.x signum_of_origin signum_of_it)
         #absolute (,// origin it)]))]
 
   [x //.x]
@@ -387,9 +393,6 @@
   [multiplication ..x   ..positive_one]
   )
 
-(the positive_signum "+")
-(the negative_signum "-")
-
 (the not_valid
   (error "Not an integer."))
 
diff --git a/stdlib/source/library/lux/math/number/integer.lux b/stdlib/source/library/lux/math/number/integer.lux
index c02fba7cf9..109704c066 100644
--- a/stdlib/source/library/lux/math/number/integer.lux
+++ b/stdlib/source/library/lux/math/number/integer.lux
@@ -21,6 +21,7 @@
  ["[0]" //
   ["[1][0]" natural]
   ["[1][0]" i64]
+  ["[0]" signum]
   [//
    [arithmetic (.only Arithmetic)]]])
 
@@ -267,8 +268,13 @@
   [minimum        ..minor (by ..interval maximum)]
   )
 
-(the -sign "-")
-(the +sign "+")
+(with_template [,character ,signum]
+  [(the ,character
+     (its signum.#character ,signum))]
+
+  [+sign signum.positive]
+  [-sign signum.negative]
+  )
 
 (with_template [<struct> <format> <error>]
   [(the .public <struct>
diff --git a/stdlib/source/library/lux/math/number/signum.lux b/stdlib/source/library/lux/math/number/signum.lux
new file mode 100644
index 0000000000..64586f434c
--- /dev/null
+++ b/stdlib/source/library/lux/math/number/signum.lux
@@ -0,0 +1,34 @@
+... This Source Code Form is subject to the terms of the Mozilla Public License, v. 2.0.
+... If a copy of the MPL was not distributed with this file, You can obtain one at https://mozilla.org/MPL/2.0/.
+
+(.using
+ [library
+  [lux (.except)
+   [data
+    ["[0]" bit]]]])
+
+(the with_template (.in_module# .prelude .with_template))
+
+... https://en.wikipedia.org/wiki/Sign_function
+(every .public Value
+  Bit)
+
+(the .public x
+  (-> Value Value
+      Value)
+  bit.xor)
+
+(every .public Signum
+  (Record
+   [#character Text
+    #value Value]))
+
+(with_template [,name ,character ,value]
+  [(the .public ,name
+     Signum
+     [#character ,character
+      #value ,value])]
+
+  [positive "+" 0b]
+  [negative "-" 1b]
+  )
diff --git a/stdlib/source/library/lux/target/jvm/attribute.lux b/stdlib/source/library/lux/target/jvm/attribute.lux
index 86ce5f9df3..fe622cdb64 100644
--- a/stdlib/source/library/lux/target/jvm/attribute.lux
+++ b/stdlib/source/library/lux/target/jvm/attribute.lux
@@ -18,9 +18,10 @@
    [math
     [number
      ["n" natural]
-     [natural
-      ["[0]" /16]
-      ["[0]" /32]]]]
+     ["[0]" /16
+      ["[1]" natural]]
+     ["[0]" /32
+      ["[1]" natural]]]]
    [macro
     ["[0]" expansion]
     ["[0]" template]]]]
diff --git a/stdlib/source/library/lux/target/jvm/attribute/code.lux b/stdlib/source/library/lux/target/jvm/attribute/code.lux
index 8834663f15..ca9e46f4ae 100644
--- a/stdlib/source/library/lux/target/jvm/attribute/code.lux
+++ b/stdlib/source/library/lux/target/jvm/attribute/code.lux
@@ -16,9 +16,10 @@
    [math
     [number
      ["n" natural]
-     [natural
-      ["[0]" /16]
-      ["[0]" /32]]]]]]
+     ["[0]" /16
+      ["[1]" natural]]
+     ["[0]" /32
+      ["[1]" natural]]]]]]
  ["[0]" ///
   [bytecode
    [environment
diff --git a/stdlib/source/library/lux/target/jvm/attribute/code/exception.lux b/stdlib/source/library/lux/target/jvm/attribute/code/exception.lux
index f4ecc0aa94..7b8f2a5bb6 100644
--- a/stdlib/source/library/lux/target/jvm/attribute/code/exception.lux
+++ b/stdlib/source/library/lux/target/jvm/attribute/code/exception.lux
@@ -13,8 +13,8 @@
    [math
     [number
      ["n" natural]
-     [natural
-      ["[0]" /16]]]]]]
+     ["[0]" /16
+      ["[1]" natural]]]]]]
  ["[0]" //
   ["//[1]" ///
    ["[1][0]" index (.only Index)]
diff --git a/stdlib/source/library/lux/target/jvm/attribute/line_number_table.lux b/stdlib/source/library/lux/target/jvm/attribute/line_number_table.lux
index 25066b25ba..28b4723194 100644
--- a/stdlib/source/library/lux/target/jvm/attribute/line_number_table.lux
+++ b/stdlib/source/library/lux/target/jvm/attribute/line_number_table.lux
@@ -16,8 +16,8 @@
    [math
     [number
      ["n" natural]
-     [natural
-      ["[0]" /16]]]]]])
+     ["[0]" /16
+      ["[1]" natural]]]]]])
 
 (every .public Entry
   (Record
diff --git a/stdlib/source/library/lux/target/jvm/bytecode.lux b/stdlib/source/library/lux/target/jvm/bytecode.lux
index 98c548b649..8948c86af9 100644
--- a/stdlib/source/library/lux/target/jvm/bytecode.lux
+++ b/stdlib/source/library/lux/target/jvm/bytecode.lux
@@ -31,11 +31,12 @@
     [number
      ["n" natural]
      ["i" integer]
-     [natural
-      ["n[0]" /08]
-      ["n[0]" /16]]
-     [integer
-      ["[0]" /32]]]]
+     ["n[0]" /08
+      ["[1]" natural]]
+     ["n[0]" /16
+      ["[1]" natural]]
+     ["[0]" /32
+      ["[1]" integer]]]]
    [macro
     ["[0]" template]
     ["[0]" expansion]]
diff --git a/stdlib/source/library/lux/target/jvm/bytecode/address.lux b/stdlib/source/library/lux/target/jvm/bytecode/address.lux
index 4c1e653ec0..05c88a6ce4 100644
--- a/stdlib/source/library/lux/target/jvm/bytecode/address.lux
+++ b/stdlib/source/library/lux/target/jvm/bytecode/address.lux
@@ -17,10 +17,10 @@
    [math
     [number
      ["n" natural]
-     [natural
-      ["[0]" /16]]
-     [integer
-      ["[0]" /32]]]]
+     ["[0]" /16
+      ["[1]" natural]]
+     ["[0]" /32
+      ["[1]" integer]]]]
    [type
     ["[0]" nominal]]]]
  [//
diff --git a/stdlib/source/library/lux/target/jvm/bytecode/environment.lux b/stdlib/source/library/lux/target/jvm/bytecode/environment.lux
index 8f13310a79..e20fef5e14 100644
--- a/stdlib/source/library/lux/target/jvm/bytecode/environment.lux
+++ b/stdlib/source/library/lux/target/jvm/bytecode/environment.lux
@@ -15,8 +15,8 @@
     ["[0]" template]]
    [math
     [number
-     [natural
-      ["[0]" /16]]]]]]
+     ["[0]" /16
+      ["[1]" natural]]]]]]
  [/
   ["/[0]" limit (.only Limit)
    ["/[0]" stack (.only Stack)]
diff --git a/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/datum.lux b/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/datum.lux
index 748610df84..567415fec2 100644
--- a/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/datum.lux
+++ b/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/datum.lux
@@ -11,8 +11,8 @@
    [math
     ["[0]" random (.only Random) (.use "[1]#[0]" monad)]
     [number
-     [natural
-      ["[0]" /16]]]]
+     ["[0]" /16
+      ["[1]" natural]]]]
    [macro
     ["[0]" template]]]]
  [/////
diff --git a/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/registry.lux b/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/registry.lux
index 83c8fbc282..2852c770a3 100644
--- a/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/registry.lux
+++ b/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/registry.lux
@@ -19,9 +19,10 @@
     ["[0]" random (.only Random) (.use "[1]#[0]" monad)]
     [number
      ["n" natural]
-     [natural
-      ["[0]" /08]
-      ["[0]" /16]]]]
+     ["[0]" /08
+      ["[1]" natural]]
+     ["[0]" /16
+      ["[1]" natural]]]]
    [macro
     ["[0]" template]]
    [type
diff --git a/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/stack.lux b/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/stack.lux
index 92dc4c7486..c50ea89de6 100644
--- a/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/stack.lux
+++ b/stdlib/source/library/lux/target/jvm/bytecode/environment/limit/stack.lux
@@ -18,8 +18,8 @@
    [math
     ["[0]" random (.only Random) (.use "[1]#[0]" monad)]
     [number
-     [natural
-      ["[0]" /16]]]]
+     ["[0]" /16
+      ["[1]" natural]]]]
    [macro
     ["[0]" template]]
    [type
diff --git a/stdlib/source/library/lux/target/jvm/bytecode/instruction.lux b/stdlib/source/library/lux/target/jvm/bytecode/instruction.lux
index 013982739d..1bd44f7de5 100644
--- a/stdlib/source/library/lux/target/jvm/bytecode/instruction.lux
+++ b/stdlib/source/library/lux/target/jvm/bytecode/instruction.lux
@@ -23,11 +23,12 @@
    [math
     [number (.only hex)
      ["n" natural]
-     [natural
-      ["n[0]" /08]
-      ["n[0]" /16]]
-     [integer
-      ["[0]" /32]]]]
+     ["n[0]" /08
+      ["[1]" natural]]
+     ["n[0]" /16
+      ["[1]" natural]]
+     ["[0]" /32
+      ["[1]" integer]]]]
    [macro
     ["[0]" template]
     ["[0]" expansion]]
diff --git a/stdlib/source/library/lux/target/jvm/bytecode/jump.lux b/stdlib/source/library/lux/target/jvm/bytecode/jump.lux
index 4abb9bc33d..e4826b7e3f 100644
--- a/stdlib/source/library/lux/target/jvm/bytecode/jump.lux
+++ b/stdlib/source/library/lux/target/jvm/bytecode/jump.lux
@@ -11,8 +11,8 @@
      [\\injection (.only Injection)]]]
    [math
     [number
-     [integer
-      ["[0]" /32]]]]]]
+     ["[0]" /32
+      ["[1]" integer]]]]]]
  [///
   [encoding
    ["[0]" signed]]])
diff --git a/stdlib/source/library/lux/target/jvm/class.lux b/stdlib/source/library/lux/target/jvm/class.lux
index 1ea2922eb8..ca0822c66a 100644
--- a/stdlib/source/library/lux/target/jvm/class.lux
+++ b/stdlib/source/library/lux/target/jvm/class.lux
@@ -23,9 +23,10 @@
     ["[0]" template]]
    [math
     [number
-     [natural
-      ["[0]" /16]
-      ["[0]" /32]]]]]]
+     ["[0]" /16
+      ["[1]" natural]]
+     ["[0]" /32
+      ["[1]" natural]]]]]]
  ["[0]" //
   ["[1][0]" modifier (.only Modifier modifiers)]
   ["[1][0]" version (.only Version Minor Major)]
diff --git a/stdlib/source/library/lux/target/jvm/constant/integer.lux b/stdlib/source/library/lux/target/jvm/constant/integer.lux
index 86436ef7af..f6ef7801c3 100644
--- a/stdlib/source/library/lux/target/jvm/constant/integer.lux
+++ b/stdlib/source/library/lux/target/jvm/constant/integer.lux
@@ -10,8 +10,8 @@
    [math
     [random (.only Random)]
     [number
-     [integer
-      ["i[0]" /32]]]]]])
+     ["i[0]" /32
+      ["[1]" integer]]]]]])
 
 (every .public Integer
   i/32.Number)
diff --git a/stdlib/source/library/lux/target/jvm/constant/pool.lux b/stdlib/source/library/lux/target/jvm/constant/pool.lux
index 379bdc6472..a221423c6f 100644
--- a/stdlib/source/library/lux/target/jvm/constant/pool.lux
+++ b/stdlib/source/library/lux/target/jvm/constant/pool.lux
@@ -25,8 +25,8 @@
     [number
      ["[0]" integer]
      ["[0]" decimal]
-     [natural
-      ["[0]" /16]]]]
+     ["[0]" /16
+      ["[1]" natural]]]]
    [macro
     ["[0]" expansion]
     ["[0]" template]]]]
diff --git a/stdlib/source/library/lux/target/jvm/constant/tag.lux b/stdlib/source/library/lux/target/jvm/constant/tag.lux
index bc608d6a26..1b54fff8de 100644
--- a/stdlib/source/library/lux/target/jvm/constant/tag.lux
+++ b/stdlib/source/library/lux/target/jvm/constant/tag.lux
@@ -15,8 +15,8 @@
      [\\injection (.only Injection)]]]
    [math
     [number
-     [natural
-      ["[0]" /08]]]]
+     ["[0]" /08
+      ["[1]" natural]]]]
    [macro
     ["[0]" template]]
    [type
diff --git a/stdlib/source/library/lux/target/jvm/index.lux b/stdlib/source/library/lux/target/jvm/index.lux
index 772df1b139..4da8416a9d 100644
--- a/stdlib/source/library/lux/target/jvm/index.lux
+++ b/stdlib/source/library/lux/target/jvm/index.lux
@@ -13,8 +13,8 @@
     ["[0]" random (.only Random) (.use "[1]#[0]" functor)]
     [number
      ["n" natural]
-     [natural
-      ["[0]" /16]]]]
+     ["[0]" /16
+      ["[1]" natural]]]]
    [type
     ["[0]" nominal]]]])
 
diff --git a/stdlib/source/library/lux/target/jvm/magic.lux b/stdlib/source/library/lux/target/jvm/magic.lux
index 69c5cc3890..f97eac408a 100644
--- a/stdlib/source/library/lux/target/jvm/magic.lux
+++ b/stdlib/source/library/lux/target/jvm/magic.lux
@@ -8,8 +8,8 @@
     ["[0]" try]]
    [math
     [number (.only hex)
-     [natural
-      ["[0]" /32]]]]]])
+     ["[0]" /32
+      ["[1]" natural]]]]]])
 
 (every .public Magic
   /32.Number)
diff --git a/stdlib/source/library/lux/target/jvm/modifier.lux b/stdlib/source/library/lux/target/jvm/modifier.lux
index 8bb64967c9..ea125b3158 100644
--- a/stdlib/source/library/lux/target/jvm/modifier.lux
+++ b/stdlib/source/library/lux/target/jvm/modifier.lux
@@ -19,8 +19,8 @@
    [math
     ["[0]" number (.only hex)
      ["[0]" i64]
-     [natural
-      ["[0]" /16]]]]
+     ["[0]" /16
+      ["[1]" natural]]]]
    [macro (.only with_names)
     ["[0]" syntax]
     ["[0]" template]]
diff --git a/stdlib/source/library/lux/target/jvm/version.lux b/stdlib/source/library/lux/target/jvm/version.lux
index ae1a35ed29..6e0602fab3 100644
--- a/stdlib/source/library/lux/target/jvm/version.lux
+++ b/stdlib/source/library/lux/target/jvm/version.lux
@@ -10,8 +10,8 @@
     ["[0]" template]]
    [math
     [number
-     [natural
-      ["[0]" /16]]]]]])
+     ["[0]" /16
+      ["[1]" natural]]]]]])
 
 (every .public Version /16.Number)
 (every .public Minor Version)
diff --git a/stdlib/source/test/lux/math/number.lux b/stdlib/source/test/lux/math/number.lux
index fed957ce2c..2753c8ad8c 100644
--- a/stdlib/source/test/lux/math/number.lux
+++ b/stdlib/source/test/lux/math/number.lux
@@ -11,6 +11,7 @@
  ["[0]" /
   ["[1][0]" digit]
   ["[1][0]" base]
+  ["[1][0]" signum]
   ["[1][0]" i64]
   ["[1][0]" natural]
   ["[1][0]" integer]
@@ -19,6 +20,15 @@
   ["[1][0]" fraction]
   ["[1][0]" complex]
   ["[1][0]" rational]
+  ["[0]" /08
+   ["[1]/[0]" natural]
+   ["[1]/[0]" integer]]
+  ["[0]" /16
+   ["[1]/[0]" natural]
+   ["[1]/[0]" integer]]
+  ["[0]" /32
+   ["[1]/[0]" natural]
+   ["[1]/[0]" integer]]
   ["[1][0]" big
    ["[1]/[0]" natural]
    ["[1]/[0]" integer]
@@ -30,6 +40,8 @@
       (all _.and
            /digit.test
            /base.test
+           /signum.test
+           
            /i64.test
            /natural.test
            /integer.test
@@ -39,6 +51,15 @@
            /complex.test
            /rational.test
 
+           /08/natural.test
+           /08/integer.test
+
+           /16/natural.test
+           /16/integer.test
+
+           /32/natural.test
+           /32/integer.test
+           
            /big/natural.test
            /big/integer.test
            /big/fraction.test
diff --git a/stdlib/source/test/lux/math/number/integer/08.lux b/stdlib/source/test/lux/math/number/08/integer.lux
similarity index 100%
rename from stdlib/source/test/lux/math/number/integer/08.lux
rename to stdlib/source/test/lux/math/number/08/integer.lux
diff --git a/stdlib/source/test/lux/math/number/natural/08.lux b/stdlib/source/test/lux/math/number/08/natural.lux
similarity index 100%
rename from stdlib/source/test/lux/math/number/natural/08.lux
rename to stdlib/source/test/lux/math/number/08/natural.lux
diff --git a/stdlib/source/test/lux/math/number/integer/16.lux b/stdlib/source/test/lux/math/number/16/integer.lux
similarity index 100%
rename from stdlib/source/test/lux/math/number/integer/16.lux
rename to stdlib/source/test/lux/math/number/16/integer.lux
diff --git a/stdlib/source/test/lux/math/number/natural/16.lux b/stdlib/source/test/lux/math/number/16/natural.lux
similarity index 100%
rename from stdlib/source/test/lux/math/number/natural/16.lux
rename to stdlib/source/test/lux/math/number/16/natural.lux
diff --git a/stdlib/source/test/lux/math/number/integer/32.lux b/stdlib/source/test/lux/math/number/32/integer.lux
similarity index 100%
rename from stdlib/source/test/lux/math/number/integer/32.lux
rename to stdlib/source/test/lux/math/number/32/integer.lux
diff --git a/stdlib/source/test/lux/math/number/natural/32.lux b/stdlib/source/test/lux/math/number/32/natural.lux
similarity index 100%
rename from stdlib/source/test/lux/math/number/natural/32.lux
rename to stdlib/source/test/lux/math/number/32/natural.lux
diff --git a/stdlib/source/test/lux/math/number/integer.lux b/stdlib/source/test/lux/math/number/integer.lux
index a40c42b071..be4a1d9233 100644
--- a/stdlib/source/test/lux/math/number/integer.lux
+++ b/stdlib/source/test/lux/math/number/integer.lux
@@ -39,10 +39,7 @@
    [//
     ["n" natural]
     ["d" decimal]
-    ["[0]" i64]]]]
- ["[0]" /08]
- ["[0]" /16]
- ["[0]" /32])
+    ["[0]" i64]]]])
 
 (the signature
   Test
@@ -282,8 +279,4 @@
 
            ..predicate
            ..signature
-
-           /08.test
-           /16.test
-           /32.test
            )))
diff --git a/stdlib/source/test/lux/math/number/natural.lux b/stdlib/source/test/lux/math/number/natural.lux
index 77222dfc38..121a560e26 100644
--- a/stdlib/source/test/lux/math/number/natural.lux
+++ b/stdlib/source/test/lux/math/number/natural.lux
@@ -38,11 +38,7 @@
  [\\library
   ["[0]" / (.only)
    [//
-    ["d" decimal]]]]
- ["[0]" /
-  ["[0]" /08]
-  ["[0]" /16]
-  ["[0]" /32]])
+    ["d" decimal]]]])
 
 (the signature
   Test
@@ -237,8 +233,4 @@
 
            ..predicate
            ..signature
-
-           /08.test
-           /16.test
-           /32.test
            )))
diff --git a/stdlib/source/test/lux/math/number/signum.lux b/stdlib/source/test/lux/math/number/signum.lux
new file mode 100644
index 0000000000..1f9d60c4ab
--- /dev/null
+++ b/stdlib/source/test/lux/math/number/signum.lux
@@ -0,0 +1,60 @@
+... This Source Code Form is subject to the terms of the Mozilla Public License, v. 2.0.
+... If a copy of the MPL was not distributed with this file, You can obtain one at https://mozilla.org/MPL/2.0/.
+
+(.using
+ [library
+  [lux (.except)
+   [abstract
+    [monad (.only do)]]
+   [data
+    ["[0]" bit]
+    ["[0]" text]
+    [collection
+     ["[0]" list (.use "[1]#[0]" functor)]
+     ["[0]" set]]]
+   [macro
+    ["[0]" expansion]
+    ["[0]" template]]
+   [math
+    ["[0]" random (.only Random) (.use "[1]#[0]" monad)]
+    [number
+     ["[0]" natural]]]
+   [test
+    ["_" property (.only Test)]]]]
+ [\\library
+  ["[0]" /]])
+
+(the random
+  (Random /.Signum)
+  (random.either (random#pure /.positive)
+                 (random#pure /.negative)))
+
+(expansion.let [,each (these /.positive /.negative)]
+  (the .public test
+    Test
+    (<| (_.covering /._)
+        (do [! random.monad]
+          [it ..random])
+        (_.for [/.Signum
+                ,each])
+        (`` (all _.and
+                 (,, (template.with [,slot ,hash]
+                       [(_.coverage [,slot]
+                          (let [each (list ,each)
+                                each_unique (|> each
+                                                (list#each (its ,slot))
+                                                (set.of_list ,hash))]
+                            (natural.= (list.size each)
+                                       (set.size each_unique))))]
+
+                       [/.#character text.hash]
+                       [/.#value bit.hash]
+                       ))
+                 (_.coverage [/.Value /.x]
+                   (and (bit.= (its /.#value /.negative)
+                               (/.x (its /.#value /.positive)
+                                    (its /.#value /.negative)))
+                        (bit.= (its /.#value /.positive)
+                               (/.x (its /.#value it)
+                                    (its /.#value it)))))
+                 )))))
diff --git a/stdlib/source/test/lux/target/jvm.lux b/stdlib/source/test/lux/target/jvm.lux
index 41b05aefc0..0243ff0915 100644
--- a/stdlib/source/test/lux/target/jvm.lux
+++ b/stdlib/source/test/lux/target/jvm.lux
@@ -37,10 +37,10 @@
      ["i" integer]
      ["d" decimal]
      ["[0]" i64]
-     [natural
-      ["n[0]" /08]]
-     [integer
-      ["[0]" /32]]]]
+     ["n[0]" /08
+      ["[1]" natural]]
+     ["[0]" /32
+      ["[1]" integer]]]]
    [macro
     ["[0]" template]]
    [test
diff --git a/stdlib/source/test/lux/target/jvm/bytecode/jump.lux b/stdlib/source/test/lux/target/jvm/bytecode/jump.lux
index d44e3bf61f..e4b3deaaef 100644
--- a/stdlib/source/test/lux/target/jvm/bytecode/jump.lux
+++ b/stdlib/source/test/lux/target/jvm/bytecode/jump.lux
@@ -16,8 +16,8 @@
     [number
      ["n" natural]
      ["i" integer]
-     [integer
-      ["[0]" /32]]]]
+     ["[0]" /32
+      ["[1]" integer]]]]
    [meta
     ["[0]" static]]
    [test
diff --git a/stdlib/source/test/lux/target/jvm/index.lux b/stdlib/source/test/lux/target/jvm/index.lux
index 7d2f0d7c66..2d28643676 100644
--- a/stdlib/source/test/lux/target/jvm/index.lux
+++ b/stdlib/source/test/lux/target/jvm/index.lux
@@ -15,8 +15,8 @@
     ["[0]" random]
     [number
      ["n" natural]
-     [natural
-      ["[0]" /16]]]]
+     ["[0]" /16
+      ["[1]" natural]]]]
    [meta
     ["[0]" static]]
    [test
diff --git a/stdlib/source/test/lux/target/jvm/magic.lux b/stdlib/source/test/lux/target/jvm/magic.lux
index 27c1c23e36..f8e1180938 100644
--- a/stdlib/source/test/lux/target/jvm/magic.lux
+++ b/stdlib/source/test/lux/target/jvm/magic.lux
@@ -13,8 +13,8 @@
     ["[0]" random (.only Random)]
     [number (.only hex)
      ["n" natural]
-     [natural
-      ["[0]" /32]]]]
+     ["[0]" /32
+      ["[1]" natural]]]]
    [test
     ["_" property (.only Test)]]]]
  [\\library
diff --git a/stdlib/source/test/lux/target/jvm/version.lux b/stdlib/source/test/lux/target/jvm/version.lux
index 36232b9589..d11ae4e459 100644
--- a/stdlib/source/test/lux/target/jvm/version.lux
+++ b/stdlib/source/test/lux/target/jvm/version.lux
@@ -16,8 +16,8 @@
     ["[0]" random (.only Random)]
     [number
      ["n" natural]
-     [natural
-      ["[0]" /16]]]]
+     ["[0]" /16
+      ["[1]" natural]]]]
    [test
     ["_" property (.only Test)]]]]
  [\\library