diff --git a/base/docs/helpdb/Base.jl b/base/docs/helpdb/Base.jl
index e065328356ff56..3f6437669cb576 100644
--- a/base/docs/helpdb/Base.jl
+++ b/base/docs/helpdb/Base.jl
@@ -2260,14 +2260,6 @@ for sets of arbitrary objects.
 """
 Set
 
-"""
-    signed(x)
-
-Convert a number to a signed integer. If the argument is unsigned, it is reinterpreted as
-signed without checking for overflow.
-"""
-signed
-
 """
     Val{c}
 
diff --git a/base/essentials.jl b/base/essentials.jl
index ac24cc77a75f4f..8a419f44e75848 100644
--- a/base/essentials.jl
+++ b/base/essentials.jl
@@ -144,20 +144,54 @@ oftype(x,c) = convert(typeof(x),c)
 
 Convert a number to an unsigned integer. If the argument is signed, it is reinterpreted as
 unsigned without checking for negative values.
+See also [`signed`](@ref).
 
     unsigned(T::Type) -> UnsignedType
 
-Return the return-type of unsigned(x::T).
+Return the return-type of `unsigned(x::T)`, so that `unsigned(x)::unsigned(typeof(x))`.
 
 ```jldoctest
 julia> unsigned(12)
 0x000000000000000c
 
-julia> unsigned(Int)
+julia> unsigned(Int64)
 UInt64
+
+julia> unsigned(2.0)
+0x0000000000000002
+
+julia> unsigned(2.2)
+ERROR: InexactError()
+[...]
 ```
 """
 unsigned(x::Int) = reinterpret(UInt, x)
+
+"""
+    signed(x)
+
+Convert a number to a signed integer. If the argument is unsigned, it is reinterpreted as
+signed without checking for overflow.
+See also [`unsigned`](@ref).
+
+    signed(T::Type) -> SignedType
+
+Return the return-type of `signed(x::T)`, so that `signed(x)::signed(typeof(x))`.
+
+```jldoctest
+julia> signed(0xc)
+12
+
+julia> signed(UInt64)
+Int64
+
+julia> signed(2.0)
+2
+
+julia> signed(2.2)
+ERROR: InexactError()
+[...]
+"""
 signed(x::UInt) = reinterpret(Int, x)
 
 # conversions used by ccall
diff --git a/base/floatfuncs.jl b/base/floatfuncs.jl
index 549e4d202b2042..90e46d2659d7fb 100644
--- a/base/floatfuncs.jl
+++ b/base/floatfuncs.jl
@@ -29,9 +29,17 @@ maxintfloat() = maxintfloat(Float64)
 
 isinteger(x::AbstractFloat) = (x - trunc(x) == 0)
 
+"""
+    hex2num(str)
+
+Convert a hexadecimal string to the floating point number it represents.
+This function, which is inherently type-unstable, returns a `Float16`,
+a `Float32`, or a `Float64` depending on the length of the string `str`
+(note that this length must hence be no greater than 16).
+"""
 function hex2num(s::AbstractString)
     l = length(s)
-    l > 16 && throw(ArgumentError("the passed string must be of length <= 16, got $l"))
+    l > 16 && throw(ArgumentError("the length of the passed string must be <= 16, got $l"))
     hex2num(l <= 4 ? Float16 : l <= 8 ? Float32 : Float64, s)
 end
 
diff --git a/base/int.jl b/base/int.jl
index 94b1202d3feb80..eba82b1dacbb7a 100644
--- a/base/int.jl
+++ b/base/int.jl
@@ -29,7 +29,34 @@ const BitReal_types  = (BitInteger_types..., BitFloat_types...)
 const BitReal        = Union{BitReal_types...}
 
 reinterpret(::Type{Unsigned}, x::BitInteger) = unsigned(x)
-
+reinterpret(::Type{  Signed}, x::BitInteger) = signed(x)
+
+unsigned(::Type{T}) where {T<:Unsigned} = T
+signed(  ::Type{T}) where {T<:Signed}   = T
+
+unsigned(::Type{<:Union{Bool,BitFloat}}) = UInt
+signed(  ::Type{<:Union{Bool,BitFloat}}) = Int
+
+
+isdefined(Main, :Base) && # zip (etc.) not available when running in Core
+    for (S,U,F) in zip(BitSigned_types, BitUnsigned_types,
+                       [nothing, Float16, Float32, Float64, nothing])
+        @eval begin
+            unsigned(::Type{$S}) = $U
+            unsigned(::Type{$U}) = $U
+            signed(  ::Type{$S}) = $S
+            signed(  ::Type{$U}) = $S
+            reinterpret(::Type{Unsigned}, ::Type{$S}) = $U
+            reinterpret(::Type{Unsigned}, ::Type{$U}) = $U
+            reinterpret(::Type{Signed}, ::Type{$S}) = $S
+            reinterpret(::Type{Signed}, ::Type{$U}) = $S
+        end
+        F === nothing && continue
+        @eval begin
+            reinterpret(::Type{Unsigned}, ::Type{$F}) = $U
+            reinterpret(::Type{Signed}, ::Type{$F}) = $S
+        end
+    end
 
 ## integer comparisons ##
 
diff --git a/base/intfuncs.jl b/base/intfuncs.jl
index 7fd538a9d99869..6c7f129ed5cb3f 100644
--- a/base/intfuncs.jl
+++ b/base/intfuncs.jl
@@ -1,5 +1,6 @@
 # This file is a part of Julia. License is MIT: https://julialang.org/license
 
+
 ## number-theoretic functions ##
 
 """
@@ -488,7 +489,7 @@ end
 
 An hexadecimal string of the binary representation of a number.
 See also the [`bits`](@ref) function, which is similar but gives
-a binary string.
+a binary string, and [`hex2num`] which does the opposite conversion.
 
 ```jldoctest
 julia> num2hex(Int64(4))
@@ -515,16 +516,9 @@ julia> hex2num(Float64, "400199999999999a")
 julia> hex2num(Int32, "fffffffe")
 -2
 ```
-
-    hex2num(str)
-
-Convert a hexadecimal string to the floating point number it represents.
-This function, which is inherently type-unstable, returns a `Float16`,
-a `Float32`, or a `Float64` depending on the length of the string `str`
-(note that this length must hence be no greater than 16).
 """
 hex2num(::Type{T}, s::AbstractString) where {T<:BitReal} =
-    reinterpret(T, parse(unsigned(T), s, 16))
+    reinterpret(T, parse(reinterpret(Unsigned, T), s, 16))
 
 const base36digits = ['0':'9';'a':'z']
 const base62digits = ['0':'9';'A':'Z';'a':'z']
diff --git a/base/multinverses.jl b/base/multinverses.jl
index be34348a1a17e1..88a07366510426 100644
--- a/base/multinverses.jl
+++ b/base/multinverses.jl
@@ -2,20 +2,10 @@
 
 module MultiplicativeInverses
 
-import Base: div, divrem, rem, unsigned
+import Base: div, divrem, rem
 using  Base: IndexLinear, IndexCartesian, tail
 export multiplicativeinverse
 
-unsigned(::Type{Int8}) = UInt8
-unsigned(::Type{Int16}) = UInt16
-unsigned(::Type{Int32}) = UInt32
-unsigned(::Type{Int64}) = UInt64
-unsigned(::Type{Int128}) = UInt128
-unsigned(::Type{T}) where {T<:Unsigned} = T
-unsigned(::Type{Float16}) = UInt16
-unsigned(::Type{Float32}) = UInt32
-unsigned(::Type{Float64}) = UInt64
-
 abstract type  MultiplicativeInverse{T} end
 
 # Computes integer division by a constant using multiply, add, and bitshift.
diff --git a/test/floatfuncs.jl b/test/floatfuncs.jl
index 96ae477f0fdbd1..7523355095d772 100644
--- a/test/floatfuncs.jl
+++ b/test/floatfuncs.jl
@@ -41,7 +41,7 @@ for elty in (Float16,Float32,Float64), _ = 1:10
     @test hex2num(num2hex(x)) ≈ x
     @test hex2num(elty, num2hex(x)) ≈ x
 end
-@test_throws ArgumentError hex2num(String(rand('a':'f', rand(17:100))))
+@test_throws ArgumentError hex2num(String(rand(['a':'f';'0':'1'], rand(17:100))))
 
 # round
 for elty in (Float32,Float64)
diff --git a/test/int.jl b/test/int.jl
index ad2e9a1e8f1f2e..71768f1616c29c 100644
--- a/test/int.jl
+++ b/test/int.jl
@@ -1,7 +1,7 @@
 # This file is a part of Julia. License is MIT: https://julialang.org/license
 
 # Test integer conversion routines from int.jl
-
+using Base: BitInteger_types, BitFloat_types
 
 for y in (-4, Float32(-4), -4.0, big(-4.0))
     @test flipsign(3, y)  == -3
@@ -18,10 +18,10 @@ for y in (4, Float32(4), 4.0, big(4.0))
 end
 
 # Result type must be type of first argument
-for T in (Base.BitInteger_types..., BigInt,
+for T in (BitInteger_types..., BigInt,
           Rational{Int}, Rational{BigInt},
           Float16, Float32, Float64)
-    for U in (Base.BitInteger_types..., BigInt,
+    for U in (BitInteger_types..., BigInt,
               Rational{Int}, Rational{BigInt},
               Float16, Float32, Float64)
         @test typeof(copysign(T(3), U(4))) === T
@@ -124,7 +124,7 @@ for T in (UInt8, UInt16, UInt32, UInt64)
 end
 
 # Test bit shifts
-for T in Base.BitInteger_types
+for T in BitInteger_types
     nbits = 8*sizeof(T)
     issigned = typemin(T) < 0
     highbit = T(2) ^ (nbits-1)
@@ -203,7 +203,34 @@ end
 @test unsafe_trunc(Int8, -129) === Int8(127)
 
 # Test x % T returns a T
-for T in [Base.BitInteger_types..., BigInt],
-    U in [Base.BitInteger_types..., BigInt]
+for T in [BitInteger_types..., BigInt],
+    U in [BitInteger_types..., BigInt]
     @test typeof(rand(U(0):U(127)) % T) === T
 end
+
+@testset "reinterpret and unsigned/signed on $T" for T in BitInteger_types
+    t = -1 % T
+    A, B =  T <: Unsigned ? (Unsigned, Signed) : (Signed, Unsigned)
+    @test reinterpret(A, reinterpret(B, t)) == t
+    @test reinterpret(A, t) == t
+    @test reinterpret(B, t) != t
+    @test (T <: Unsigned ? unsigned : signed)(T) === T
+    @test unsigned(T) <: Unsigned
+    @test signed(T) <: Signed
+end
+
+@testset "unsigned/signed(::Bool)" begin
+    @test signed(true) === 1
+    @test signed(false) === 0
+    @test unsigned(true) === 1 % UInt
+    @test unsigned(false) === 0 % UInt
+    @test signed(Bool) === Int
+    @test unsigned(Bool) === UInt
+end
+
+@testset "unsigned/signed(::$T)" for T in BitFloat_types
+    @test signed(T(1)) === Int(1)
+    @test unsigned(T(1)) === UInt(1)
+    @test_throws InexactError signed(T(1.1))
+    @test_throws InexactError unsigned(T(1.1))
+end