diff --git a/base/irrationals.jl b/base/irrationals.jl
index 6513e3269a4d7..99915eb699966 100644
--- a/base/irrationals.jl
+++ b/base/irrationals.jl
@@ -48,8 +48,7 @@ AbstractFloat(x::AbstractIrrational) = Float64(x)::Float64
 Float16(x::AbstractIrrational) = Float16(Float32(x)::Float32)
 Complex{T}(x::AbstractIrrational) where {T<:Real} = Complex{T}(T(x))
 
-# XXX this may change `DEFAULT_PRECISION`, thus not effect free
-@assume_effects :total function Rational{T}(x::AbstractIrrational) where T<:Integer
+function _irrational_to_rational(::Type{T}, x::AbstractIrrational) where T<:Integer
     o = precision(BigFloat)
     p = 256
     while true
@@ -63,13 +62,16 @@ Complex{T}(x::AbstractIrrational) where {T<:Real} = Complex{T}(T(x))
         p += 32
     end
 end
-Rational{BigInt}(x::AbstractIrrational) = throw(ArgumentError("Cannot convert an AbstractIrrational to a Rational{BigInt}: use rationalize(BigInt, x) instead"))
+Rational{T}(x::AbstractIrrational) where {T<:Integer} = _irrational_to_rational(T, x)
+_throw_argument_error_irrational_to_rational_bigint() = throw(ArgumentError("Cannot convert an AbstractIrrational to a Rational{BigInt}: use rationalize(BigInt, x) instead"))
+Rational{BigInt}(::AbstractIrrational) = _throw_argument_error_irrational_to_rational_bigint()
 
-@assume_effects :total function (t::Type{T})(x::AbstractIrrational, r::RoundingMode) where T<:Union{Float32,Float64}
+function _irrational_to_float(::Type{T}, x::AbstractIrrational, r::RoundingMode) where T<:Union{Float32,Float64}
     setprecision(BigFloat, 256) do
         T(BigFloat(x)::BigFloat, r)
     end
 end
+(::Type{T})(x::AbstractIrrational, r::RoundingMode) where {T<:Union{Float32,Float64}} = _irrational_to_float(T, x, r)
 
 float(::Type{<:AbstractIrrational}) = Float64
 
@@ -107,14 +109,18 @@ end
 <=(x::AbstractFloat, y::AbstractIrrational) = x < y
 
 # Irrational vs Rational
-@assume_effects :total function rationalize(::Type{T}, x::AbstractIrrational; tol::Real=0) where T
+function _rationalize_irrational(::Type{T}, x::AbstractIrrational, tol::Real) where {T<:Integer}
     return rationalize(T, big(x), tol=tol)
 end
-@assume_effects :total function lessrational(rx::Rational{<:Integer}, x::AbstractIrrational)
-    # an @assume_effects :total version of `<` for determining if the rationalization of
-    # an irrational number required rounding up or down
+function rationalize(::Type{T}, x::AbstractIrrational; tol::Real=0) where {T<:Integer}
+    return _rationalize_irrational(T, x, tol)
+end
+function _lessrational(rx::Rational, x::AbstractIrrational)
     return rx < big(x)
 end
+function lessrational(rx::Rational, x::AbstractIrrational)
+    return _lessrational(rx, x)
+end
 function <(x::AbstractIrrational, y::Rational{T}) where T
     T <: Unsigned && x < 0.0 && return true
     rx = rationalize(T, x)
diff --git a/base/mathconstants.jl b/base/mathconstants.jl
index 4bb8c409acf00..de6b98cea634d 100644
--- a/base/mathconstants.jl
+++ b/base/mathconstants.jl
@@ -16,6 +16,26 @@ Base.@irrational γ        euler
 Base.@irrational φ        (1+sqrt(big(5)))/2
 Base.@irrational catalan  catalan
 
+const _KnownIrrational = Union{
+    typeof(π), typeof(ℯ), typeof(γ), typeof(φ), typeof(catalan)
+}
+
+function Rational{BigInt}(::_KnownIrrational)
+    Base._throw_argument_error_irrational_to_rational_bigint()
+end
+Base.@assume_effects :foldable function Rational{T}(x::_KnownIrrational) where {T<:Integer}
+    Base._irrational_to_rational(T, x)
+end
+Base.@assume_effects :foldable function (::Type{T})(x::_KnownIrrational, r::RoundingMode) where {T<:Union{Float32,Float64}}
+    Base._irrational_to_float(T, x, r)
+end
+Base.@assume_effects :foldable function rationalize(::Type{T}, x::_KnownIrrational; tol::Real=0) where {T<:Integer}
+    Base._rationalize_irrational(T, x, tol)
+end
+Base.@assume_effects :foldable function Base.lessrational(rx::Rational, x::_KnownIrrational)
+    Base._lessrational(rx, x)
+end
+
 # aliases
 """
     π