diff --git a/src/lib/number.jl b/src/lib/number.jl
index ced2d0ca9..b05100b40 100644
--- a/src/lib/number.jl
+++ b/src/lib/number.jl
@@ -20,8 +20,13 @@ end
 
 @adjoint (T::Type{<:Complex})(re, im) = T(re, im), c̄ -> (nothing, real(c̄), imag(c̄))
 
+# we define these here because ChainRules.jl only defines them for x::Union{Real,Complex}
+
+@adjoint abs2(x::Number) = abs2(x), Δ -> (real(Δ)*(x + x),)
 @adjoint real(x::Number) = real(x), r̄ -> (real(r̄),)
 @adjoint conj(x::Number) = conj(x), r̄ -> (conj(r̄),)
 @adjoint imag(x::Number) = imag(x), ī -> (real(ī)*im,)
 
-@adjoint abs2(x::Number) = abs2(x), Δ -> (real(Δ)*(x + x),)
+# for real x, ChainRules pulls back a zero real adjoint, whereas we treat x
+# as embedded in the complex numbers and pull back a pure imaginary adjoint
+@adjoint imag(x::Real) = zero(x), ī -> (real(ī)*im,)