diff --git a/src/algebra.jl b/src/algebra.jl
index 669cc4a..7cdd946 100644
--- a/src/algebra.jl
+++ b/src/algebra.jl
@@ -17,9 +17,14 @@
 -(::InfiniteCardinal) = -∞
 
 
+# The direction of an infinity in half turns, which for a real is its sign bit.
+@inline _halfturns(x) = signbit(x)
+@inline _halfturns(x::Complex) = angle(x)/π
+
+
 # addition
 @inline toinf(x) = RealInfinity(signbit(x))
-@inline toinf(x::Complex) = ComplexInfinity(angle(x))
+@inline toinf(x::Complex) = ComplexInfinity(_halfturns(x))
 @inline toinf(x::ComplexInfinity) = x
 
 @inline _infadd(x, y) = angle(x) == angle(y) ? y : throw(ArgumentError("Angles must be the same to add ∞"))
@@ -44,13 +49,10 @@
 
 # multiplication
 
-@inline _sb(x) = signbit(x)
-@inline _sb(x::Complex) = angle(x)/π # overloading `signbit` causes type piracy 
-
-@inline __mul(x, y::AllInfinities) = RealInfinity(_sb(x) ⊻ _sb(y))
-@inline __mul(x, y::ComplexInfinity) = ComplexInfinity(_sb(x) + _sb(y))
-@inline __mul(x, y::ComplexInfinity{Bool}) = ComplexInfinity(_sb(x) ⊻ _sb(y))
-@inline __mul(x::Complex, y::ComplexInfinity{Bool}) = ComplexInfinity(_sb(x) + _sb(y))
+@inline __mul(x, y::AllInfinities) = RealInfinity(_halfturns(x) ⊻ _halfturns(y))
+@inline __mul(x, y::ComplexInfinity) = ComplexInfinity(_halfturns(x) + _halfturns(y))
+@inline __mul(x, y::ComplexInfinity{Bool}) = ComplexInfinity(_halfturns(x) ⊻ _halfturns(y))
+@inline __mul(x::Complex, y::ComplexInfinity{Bool}) = ComplexInfinity(_halfturns(x) + _halfturns(y))
 @inline __mul(x::Integer, y::InfiniteCardinal) = x > 0 ? y : throw(ArgumentError("Cannot multiply $x * $y"))
 
 @inline _mul(x, y) = iszero(x) ? throw(ArgumentError("Cannot multiply $x * $y")) : __mul(infpromote(x, y)...)
diff --git a/test/runtests.jl b/test/runtests.jl
index a1d5069..e51b1e5 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -283,12 +283,36 @@ Base.iterate(s::CharString, i::Integer=1) = i ≤ length(s.chars) ? (s.chars[i],
              ComplexInfinity() * ∞ ≡ ComplexInfinity() * RealInfinity() ≡ ComplexInfinity()
 
         @test  2.0im*∞ ≡ ∞*2.0im ≡ 2.0im * RealInfinity() ≡ RealInfinity() * 2.0im ≡ ComplexInfinity(1/2)
+        # On an axis `angle(x)/π` is exact, `angle` giving exactly `0`, `±π/2` or `π` and halving being exact.
+        for (z, s, w) in ((2.0+0.0im, 0.0, complex(Inf, 0.0)), (2.0im, 0.5, complex(0.0, Inf)),
+                          (-2.0+0.0im, 1.0, complex(-Inf, 0.0)), (-2.0im, -0.5, complex(0.0, -Inf)))
+            @test z*∞ ≡ ComplexInfinity(s)
+            @test z*∞ == w
+        end
         @test 2ComplexInfinity() ≡ ComplexInfinity()*2 ≡ ComplexInfinity()
 
-        @test exp(im*π/4)*∞ == Inf+im*Inf
+        # Off the axes it is not exact in general: `angle(exp(im*π/8))/π` gives 0.12500000000000003.
+        for θ in (π/8, π/4, 0.3, -2.4)
+            @test angle(exp(im*θ)*∞) ≈ angle(∞*exp(im*θ)) ≈ θ
+        end
+        # The direction is the one the operand actually has, so rescaling keeps it only while it is exact.
+        @test 4*(0.3+0.1im)*∞ ≡ (0.3+0.1im)*∞
+        @test 3*(0.3+0.1im)*∞ ≢ (0.3+0.1im)*∞
         @test exp(im*π/4)+∞ == ∞
         @test Inf + im + ∞ ≡ ComplexInfinity()
 
+        # An infinite summand reaches `_infadd` through `toinf`, which has to answer in half turns.
+        @test complex(Inf, 0.0) + ∞ ≡ ComplexInfinity()
+        @test complex(-Inf, 0.0) + (-∞) ≡ -ComplexInfinity()
+        @test complex(0.0, Inf) + ComplexInfinity(1/2) ≡ ComplexInfinity(1/2)
+        @test complex(0.0, -Inf) + ComplexInfinity(-1/2) ≡ ComplexInfinity(-1/2)
+        @test_throws ArgumentError complex(0.0, Inf) + ∞
+
+        # Two infinite parts are the only way an infinite `Complex` points off the axes.
+        for (x, y, s) in ((Inf, Inf, 1/4), (-Inf, Inf, 3/4), (-Inf, -Inf, -3/4), (Inf, -Inf, -1/4))
+            @test complex(x, y) + ComplexInfinity(s) ≡ ComplexInfinity(s)
+        end
+
         @test Inf == ComplexInfinity()
         @test ComplexInfinity() == Inf