diff --git a/spec/std/complex_spec.cr b/spec/std/complex_spec.cr
index 65add18f8533..2b90239d0796 100644
--- a/spec/std/complex_spec.cr
+++ b/spec/std/complex_spec.cr
@@ -265,6 +265,12 @@ describe "Complex" do
     it "complex / complex" do
       ((Complex.new(4, 6.2))/(Complex.new(0.5, 2.7))).should eq(Complex.new(2.485411140583554, -1.0212201591511936))
       ((Complex.new(4.1, 6.0))/(Complex.new(10, 2.2))).should eq(Complex.new(0.5169782525753529, 0.48626478443342236))
+
+      (1.to_c / -1.to_c).should eq(-1.to_c)
+      assert_complex_nan 1.to_c / Float64::NAN
+
+      (1.to_c / 0.to_c).real.abs.should eq(Float64::INFINITY)
+      (1.to_c / 0.to_c).imag.nan?.should be_true
     end
 
     it "complex / number" do
diff --git a/src/complex.cr b/src/complex.cr
index 65fbc9204b59..e2a5830b395a 100644
--- a/src/complex.cr
+++ b/src/complex.cr
@@ -237,14 +237,28 @@ struct Complex
 
   # Divides `self` by *other*.
   def /(other : Complex) : Complex
-    if other.real <= other.imag
-      r = other.real / other.imag
-      d = other.imag + r * other.real
-      Complex.new((@real * r + @imag) / d, (@imag * r - @real) / d)
-    else
+    if other.real.nan? || other.imag.nan?
+      Complex.new(Float64::NAN, Float64::NAN)
+    elsif other.imag.abs < other.real.abs
       r = other.imag / other.real
       d = other.real + r * other.imag
-      Complex.new((@real + @imag * r) / d, (@imag - @real * r) / d)
+
+      if d.nan? || d == 0
+        Complex.new(Float64::NAN, Float64::NAN)
+      else
+        Complex.new((@real + @imag * r) / d, (@imag - @real * r) / d)
+      end
+    elsif other.imag == 0 # other.real == 0
+      Complex.new(@real / other.real, @imag / other.real)
+    else # 0 < other.real.abs <= other.imag.abs
+      r = other.real / other.imag
+      d = other.imag + r * other.real
+
+      if d.nan? || d == 0
+        Complex.new(Float64::NAN, Float64::NAN)
+      else
+        Complex.new((@real * r + @imag) / d, (@imag * r - @real) / d)
+      end
     end
   end