diff --git a/core/kernel/Float_spec.rb b/core/kernel/Float_spec.rb
index 1705205996..e00fe81572 100644
--- a/core/kernel/Float_spec.rb
+++ b/core/kernel/Float_spec.rb
@@ -163,6 +163,7 @@ def to_f() 1.2 end
       -> { @object.send(:Float, "+1.") }.should raise_error(ArgumentError)
       -> { @object.send(:Float, "-1.") }.should raise_error(ArgumentError)
       -> { @object.send(:Float, "1.e+0") }.should raise_error(ArgumentError)
+      -> { @object.send(:Float, "1.e-2") }.should raise_error(ArgumentError)
     end
   end
 
@@ -172,6 +173,7 @@ def to_f() 1.2 end
       @object.send(:Float, "+1.").should == 1.0
       @object.send(:Float, "-1.").should == -1.0
       @object.send(:Float, "1.e+0").should == 1.0
+      @object.send(:Float, "1.e-2").should be_close(0.01, TOLERANCE)
     end
   end
 
diff --git a/core/string/to_f_spec.rb b/core/string/to_f_spec.rb
index a91ccc168e..abfd2517b6 100644
--- a/core/string/to_f_spec.rb
+++ b/core/string/to_f_spec.rb
@@ -127,4 +127,16 @@
       }.should raise_error(Encoding::CompatibilityError, "ASCII incompatible encoding: UTF-16")
     end
   end
+
+  it "allows String representation without a fractional part" do
+    "1.".to_f.should == 1.0
+    "+1.".to_f.should == 1.0
+    "-1.".to_f.should == -1.0
+    "1.e+0".to_f.should == 1.0
+    "1.e+0".to_f.should == 1.0
+
+    ruby_bug "#20705", ""..."3.4" do
+      "1.e-2".to_f.should be_close(0.01, TOLERANCE)
+    end
+  end
 end