diff --git a/scan.go b/scan.go
index 69f9a25..3199ef7 100644
--- a/scan.go
+++ b/scan.go
@@ -268,6 +268,11 @@ Loop:
 		}
 	}
 
+	// A lone "." has no digits. The numeric grammar requires at least one.
+	if len(buf) == 0 {
+		return ConversionSyntax
+	}
+
 	if big || z.compact == c.Inflated {
 		z.unscaled.SetString(string(buf), 10)
 		z.compact = c.Inflated
@@ -300,6 +305,10 @@ func (z *Big) scanExponent(r io.ByteScanner) error {
 
 	ch, err = r.ReadByte()
 	if err != nil {
+		if err == io.EOF {
+			// "1e" has an exponent mark and no digits.
+			return ConversionSyntax
+		}
 		return err
 	}
 	var neg bool
@@ -318,6 +327,7 @@ func (z *Big) scanExponent(r io.ByteScanner) error {
 	}
 
 	var exp uint64
+	digits := 0
 	for {
 		ch, err := r.ReadByte()
 		if err != nil {
@@ -347,6 +357,12 @@ func (z *Big) scanExponent(r io.ByteScanner) error {
 			return Overflow
 		}
 		exp = v
+		digits++
+	}
+
+	// "1e+" and "1e-" have a sign and no digits.
+	if digits == 0 {
+		return ConversionSyntax
 	}
 
 	if neg {
diff --git a/scan_test.go b/scan_test.go
index 7d32bb1..a5c06aa 100644
--- a/scan_test.go
+++ b/scan_test.go
@@ -52,8 +52,6 @@ func BenchmarkBig_SetString(b *testing.B) {
 	globOk = ok
 }
 
-
-
 func TestUnmarshalTextTwice(t *testing.T) {
 	x := &Big{}
 
@@ -64,6 +62,35 @@ func TestUnmarshalTextTwice(t *testing.T) {
 	}
 }
 
+func TestSetStringTruncatedNumeric(t *testing.T) {
+	for _, s := range []string{"1e", "1E", "1e+", "1e-", "-1e", "1.2e", "."} {
+		x, ok := new(Big).SetString(s)
+		if !ok {
+			t.Fatalf("%s: SetString returned false", s)
+		}
+		if !x.IsNaN(0) {
+			t.Fatalf("%s = %s, want NaN", s, x)
+		}
+		if x.Context.Conditions&ConversionSyntax == 0 {
+			t.Fatalf("%s: missing ConversionSyntax", s)
+		}
+	}
+
+	kept := []struct{ in, want string }{
+		{"1e2", "1E+2"},
+		{"1e+2", "1E+2"},
+		{"1e-1", "0.1"},
+		{"1.", "1"},
+		{".5", "0.5"},
+		{"10.00", "10.00"},
+	}
+	for _, tc := range kept {
+		x, ok := new(Big).SetString(tc.in)
+		if !ok || x.String() != tc.want {
+			t.Fatalf("%s = %s ok=%v, want %s", tc.in, x, ok, tc.want)
+		}
+	}
+}
 
 func TestSetStringTwice(t *testing.T) {
 	x := &Big{}