Documentation improvements for rounding methods

src/sage/rings/real_arb.pyx

@@ -2067,7 +2067,7 @@ cdef class RealBall(RingElement):
 
         EXAMPLES:
 
-        It is possible to create balls whose midpoint is more precise that
+        It is possible to create balls whose midpoint is more precise than
         their parent's nominal precision (see :mod:`~sage.rings.real_arb` for
         more information)::
 

src/sage/rings/real_double.pyx

@@ -1521,15 +1521,20 @@ cdef class RealDoubleElement(FieldElement):
 
     def round(self):
         """
-        Given real number `x`, rounds up if fractional part is greater than
-        `0.5`, rounds down if fractional part is less than `0.5`.
+        Rounds ``self`` to the nearest integer. Uses the convention of rounding
+        half to even (i.e. if the fractional part of ``self`` is `0.5`, then it
+        is rounded to the nearest even integer).
 
         EXAMPLES::
 
             sage: RDF(0.49).round()
             0
             sage: a=RDF(0.51).round(); a
             1
+            sage: RDF(0.5).round()
+            0
+            sage: RDF(1.5).round()
+            2
         """
         return Integer(round(self._value))
 

src/sage/rings/real_mpfr.pyx

@@ -2990,8 +2990,10 @@ cdef class RealNumber(sage.structure.element.RingElement):
 
     def round(self):
         """
-         Rounds ``self`` to the nearest integer. The rounding mode of the
-         parent field has no effect on this function.
+         Rounds ``self`` to the nearest representable integer, rounding halfway
+         cases away from zero. 
+         Note : the rounding mode of the parent field has no effect on this
+         function.
 
          EXAMPLES::