David Fowler is correct: I have no evidence that Archimedes knew about Babylonian notation. In fact, he probably didn't. I was probably thinking of his "exponential" notation in the "Sand Reckoner." Sorry for the slip.
But David Fowler's remarks about area simply reinforce my question about units. Even if the Greeks used Egyptian fractions in their area calculations, one must still ask: "What were the units of area they were denoting?" It's exactly because I am aware of the various "formulas" for area (some correct, others mostly incorrect) known to the Egyptians and Greeks that I ask the question. If you think the area of a quadrilateral is the product of the averages of the opposite sides, you still can get non-integer values for the area; however, even if everything comes out even, WHAT ARE THE UNITS? Is there a "square cubit" and is there, in the record, any mention of an area being, say 3 + 1/4 square cubits or somesuch?
One of the reasons I am interested in this is because if area was commonly measured in square units of some sort, the existence of irrationals would be a great shock. A rectangle with base 1 and height the diagonal of the unit square could not be decomposed into congruent squares, hence its area could not be expressed in terms of square units or squares 1/N th of a unit on a side (for any N).
(Of course, this is more of a shock for us, since we use square units all the time; in fact, that's how we think of area or measure once we've chosen a unit. It forces us to think in terms of limits even for the area of rectangles...)