Maeve, 

     This is a mostly good discussion of Egyptian mathematics and 
   a consideration of whether what they did constitutes algebra
   or not.  As you say, there are definitely differences of opinion,
   but I think that if you look at the solutions presented to some
   of the problems in the Rhind and Moscow papyri, then Joseph's
   "rhetorical algebra" suggestion certainly seems plausible.  

   I have some comments for you, though:

   (1) The question of whether the Egyptians really understood
       that every integer could be represented in binary or 
       base 2 form is one of those semantic questions that 
       is hard to answer.  On the other hand, I think an example
       or two would have made it clear that their whole system
       of multiplication was based on the fact that it is possible(!)
       So the "inkling" is probably much more than that.  Here's 
       what they would do to multiply 9 x 12 for instance.  I'll 
       pick the factor 12 and start doubling:

                   1 x 12 = 12 <=
                   2 x 12 = 24
                   4 x 12 = 48
                   8 x 12 = 96 <=

       (They wouldn't need to double again, since 2 x 8 = 16 is 
       larger than 9.)  Now to make 9, you need 1 + 8, so mark
       those rows above.  Adding, 12 + 96 = 108, and that is 9 x 12.  
       As you can see, you really need to be making any multiple
       of 12 out of these steps of the repeated doubling to do 
       a general product n x 12.  So the Egyptians are really 
       relying on the fact that n can always be expressed as a
       sum of powers of 2.  It would have been good to present
       example(s) like this!  Note that the two equations at the 
       top of page 3 are not quite right, though.  They should 
       say:

             10 = 0 x 1 + 1 x 2 + 0 x 4 + 1 x 8
             12 = 0 x 1 + 0 x 2 + 1 x 4 + 1 x 8

   (2) The discussion of the Egyptian way of working with fractions
       doesn't quite go far enough, and one of the things you said
       is not right.  It's true that the Egyptians wanted to write
       fractions using numerators of 1 always, so they would not
       write 2/5.  However, 1/5 + 1/10 = 3/10, which is not the 
       same as 2/5.  To get 2/5, you need 1/3 + 1/15 = 5/15 + 1/15 = 6/15.
       There are many ways to generate these sums.  The Egyptians
       used tables of standard forms for the common ones (pretty
       much the same idea as the Babylonians' tables for various
       things).  

   Grade: B+