Lecture: More on Egyptian

Since the Egyptians did not like to used complex fractions, except 2/3, it would be convenient to have a simple way to express them as a unit fractions. As an example take 5/6

5/6 = (2+3)/6
      = 2/6 + 3/6
      = 1/3 + 1/2

As a more complicated example try 2/35. One would expect  1/35 +1/35 however the Egyptians did not like to have repeated fractions most likely due to the argument that any fraction can be written as repeating unit fractions and that mathematically it would not get them any where. Instead


2/35 = (2/35) (18/18)
        = 35/ (35*18)  +  1/ (35*18)
        = 1/18 + 1/ 630   [Final answer]

They multiplied by 1 (18/18 = 1) and manipulated the equation to get a sum of units.

Another math problem the Egyptians might run into is that they are presented with a list of unit fractions and they would need to figure out what other unit fractions will be needed to have a sum of 1. They would follow the steps below.

Problem: Complete the list to 1:  2/3, 1/15
Starting off with  2/3     1/15   
                                             Multiply by 15 to each fraction (multiply by the largest denominator)
                          30/3    15/15
                                                   Simplify
                          10       1

                           Add 10 +1 = 11

                           15 -11 = 4         This is the remainder
Then,
           4/15 =  in unit fractions?
                   = 1/15 + 3/15
                   = 1/15 + 1/5

Thus,  2/3 +1/15 + 1/15 + 1/5 = 1.

Unanswered Questions:
1) Was there any practical uses knowing how to add to a list of unit fractions to have a sum of 1? How did they come up with those steps since the modern way seems more intuitive?

Evaluation:
Interesting: 8
Quality: 7
Complexity: 5