Conference rusure::math

    
    Egyptian Arithmetic.
    
    Numbers where represented as sums of units, tens, hundreds, thousands,
    etc.. The Egyptians however did not use radix based arithmetic. Units
    were represented as a grouping of 1 to 9 vertical strokes. Tens were
    represented by a grouping of 1 to 9 arch-shaped symbols, thousands by a
    group of 1 to 9 squiggles that looked somewhat like our symbol for 9.
    and so on. The Egyptians did not have a 0 place holder so they needed a
    new symbol for each new order of magnitude.
    
    The numbers were written from right to left so that the higher order of
    magnitude is to the right of the lower order of magnitude which is the
    opposite of the way we write our numbers. However the addition had to
    start with the lower order of magnitude so addition proceeded from left
    to right. Ours proceeds from right to left for the same reason.
    
    Since I can't reproduce the shapes of the Hieratic symbols as they
    appear, let me represent units by a grouping of 1 to 9  "|"; tens by a
    grouping of 1 to 9 "^"; hundreds by a grouping of 1 to 9 "@".
    
    Let us add 37 to 46 Egyptian style:
    
    	|||||||    ^^^  (37)
    
    	||||||     ^^^^ (46)
    
    
    to get
    
    	|||	   ^^^^ (83)
    		   ^^^^
    
    Subtraction of a lesser number from a greater would have proceeded in a
    similar fashion with a "borrow" from the next higher order of magnitude
    done as necessary. Gillings speculates that the Scribes may have had
    addition tables (which also would have been subtraction) tables to
    faciliate their computations. 
    
    

    
    Next let us consider multiplication and division.  
    
    The Egyptians found that a times-two table was quite sufficient for
    multipication. Let us consider the product 23 x 8.
    
    The scribe did it thus:
    
    	1			8
    	2			16
    	4			32
    	8			64
    	16			128
    
    23 is 1 + 2 + 4 + 16 so we add together 8 + 16 + 32 + 128 to get 84.
    
    Thus the Egyptians were doing binary arithmetic many millenia prior to
    electronic computers.
    
    How did they do division?  Consider the inverse problem 184 divided by
    8. If you look at the right hand column you will see that 8 + 16 + 32 +
    128 adds up to 184. By adding the corresponding powers of two in the
    left hand column, 1 + 2 + 4 + 16 you get 23 which is the correct
    answer.
    

    
    Next we consider fractions. Here is where things become "strange" from
    our modern point of view. The Egyptian wrote fractions as sums of
    reciprocals, i.e. sums of fractions of the from 1/n, with the notable
    exception of the fraction 2/3 which they permited as a component
    summand for fractions. 
    
    Consider the fraction 3/5. How might we represent such a fraction?
    
    	3/5 = 1/2 + 1/10 is one representation. 
    
    We could also write  
    
    	3/5 = 1/3 + 1/5 + 1/15.
    
    In general a fraction could have many representations as sums of
    reciprocals. There appears to be some kind of precedence rules to
    select an easier to use representation. The rules, according to
    Gillings went like this:
    
    	A. A representation with fewer terms is better than a
    	   representation with more terms.
    
    	B. Of two representations of the same length, choose the
    	   one whose smallest fraction has the smaller reciprocal.
    
    	C. Whenever possible keep the denonminators even, so the twice two
    	   table can be used.
    
    There sometimes was a conflict between B and C and the scribe
    occasionally took a longer representation to keep all the denominators
    even. It wasn't hard and fast however.
    
    We now can raise some problems of interest to us moderns:
    
    	Given that the order of applying the rules is as above:
    
    	1. Give an algorithm for generating the best representation of a 
           fraction p/q where p < q and p, q relatively prime.
    
    	2. How many representations of whatever quality are there for a
    	   fraction. 
    
    	3. How many representations of a given length.
    
    You may ask why did the Egyptians insist on this way of representing
    general fractions. One of the answers suggested by Gillings is the way
    of subdividing, say loaves of bread. If you had two loaves to be
    divided amoung three people, you might divide the loaves (2/3,1/3), 
    (2/3, 1/3). This would not have pleased the Egyptians since two persons
    would get a 2/3 length piece and the third would get two 1/3 pieices. 
    
    The Egyptians apparently preferred that not only does every body get an
    equal amount but that the shares whould consist of the same number of
    pieces.  This "ethical" requirement made the reciprocal form of
    fractions more useful for their purposes.
    
    In the next replies we will do some arithmetic with fractions.
    

	I haven't read the book (or the 4 original manuscripts) but I wonder if
they aren't a rather small sample on which to base the conclusions that 
Egyptian fractions were central to Egyptian mathematics.   Or is there in fact
a lot more ancient mathematical graffitti daubed on pyramids, sphinxes,
sarcophagi, etc?

	Imagine if human civilization were blasted off the surface of the 
planet, and all that were left of human mathematical wisdom were a few pages
from RUSURE::MATH.   Aliens would perhaps get a rather distorted impression of
human mathematical accomplishment and preoccupations.

	Presumably the library at Alexandria vacuumed up all accessible 
mathematical documents for centuries, before finally being incinerated.   [I 
read recently that one of the Ptolemies insisted that all visitors to Alex
were searched for documents.   If *any* were found, they were copied 
immediately, and the visitor was given the copy, whilst the original went to
the library.]

	There's another couple of interesting puzzles involving Egyptian
fractions, which I don't think are in the Notesfile.

	(4) What natural numbers cannot be expressed as the sum of natural
numbers whose reciprocals sum to 1?

	(5) What natural numbers cannot be expressed as the sum of distinct 
natural numbers whose reciprocals sum to 1?

Cheers,
Andrew.

    rep .4
    
    Quite right. Unfortunately, scholars must work with the materials they
    have. The destruction of the Library at Alexandria was a *major* loss.
    We can only speculate on how much the Ancients really knew.
    
    I have this fantasy. There in the basement of the library, with the
    flames licking about, was the definitive paper (papyrus) with the proof
    that P = NP. 8>:)