Calvin C. Clawson's latest book, MATHEMATICAL SORCERY, The
Secrets of Number (Plenum Press, 1999) requires additional
explanations. Within the "The Wonderful Egyptian" section
in chapter one, Clawson primarily offers the 1925 suggestions
of David E. Smith, in the following words:
"...
We know that the Egyptians used fractions, but they were only
unit fractions, where the numerator is a 1 while the denominator
can be any number. For example, to represent the fraction 3/7
the Egyptians wrote it as the sum of fraction 1/3 + 1/11 + 1/231.
We see at once the great awkwardness of this system for it requires
us to compute cumbersome calculations just to add or subtract
simple fractions. To make their work easier, the Egyptians used
extensive tables of unit fractions in performing computations
on fractions.
The Egyptians didn't evolve a more efficient system for recording
fractions because of their system symbols. To represent a unit
fraction they drew a small oval or do over a whole number that
represented the fraction's denominator. This made it easy to
write a fraction with a numerator less than 1 unless a symbol
was invented. Hence it was the Egyptian symbolism that prevented
them from exploring a more useful form (of mathematics)"
There are five intellectual and historical problems with the
above analysis.
1. Clawson's 3/7 = 1/3 + 1/11 + 1/231 is a red herring, not
even historical series (as I mentioned a couple of days ago).
2. Who is we, that " see at once the great awkwardness of this
system"? DE Smith surely was one, since he only saw the
additive side of Egyptian fractions, and totally ignored
the well thought out 1895 analysis provided by Hultsch.
Hultsch shows that all but one RMP 2/p series used the
rule,
2/p - 1/A = (2A -p)/Ap
where A could be any highly divisible number, selected in
the range p/2 < A < p, can be chosen such that its divisors
additively compute 2A - p, and thereby ALWAYS partitions 2/p.
2/89 is a wonderful example, the RMP selected A = 60
with 2/89 - 1/60 = (120 - 89)/(60*89) with 120 - 89 = 31
being divisors of 60, by selecting 15 + 6 + 5, or
2/89 = 1/60 + (15 + 6 + 5)/(60*89)
= 1/60 + 1/356 + 1/534 + 1/890
(was a relatively advanced series of mental calculations).
3. Egyptians often did not use tables to make their initial
calculations. Tables were used for other purposes, such
as listing the optional values for 2/p and 2/pq series
(which, by the way were computed by different algebraic
identities - the second one being 2/pq = 2/A x A/pq
with A = (p + 1 and p + q).
4. Smith and Clawson's suggestion that Egyptians lacked 'proper'
symbols for writing unit fraction is another red herring.
I would like to see Clawson or anyone else's proof that
Egyptian symbolism limited their mathematical development.
For example, as easily shown below, Greeks used Egyptian
unit fraction symbols, as the same notation, with only one
minor substitution. The Egyptian oval or dot was replaced
by ( ') such that Greeks wrote 1/p = p'.
5. Clearly the number theory side of Egyptian fraction has
been thrown out, like the baby and the bath water, by
Smith's 1920's thinking.
One day a fresh reading of the ancient texts will show large
chunks of number theory, in both Greece and Egypt, written
in equivalent notations. But that day occurs, will the 1920's
type of thinkers still ignore the hard facts. Will they continue
to conclude that Greeks first developed number theory, and that
any connection to Egyptian arithmetic is only a coincidence?
Regards to all,
Milo Gardner