Subject: Re: [HM] Any views on 'Number from Ahmes to Cantor'?
From: Milo Gardner (milogardner@juno.com)
Date: Mon Sep 18 2000 - 16:58:05 EDT
I thank David from requesting a brief review of this book. My view
is guarded as I will detail. Gazale set out to discuss Egyptian
fractions early in his book; however deferred it, saying it would be
discussed later. When the later time arrived Gazale discussed
an interesting but none historical view of the subject, as he freely
concluded. In the end, nothing of historical value was concluded
concerning reasons why Egyptians like Ahmes wrote in in exact form of
rational number conversions, nor the methods that were employed to
achieve concise series, as Greeks later adopted as central to their
view of number.
Other aspects of Gazale's book are not challenged in any way. My
primary question is: with a book titled from "Number from Ahmes to
Cantor" why was not more time and details of Egyptian number spent on
discussing Ahmes?
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Subject: Number from Ahmes to Cantor, added views
David's review of Gazale's "Number from Ahmes to Cantor" might like to
consider the RMP as read by Robin's Shute. It is difficult to completely
summarize Ahmes's view of number. Aspects of this subject are included
in the RMP 2/nth table as Robins-Shute's "The Rhind Mathematical Papyrus"
covered 2-3 pages to outline the Hultsch-Bruins method, as Gazale did
not consider. I suggest that the historical RMP 2/p series used aliquot
parts of "think of a highly divisible number between p/2 and p", as
suggested by Hultsch, and later independently confirmed by Bruins. The
Robins-Shute book was first published by the British Museum, and later
reprinted by Dover in 1987, covers this and many related number ideas of
Ahmes.
On one level Shute's summary on page 33 of Hultsch-Bruins introduces
several general difficulties in reading the RMP as a historical document
by saying:
"In considering what methods in the Egyptians may have used to double
unit fractions, it must be emphasized again that it is essential to
take note of such working as included in the text, and be cautious
about erecting any theory that does not have some attestation. On the
other hand, it must be borne in mind that RMP entries may serve
different purposes. They may include the actual methods employed by
the scribe, they may offer formal proofs of what he had has previously
worked out by other means, or they may represent the procedures that
an apprentice was expected to go through as part of his scribal
training so as to engender in him a better understanding of
mathematical principles and practices."
These are serious difficulties in reading the RMP, so I can understand
one reason for Gazale not discussing alternative ways to view 2/p series,
as an apprentice or scribal proof, all additive methodologies for all but
2/29, 2/37 and 2/41, as Shute cites. However, as one of several knotty
issues: why were all the so called 'proofs' listed only in the additive
duplation context, and not in the 'algebraic identity' aspect of
Hultsch-Bruins includes fewer than seven steps, points that a serious
look at Ahmes' view of number might like to ponder?
Taking a broader of Ahmes' number education than Gazale considered,
Ahmes would have known that from the Old Kingdom to the Middle Kingdom
major changes in numeration had taken place. One was the implementation
of ciphered numerals, a critical historical step, as Boyer analyzes the
innovation. On the ciphered numeral level Greeks also ciphered their
numerals, onto their alphabet, in a manner that Middle Kingdom Egyptians
would have recognized; while Romans followed the more cumbersome Old
Kingdom method of non-ciphered numerals.
To review the 'algebraic identity' content of the Hultsch-Bruins method,
rather than Shute's additive seven (7 ) steps, Ahmes was working with
data created from:
2/p - 1/A = (2A -p)/Ap
where A, a highly composite number selected in the range p/2 < A < p, as
a 'pick a number' game.
That is, Gazale could have spent a little time discussing the Middle
Kingdom practice of entally computing algebraic values, as noted by
'false position', a related technique. Note that for converting 2/p to
a concise unit fraction series a highly composite number A can range
anywhere between p/2 and p, and not only near p/2 as Shute suggests.
The key to selecting A is, and was, the lowest last term in the final
series, following these steps:
1. select a highly composite A, that minimizes the last term of the
series, such as A = 12 for 2/19, the first three term series in the
RMP table,
2. compute 2/p - 1/A, such as 2/19 - 1/12 = 5/(12*19)
3. find (2A -p), 5 in this case, as aliquot parts of 5, noting that
(4 + 1) and (3 + 2) are possible allows (3 + 2) to be chosen
(and selected by an LCM 'red auxiliary' number technique?)
4. write out the answer,
2/19 = 1/12 + ( 3 + 2)/(12*19)
= 1/12 + 1/76 + 1/114
This procedure could have generally taken at few as five steps when step
one was expanded to consider than A = 12 as not optimal. Other iterations,
as 'false position' employed, guessing at other final answers would have
a. shown that A = 10, or
2/19 - 1/10 = 1/190
produces a much larger last term than 1/.114
and that,
B. A = 14
2/19 - 1/14 = 9/(14*19)
2/19 = 1/14 + (7 + 2)/(14*19)
= 1/14 + 1/38 + 1/133
(note that A = 15 could be been tested, though not one odd value for
A has found in the RMP - though the ELMR clearly used A = 5, 25 for
a related 1/pq method. Had A = 15 been used
2/19 - 1/15 = 11/(15*19)
2/19 = 1/15 + (5 + 3 + 1) reaches only 10, and not the needed 11)
c. A = 16
2/19 - 1/16 = 13/(16*19)
2/19 = 1/16 + (8 + 4 + 1)/(16* 19)
or,
2/19 = 1/16 + 1/38 + 1/76 + 1/304
d. A = 18
2/19 - 1/18 = 17/(18*19)
2/19 = 1/18 + (9 + 3 + 1). again insufficient to reach 17, and
therefore A = 18 would have been excluded.
In conclusion this form of Hultsch-Bruins method uses an expanded 'false
position' guessing game, one that should be considered when attempting
to read an 'attested' historical Middle Kingdom 2/p series. Note that
for 2/43, A = 42 was used by Ahmes. Why not try calculating A = 22, 24,
26, 28, 30, 32, 34, 36, 38 and 40, and see why Ahmes excluded them in
favor of 42?
Milo Gardner
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