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p/q converted to Egyptian fractions in 2,000 BCE?
Posted:
Mar 1, 1996 12:04 PM
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Final comments are requested to iron out obvious historical and/or
mathematical errors contained in the following summary of Middle Kingdom
Egyptian fractions, sometimes called hieratic fractions.
As an introduction this paper proposes to historically combine the
EMLR and RMP 2/nth table as one document in a manner that shows that
Old Kingdom Horus-Eye fractions and Middle Kingdom hieratic fractions
were closely related, but dissimilar in several important respects. That is
to say, this one document proposal is new and should be openly discussed
as plausible, worth of a serious investigation, or clearly a risky
historical mixing of apples and oranges, as proven here on this discussion
group.
One theme of this paper, if the EMLR and RMP can not be refuted as belonging
to the same Middle Kingdom hieratic fraction tradition, strongly suggests
that p/q, as rational numbers, were easily converted into exact unit
fraction series as early as 2,000 BCE, much as Horus-Eye fractions converted
p/q into its inexact decimal fraction series. The Moscow Papyrus, written
on or about 2,000 BCE, provides evidence, such as its writing of 2/5 by
the same algorithm that is found in the improved EMLR and RMP, that hieratic
fractions were not always computed by the Horus-Eye duplation multiplication
paradigm, as now accepted by the Egyptology community (Shute). A new and or
historically compatible/alternate Middle Kingdom hieratic paradigm is
proposed by:
1. EMLR, The Egyptian Mathematical Leather Roll, 26 equations (Gillings)
a. 1/p = 1/2p + 1/2p and 1/p = 1/3p + 1/3p + 1/3p
b. 1/p(1/2) = 1/p(1/3 1/6) and 1/pq(1/3 1/6)
c. 1/p(1) = 1/p(1/2 1/3 1/6) and 1/pq(1/2 1/3 1/6)
d. 1/p(2/q) where 2/q was taken from the RMP 2/nth table such as
(1) 1/p(2/7) = 1/p(1/4 + 1/28) and
(2) line 17's obvious error of 1/13 = 1/28 1/49 1/96, surely was not
(as Gillings suggested) an attempt to write -
(a) 1/13(1) = 1/13(1/2 1/3 1/6) = 1/26 1/39 1/78, using rule 1.c, but
(b) 1/13= 1/3(3/13) = 1/3(1/8 1/17 1/52 1/104) = 1/24 1/39 1/156 1/312
since 2/13 = 1/8 1/52 1/104 from the RMP, as also hinted by
e. Lines 1, 2 and 3 from the EMLR which shows
(1) 1/8=1/10 1/40 = 1/10(1/1 1/4) = 1/10(5/4)=1/pq= 1/p(1/(q+1)(1/1 1/q)
(2) 1/4 = 1/5 1/20 = 1/5(1/1 1/4) = 1/5(5/4) =1/pq =1/(pq+1)(1/1 + 1/pq)
(3) 1/3 = 1/4 1/12 = 1/4(1/1 1/3) = 1/4(4/3) = 1/p = 1/(p+1)(1/1 + 1/p)
2. RMP, 2/nth table (Shute, Gillings, Chace and others)
a. 2/p - 1/a = (2a -p)/ap where a is a highly divisible number,
usually about 2/3rd of p; with 2a -p additively composed of divisors of a.
Note: Every 2/nth table 2/p unit fraction series follows this one rule
much as Hultsch, Bruins and several others have proposed over the
last 100 years.
b. Otto Neugebauer, the dominate scholar on which the current Egyptology
view of Egyptian fractions rests, reported only a muddled version of
the easy to read composite case:
(1) 2/pq = (1/p + 1/pq)2/(p + 1),
than computes all composite 2/nth table member except 2/35, 2/91, 2/95.
It is important to note that the only positive Egyptian fractions view
that Neugebauer included in his Exact Sciences in Antiquity analysis
is the acceptance of algorithm(1) with the form:
"2/n=1/3(1/n) + 5/3(1/n)... (with the comment)...
in this way, more and more cases of the table can be reached
and it appears to me there is little doubt that we have found in essence
the procedure which has lead to these rules of replacement of 2/n by
the sum of unit fraction." As a counter example to Neugebauer note that
(2) 2/pq =(1/p + 1/q)2/(p + q)
is clearly present in the 2/35 and 2/91 cases, as not seen by Neugebauer,
even though algorithm 2.b.(2) is simply read as the product of the
arithmetic mean (A) and harmonic mean (H), seen in the form 2/AH,
a common Ancient Near East pattern.
(3) Wrapping up the final exception, 2/95, is achieved by the trivial
form 2/95 = 2/19(1/5) where 2/19 was taken from equation 2.b.(1).
c. In conclusion, the EMLR and RMP as proposed as one document presents an
interesting set of patterns. Two clues that tend to closely link the EMLR
and RMP, mathematically and historically, beyond Henry Rhind bringing
both back to England in 1855, can be summarized by:
(1) Three of the 26 EMLR lines contain RMP 2/nth table members. One error
contained on line 17 is interesting in that a student may have been
confused in the writing of 1/13 as 1/3(3/13) as also suggested by lines
1: 1/8 = 1/10 1/40 = 1/10(1/1 1/4) = 1/10(5/4)
2: 1/4 = 1/5 1/20 = 1/5(1/1 1/4) = 1/5(5/4)
3: 1/3 = 1/4 1/12 = 1/4(1/1 1/3) = 1/4(4/3)
For some unknown reason the EMLR stated 1/13 = 1/7(3/7) = 3/49 rather
than the correct value 3/39. Considering several EMLR rules available the
student may have simply been confused.
(2) One RMP 2/nth table line, 2/101, contains an EMLR type algorithm, as
noted by:
2/101 = 1/p(1/1 + 1/2 + 1/3 + 1/6).
I thank one and all for the many supportive comments, posted to MAA and
to my private email box. Closing out this phase of my investigation has
not been easily. The subtle aspects of Egyptian fractions that have long
confused ancient students and modern scholar alike, including myself on
many points, may now be clearing up.
References:
Boyer, C.B., 1968, History of Mathematics, John Wiley, 1985 re-print
Princeton University Press.
Bruckheimer, M, and Salomon Y., The RMP Unit Fraction System,
Historia Mathematica, Nov. 1977.
Chace, A. B., 1927, Rhind Mathematical Papyrus, National Council of
the Teachers of Mathematics, 1979 reprint.
Gillings, Richard J., 1972, Mathematics in the Time of the Pharaoh's,
Dover Publications, 1982 re-print.
Klee, Victor and Wagon, Stan, 1991, Old and new Unsolved Problems
in Plane Geometry and Number Theory, Mathematical Association of
America, Dolciani Mathematical Expositions-No. 11.
Knorr, Wilbur, Historia Mathematica, HM 9, "Fractions in Ancient Egypt
and Greece, 1982.
Neugebauer, Otto, 1962, Exact Sciences of Antiquity, Harper and Rowe.
Ore, Oystein, 1948, Number Theory and its History, McGraw Hill
(Dover reprint is available).
Robins, Gay and Shute, Charles, The Rhind Mathematical Papyrus, Dover
Publications (a reprint of a 1987 British Museum publication).
Author:
Milo Gardner
Cryptanalyst
Sacramento, CA
March 1, 1996
Qualifications
==============
As background information concerning my cryptanalysis qualifications to
discuss the mathematical side of Egyptian fractions it may be important to
note that I worked for several years as a Cryptanalytical Specialist.
My daily and ongoing assignments were twofold:
1. To sort through encoded electronic messages and place practice
traffic in one pile and language text traffic in another.
2. To attempt to predict the next day's practice traffic, which
my partner(s) and I achieved from time to time, and secondarily
process ciphered messages using the latest cryptanalysis
techniques, frequently obtaining readable messages.
References:
Friedman, William F., Military Cryptanalyis, Parts I, II, III and IV
reprints available from Aegean Park Press, PO Box 2837, Laguna Hills, CA
92654, (800) 736-3587, USA and Canada, FAX (714) 586-8269
Friedman, William F. and Callimahos, Lambros, Military Cryptanalytics,
Part I, Vol. 1, 2 and Part II, Vol. 1, 2. Aegean Park Press.
Friedman, William F., Elementary Military Cryptography, Aegean Park Press.
Kahn, David, Codebreakers, a none technical book that went through
several editions, the last one in 1974. Kahn discusses Friedman
and modern cryptanalysis as well as Roman and Egyptian ciphers.
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