Egyptian Arithmetic (Appendix)

INTRODUCTION: Defining any arithmetic system, at any time,
or anywhere, requires the use of numbers. In the Egyptian
case, Old Kingdom hieroglyphic numbers were many to one,
such as listing the symbol one(1) four times to show that the
number four (4) was being listed. This numeral system was
superceded by ciphered numerals at the beginning of the Middle
Kingdom, about 2200 BC. The ciphered Hieratic numerals began
with one (1), and proceeded to over one million, as was the case
forthe its predecessor. Hieratic numbers were represented within a
base 10 system, mapping each number onto an alphabetical symbol.
To save space the actual listing of numbers versus letters
will not be cited here.

Zero itself was known and used for many situations, in the Old and
New Kingdom. However, zero did not appear in either of the
Egyptian numeration systems, nor any Ancient Near East numeration
system.The first use of a numeration based zero, outside of Europe
and its base 10 application, was in ancient Mexico, and its fully
positional base 20 numeration system (the later being bases 4 and
5, not a pure base 20 system, counting only 0-19). More on zero
later. It is a topic worthy of study, noting that many books have
been written on the subject. I too may add a blog in the near future.

But, getting back to Western numeration, after number itself was
defined, following the Greek tradition, Europeans began to accept
Vedic- Arabic base 10 numerals, around 900 AD, though zero was
not formalized within a numeration system in that wave of diffusion.
Around 1200 AD, zero did arrive in Europe, first in Germany, and
then it began to formalized in our base 10 decimal positional
system and its four functions of arithmetic, addition, subtraction,
multiplication and division, as we know them today. Given this
incomplete history, other math subjects need to be brought into
the discussion.

THE BASE 10 DECIMAL EXAMPLE

Stevins wrote two books in 1585 AD, one for business and one
for science, that may have first formalized base 10 decimals, using
zero as a place holder. Addition began with the use of rational
numbers, usually a counting number, or integer, a, b, c, and d,
all unequal such that:

1. addition: a + b = c

Addition was straight forward, when all the numbers involved
were > 1. However, Greeks and many cultures had trouble
accepting the general use of a negative number, so even here
problems had arisen in certain cultures. Stevin's work resolved
a range of issues with his definition of our base 10 decimal system.

2. subtraction: a - b = d

Subtraction was apparently straight forward when a > b. However,
when a remainder involved a fraction, a translation to base 10
decimals took place, defining a method of round off was sometimes
required. It should be recalled that Arabic-Vedic history had been
closely linked to the Babylonian infinite series numeration, with
several of base 60 thinking ending up in Stevin's analysis and
final product. Citing an extreme Babylonian case, subtraction base
remainders, stated as fractions, were rounded off when certain
situations took place. For Babylon, one of their cases cited
d = 1/91. Round off too place such that only multiples of 2, 3 and 5
were used as denominators, usually rounding off to 1/90. Stevins
view of base 10 round off was not than complicated. We round off
today based on a predetermined number of significant digits, so
accuracy has been left to the user, to the businessman and to the
scientist.

3. Multiplication: a x b = ab = e

Multiplication was seen as repeated addition, such as
5 x 7 = 35 as, 5 x 1 = 5, added seven time, or 7 x 1 = 7,
added together 5 times. Tables have been memorized
over the years, the easiest ones being based on prime
numbers.

4. Division: a divided by n meant = q + r/n (but from this point
a general notation of A/n = Q + R/n will be used, following the
shorthand notation presented in three Middle Kingdom
mathematical texts, Akhmim Wooden Tablet, 2000 BCE:
Reisner Papyri, 1800 BCE and the RMP or Ahmes
Papyrus, 1650 BCE. These texts defined R = remainder,
somtimes allowing 0, an integer or a fraction, preceded
by Q = quotient.

A rigorous definition of the factor and remainder theorem, a method
that is reported only in terms of algebra, will not be offered at this
time. At some future point the pure arithmetic arithmetic aspects
will be compared to the ancient versions of Egyptian two-part
numbers. However, to fill a logical hole, I offer Oystein Ore's
analysis on this general history of arithmetic (number theory)
subject, as his history of number theory developed in Greece and
elsewhere, based on LCM's, GCD's, aliquot parts and a
range of other ideas.

Concerning the sophistication of Egypt arithmetic or any cultural
view of arithmetic, the manner in which remainders were handed
offers significant insight into the depth their mathematical thinking.
To understand Egypt's Middle Kingdom arithmetic, a close look
at its small set of mathematical texts need to take place.

MIDDLE KINGDOM ARITHMETIC

The four arithmetic operations will be defined in reverse order,
division, multiplication, subtraction, and addition. In this
manner the interesting aspects of division can be high-lighted,
revealing a few of the complexities of Egyptian thinking. As a
basic point, Egyptian math tended to be limited to the domain of
positive rational numbers, with only a small number of exceptions.
For the purposes of this discussion, only positive rational numbers
will be considered.

1. Division

The AWT lits five problems of the division of unity (64/64) by 3, 7, 10,
11 and 13. A shorthand notation was developed that we can recognize
as following our modern base 10 structure, as given by:

(64/64)/n = Q/64 + R/64.

A simplier Reisner and RMP version says

p/n or pq/n = Q + R/n

with p and q being primes, without the additional hekat
unity substitution.

Divisors n were easily read in exact ways for both forms of
arithmetic when remainders were converted to vulgar fractions
and finally to an Egyptian fraction series.

This shorthand notations is explained by one division example, let n = 3,
then Q = 21, R= 1 such that a final answer in our modern base 10 would
have been:

64/64 divided by 3 = 21/64 + 1/(3*64) = 320/960

This form of thinking lies at the center of the discussion of
the AWT and its five division problems. The most complicated
division was dividing a fraction by a fraction, in these cases
1/64 divided by 1/n (with n < 64). The Akhmim Wooden Tablet
lists five of these divisions. The division process was simplified
by a mental process that first divided 64 by n, producing a two
part answer, and quotient (Q) and a remainder(R). Suspecting
that the history of Egyptian numeration was wished to be closely
associated with the Old Kingdom, the Q portion was stated as
a Horus-Eye series. R was stated in the Middle Kingdom mode
of numeration, Hieratic fractions.

In the case of 64/3, Q = 21, such that the question, which
divisors of 64 (1, 2, 4, 8, 16, 32, 64) added up to 21 was asked?
To an ancient scribe 16 + 4 + 1 achieved 21, meaning that 1/64th
fractional inverses of 16, 4 and 1, were easily written out as
(1/4 + 1/16 + 1/64) hekat.

Concerning the 64/3 remainder, R = 1, meant that 1/3 our simple
answer of 1/(3*64) = 1/192 was not written. What was written
can be explained by 1/64 being replaced by 5/320, such that
1/3 x 5/320 = 5/960, or (1 + 2/3) ro was written as the remainder
portion, or totally,

(1/4 + 1/16 + 1/64) hekat (1 + 2/3)ro.

Continuing with . the 64/3 case, 1/192 was written out as 5/3 ro or
(1 + 2/3)ro using Hieratic script, since ro = 320, which also meant
that:

(5*Q*n + 5*R)

was the numerator, which always added up to 320, or generally

320/(n*320) = 1/n

In that way 1/n was multiplied by n, as the AWT did with its
five division problems, thereby finding and confirming that
a 100% accurate hekat was found in each case.

The AWT data is confirmed in the Rhind Mathematical Papyrus,
or the Ahmes Papyrus, if you prefer, by ro being used several
times, always mistranlated by historians, for one myopic reason or
another. Strange but true.

2. Multiplication: repeated additions can be used to describe the
operation, as an introduction. Duplation tables were listed for one
of the numbers being multiplied such that the final answer was picked
from the table. At this time, a rigorous discussion of multiplication
will not be offered.

The AWT proved each of its 1/3, 1/7, 1/10, 1/11, and 1/13 valuations
by multiplying by 3, 7, 10, 11 and 13, respectively. Examples taken
from the AWT will be listed at a later time.

3. Subtraction: This operation can be seen as a reminder, much as
the division operation calculates a remainder. Seen as a remainder,
Egytians did not allow round off, except when irrational numbers like
pi were being discussed, no matter how small the fractions that were
involved. The exact remainder for all rational number substractions
sets Egyptian method at a higher level than its Babylonian counter
parts. Five examples follow, as variations of the AWT theme,
beginning with unity (1), and subtracting: 1/3, 1/7, 1/10, 1/11 and 1/13.

The Hultsch-Bruins 2/p method will be used, as first re-discovered in
1895, as easily generalized to the n/p case:

1. 1 - 1/3 = 2/3

(The EMLR lists 1/3 + 1/3, but usually 2/3 was considered prime,
and not reduced to lower partitions)

2. 1 - 1/7 = 6/7

= 1/2 + (12 - 7)/14, since 5 can not be found amidst the divisors of 14,

= 1/2 + 1/3 + (15 - 14)/(3*14)

= 1/2 + 1/3 + 1/42

(check 2/3 for the first partition, an interest series)

3. 1 - 1/10 = 9/10

= 1/2 + (18 - 10)/20, with 8 being found amidst the divisors of 20

= 1/2 + (5 + 2 + 1)/20

= 1/2 + 1/4 + 1/10 + 1/20

( try 2/3 as the first partition, it is also interesting)

4. 1 - 1/11 = 10/11
= 1/2 + (20- 11)/22, with 20-11 = 9 not amidst the divsors of 22, therefore
= 1/2 + 1/3 + (27 - 22)/66, with 27 - 22 = 5 admidst divisors of 66, such that
= 1/2 + 1/3 + (3 + 2)/66
= 1/2 + 1/3 + 1/22 + 1/33

5. 1 - 1/13 = 12/13
= 1/2 + (24 - 13)/26, with 11 not being amidst the divisors of 26, therefore
= 1/2 + 1/3 + (33 - 26)/72, with 7 being amidst the divisors of 75, such that
= 1/3 + 1/3 + 1/13 + 1/39

(of course 2/3 can be a first partition, revealing 2/3 + 1/4 + 1/156, as an
alternative)

4. Addition: a + b stated as unit fraction series were added, with
ease, allowing for the understanding of the general conversion
of n/p and n/pq tables of unit fraction series, all providing
guidance to each scribe, to know the range of the expected
final answers.

Conclusion: Completely reading Egyptian mathematical texts has
always been a difficult task. The primary reason is that scribes left
only brief outlines of their work, and therefore modern scholar,
until very recently, have not fully translate Middle Kingdom data
fully into modern base 10 fractions. This ancient hieratic shorthand
has been called many things, including only a form of hieroglyphc
math from an older era. Clearly, hieratic shorthand contained at
least two forms of remainder arithmetic, as modern researchers
have been detailing in 2005. This researcher has followed faint
clues by fully translating data into modern base 10 fractions, by
finding and explaining the scribal conversion used in the missing
vulgar fraction step. This extra work has been necessary since
scribes usually thought of in a form of mental arithmetic, and
when they did not, scribal notes were very brief.

In addition it had been shown that the word ro, 1/320, also meant
common divisor. Common divisors is a concept long over looked,
one that sheds new light on this 80- 120 year old question.

Irrespective of anyone's particular view of a particular ancient
problem, a clear translation to modern base 10 decimal fractions must
take place at some point. Fuzziness aside, to achieve that goal, as
best as anyone is able, all of the knowable arithmetic steps
mentioned and omitted in the ancient texts must be discussed, and
compared to the modern base 10 decimal fraction version of the
problem. 'No stone or text should be left unturned or unread',
is one way to summarize the operational aspect of this task.
Skipping over one ancient or modern arithmetic issue, or
another, has caused the great confusion that still exists in
the modern reading of the ancient mathematical texts. However,
given a little patience and humility, a better reading will be
appearing over the next few years. Adding a vulgar fraction step
to both forms of remainder arithmetic has greatly assisted in
translating scribal arithmetic into our modern base 10 arithmetic.

There is still additional work to do in reading Middle Kingdom
mathematical texts. But 2005 has been a productive year.

June 2006 has been even more productive, noted by the obvious proof
that 10/n hin was used in 29 RMP 80 cases, a method that could have
been easily extended to 320/n ro, and any other volume unit. The
problems with proving the exact value of a dja, oipe, or any other
infrequently used volume measure is to directly link it to a known
value, hopefully the hekat itself.