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Towards a Digital Mathematics Library

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Towards a Digital Mathematics Library

The one hour Math 2.0 Webinar is being prepared that may be of interest.

A taped version of the presentation will be available. A follow up note will include a date when the tape version is posted.

In summary, the Webinar will discuss Fibonacci and Ahmes' related rational number conversion methods. Both scribes converted 4/13 to unit fraction series and began with 1/4 as the first partition. Fibonacci selected 1/18 for a second partition detailed by:

A. Fibonacci (1202 AD) per

1. (4/13 - 1/4) = (16 - 13)/52

2. (3/52 - 1/18)= (54 - 52)/936 = 1/468

3. 4/13 = 1/4 + 1/18 + 1/468

Fibonacci's notation recorded the unit fraction series from left to right.


and summarized by distinction seven mentioned in:

B. Ahmes

1. 4/13 x (4/4) = 16/52

2. considered the the divisors of 52: 52, 26, 13, 4, 2 and 1 that summed to numerator 16

3. (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52

Ahmes' unit fraction series was recorded from right to left, absent the + sign.


The scope of the Webinar offers an alternative to BBC and the British Museum's 15 minute RMP broadcast ... aired on Feb. 9, 2010 ...

as well as an alternative to 20th century views of Egyptian mathematics contained in Joran Friberg's recent book:

Skip down to Friberg's view of RMP 36 and Ahmes' 2/n table. A clear contrast will be provided in the Webinar. Friberg accepts 'single false position' as Ahmes division method. The Webinar will show that Ahmes nor Fibonacci used guess-work to generally convert rational number n/p to concise unit fraction series.

Best Regards,

Milo Gardner

The blog world is changing. Maria D, of Math 2,0 runs open forum math education Webinars. An invitation to present an Egyptian math webinar was accepted with the following agenda outlining an event that Planetmath members interested in the history of mathematics may wish to view (in taped mode after July 21, 2010).

Math 2.0 brings the blog world into direct connect with small groups of interested people into a Webinar room. Chat conversations run concurrently with co-hosted webinar White Boards. My co-host Bruce Friedman lives in NYC and runs Our White Board will offer blog information connecting 15 years of internet activity.

On July 21, 2010, the date of the Webinar, two ancient Egyptian math onions will be peeled, revealing two proto-number theory cores.

A. The easiest arithmetic onion, the "Liber Abaci" was written in Latin, and translated by LE Sigler in 2002.

One of its three arithmetic notations reveals a proto-number theory method that converted 4/13 by selecting three least common multiples 4 and 18, written as

1. (4/13 - 1/4) = (16 - 13)/52

2. (3/52 - 1/18) (53 - 52/936

such that:

3. 4/13 = 1/4 + 1/18 + 1/468

was recorded from right to left, discussed by:

As Planetmath members may know Arabs re-wrote Greek arithmetic into Vedic 1-9 numerals around 800 AD. Pope Sylvester in 999 AD required Latin speaking/writing Europeans to adopt the updated Arab (and Greek) mathematics. By 1202 AD Leonardo de Pisa (Fibonacci) wrote "Liber Abaci", a 500 pape book (using Sigler's page counting)/ Leonardo updated the book several times before his death around 1250. The book served as Latin speaking Europe's arithmetic book for another 200 years. Its replacement by modern base 10 decimals in 1585 AD is a topic that deserves its own book. Today five copies of the Latin book remain.

B. The most difficult onion to peel covers the 2050 BC to 1550 BCE Egyptian Middle Kingdom era. The Rhind Mathematical Papyrus 2/n table and RMP 36 detailed a proto-number theory method that awas slightly altered from the one used by Fibonacci. Rather than a subtraction context, Ahmes in 1650 BCE used a multiplication context to scaled 4/13 to a unit fraction series using one least common multiple 4 using these steps:

1. 4/13 x (4/4) = 16/52

2. find the divisors of 52: 52, 26, 13, 4, 2, 1 that summed to numerator 16

3. 4/13 = (13 + 2 + 1)/52

such that:

4. 4/13 = 1/4 + 1/26 + 1/52

recorded from right to left, without (+) signs, discussed by:

Expanded Webinar agenda:
We will be discussing Egyptian math as two onions. The oldest onion consisted of eight layers: numeration, arithmetic, algebra, geometry, arithmetic progressions, arithmetic proportions, recreational problems and weights and measures. Any layer could be dedicated to an interesting one hour Webinar.

The Webinar covers Middle Kingdom arithmetic connections to Greek, Arab and medieval arithmetic. Webinar participants may feel their modern expectations of 800 year old to 4,000 year old Egyptian fraction arithmetic methods peeled from their heads. Relax, and enjoy two jumps back into time.

Egyptian, Greek, Arab and medieval arithmetic operations were written in proto-number theor. The Webinar begins with Fibonacci's 1202 AD "Liber Abaci", a book that was Europe's arithmetic book for 250 years (ending in 1454 AD with the closure of the Silk Road ... with the Ottoman Empire ending the Byzantine Empire?).

The Latin onion was peeled in 2002 by Sigler (into English). Fibonacci demonstrated seven distinctions (Sigler's terminology) in one of three rational number conversion methods. The first six distincitons created 2-term unit fraction series. The 7th distinction created 3-term or longer series that connect to 2,050 BCE.

Fibonacci 's scaled n/p was recorded as a 2-term unit fraction series by finding an LCM m, written as 1/m. The first six distinctions were discussed over 120 pages of a 500 page book. For example 1/7 was scaled to 8/56 by selecting LCM m as (p + 1) = m such that:

(1/7 - 1/8) = (8 - 7)/56


1/7 = 1/8 + 1/56

Fibonacci's 7th distinction covered a case that a 2-term series could not found. For example, the first available LCM is 4 that solved 4/13 in two phrases, first writing

(4/13 -1/4) = (16 - 13)/52

and second writing

(3/52 - 1/18) = (54 -52)/936

creating a 3-term series

4/13 = 1/4 + 1/18 + 1/468

With those notes in hand, lets jump back 5,000 years to the Egyptian Old Kingdom and it numeration and algorithmic arithmetic systems that employed clones to "Russian Peasant" multiplication (Bruce's 2007 story?). For 1,000 years Egyptian used duplation multiplication, a form of thinking that was modified in 2050 BCE.

To decode Middle Kingdom arithmetic it number theory will be peeled to disclose arithmetic operations that did not employ algorithms. The MK arithmetic was finite with operations that looked and acted like modern arithmetic operations. Ahmes would have converted 4/13 by the same LCM 4, considering 16/52 before finding divisors of 52 that summed to 16, namely (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52, a one LCM problem.

Concerning, multiplication and division operations, this pair were demonstrated in RMP 38: 320 (ro) was multiplied by 7/22 obtaining 101 9/11. The 101 9/11 information was recorded as a unit fraction series, and proven correct by being multiplying by 22/2 obtaining 320 (QED).

Retrogressing further to the Old Kingdom (OK) numeration, a many-to-one structure, were facts that Romans used to develop Roman numerals. OK cursive arithmetic defined one (1) as an infinite series, rounded off to 6-terms1/2 1/4 1/8 1/16 1/32 1/64. Upto a 1/64 unit was thrown away, an inaccuracy that scribes corrected after 2050 BCE in a finite system of mathematics, such as weights and measures

Quoting from "RMP 36 and the 2/n table", the blog contents can be condensed in the Webinar ... the wider threads can be read as participants have time.

" The Rhind Mathematical Papyrus (RMP) is an Ancient Egyptian (AE) hieratic text that has been dated to 1650 BCE. The papyrus has been housed in the British Museum since 1863. The math text contains a series of equalities called the "2/n table" which represents about a third of the papyrus. The remainder of the papyrus contains 87 loosely grouped arithmetic problems.

The scribe, Ahmes, converted rational numbers to unit fraction series by a hard-to-read scaling method. The 2/n table method is disclosed in RMP 36 and 37 a fact that was not recognized by 20th century scholars. Ahmes converted 2/3, 2/5, 2/7, ..., 2/101 to concise unit fraction series as handy references to solve difficult rational numbers conversions. In RMP 36 Ahmes substituted 28/53 + 2/53 for 30/53 and in RMP 31 substituted 26/97 + 2/97 for 28/97. The relatively easy 2/n table entries were used to solve difficult n/p rational number conversion problems by selecting a least common multiple (LCM) m to solve mn/mp rational numbers.

For example, to convert 26/97 Ahmes selected LCM 4 considering 104/388. To convert 2/97 Ahmes selected LCM 56 considering 112/5432. Ahmes solved 30/53 and 29/97 (difficult rational numbers) by substituting (n-2)/p + 2/p before inspecting the divisors of mp that best summed to numerator mn.

Ahmes solved all 87 problems by applying the 2/n table unit method many times. Modern looking arithmetic operations, rhetorical algebraic, modern-looking arithmetic proportions and geometric methods were discussed by Ahmes, on the second level of Ahmes ancient encoded document. The primary 2/n table method used red number divisors of mp, detailed in RMP 36 that explicitly solved 2/53, 3/53, 5/53, 15/53 and 28/53, as Ahmes' 2/n table notes had not reported. The entire 2/n table and each LCM m is published on-line in: 2008.

The RMP's 87 problems also report fragmented initial, intermediate, final, and proof information. The majority of the information has been decoded and translated into modern arithmetic statements by adding back missing initial and intermediate facts. The majority of the RMP translations were posted online in 2009.

The majority of RMP and Kahun Papyrus(KP) translation problems have been decoded by opening new doors to understanding MK arithmetic. Once muddled RMP and KP texts misunderstood by Marshall Clagett, Ancient Egyptian Science, Vol III have 1999 and other 20th century scholars are being corrected by new decoding methods. One of the least understood aspects concern two 2/n tables, and the scaling method used by Ahmes and the Kahun P. scribe. The RMP and KP problems used auxiliary numbers in a fragmented manner.

Red numbers were not recorded in the Kahun 2/n table or the Ahmes 2/n table. However, red numbers were frequently listed in Ahmes' 87 problems and elsewhere in the KP, facts that were required to calculate 2/n tables in the ancient manner.

For example, RMP 36 offers explicit evidence of Ahmes red number method. Ahmes' red numbers clues detail rational number conversions that calculated the 2/n table series in optimized, but not optimal, ways by selected LCMs.

But, how were the 2/n table table unit fraction series and Ahmes' 87 unit fraction answers calculated? To decode those questions, by following the methods reported in the ancient texts, a discussion of red auxiliary numbers is required. One RMP problem, RMP 36, will be used to establish Ahmes red auxiliary number method that also calculated 2/n table unit fraction series data reported in Ahmes 87 problems.

RMP 36 solved

3x+(1/3+1/5)x = 1 hekat

a simple algebra problem that Ahmes solved in a weights and measures context.

A duplation proof of the problem considered LCM 15 by solving:


such that

x= 15/53

Ahmes converted 15/53 to a unit fraction series by thinking:


by writing:


with 4 + 2 + 1 implicitly recorded in red,

Ahmes also considered

106 times 15/53 = 30

With 30 the LCM that converted 2/53 to a unit fraction series


30 times 53/15 = 106

the GCD 106 was found by scaling 28/53 by LCM 2 translated as


Ahmes' primary division method inverted 15/53 to 53/15, a property of modern division, an arithmetic operational fact also reported in RMP 38. Scholars writing on this topic in the 19th and 20th century falsely concluded that 'single false position', a medieval method for finding roots, was Ahmes' primary division operation.

Ahmes converted 30/53 as 2/53 + 28/53. RMP 36 explicitly converted 3/53 to unit fraction series by a 2/n table red auxiliary number proof.

1 Implicit 2/n table proofs include 2/53, 5/15, 15/53 and 28/53:

a. 2/53=2/53(30/30)=60/1590=(53+5+2)/1590= 1/30 + 1/318 + 1/795

b. 5/53=(5/53)(12/12)=60/636=(53+4+2+1)/1253=1/12+1/159+1/318+1/636

c. 15/53(4/4)=60/212=(53+4+2+1)/212=1/4+1/53+1/106+1/212 .

d. 28/53(2/2)=56/106=(53+2+1)/106= 1/53+1/318+1/795+1/2+1/53+1/106

e. sum: 28/53+15/53+5/53+3/53+2/53=53/53= one hekat(unity)

2. The second proof scaled 28/53, 15/53, 5/53 and 3/53 (hekat) to 60/53m by selecting divisors of 53m (53 + 4 + 2 + 1) written in red numbers to equal numerator 60 stressing:


The second RMP 36 proof listed duplation proofs by a beginning statement:

15/53 = (1/4 + 1/53 + 1/106 + 1/212) hekat that were re-written as

1. 15/53*(4/4) = (60/212 - 1/4) = 7/212 = (4 + 2 + 1)/212 with 4 + 2 + 1 recorded in red ink.

2. 28/53*(2/2) = (56/106 - 1/2) = 7/106 = (4 + 2 + 1)/106 with 4 + 2 + 1 recorded in red ink

3. 5/53*(12/12) = (60/636 - 1/12) = 7/53 = (4 + 2 + 1)/53 with 4 + 2 + 1 recorded in red ink

4. 3/53*(20/20)= 60/1060 - 1/20 =7/1060 = (4 + 2 + 1)/1060 with 4 + 2 + 1 recorded in red


Note that the step introduced a a subtraction statement. the only context used by Fibonacci 2850 years later to work the same class of problems in the Liber Abaci.

Ahmes' arithmetic also scaled


solved 30/53 = 28/53 + 2/53 by applying a red auxiliary method that was implicitly used in the 2/n table. Note that 30/53 nor 28/97 in RMP 31 could not be solved by an LCM, hence the substitution of 28/53 + 2/53 for 30/53 and 28/97 + 2/97 for 28/97.

Concerning the RMP's 87 problems Ahmes rarely calculated beginning, intermediate, answers, and duplation proofs as modern mathematicians understand Middle Kingdom arithmetic and mathematics. To fairly parse Middle Kingdom arithmetic explicit details are required.

In RMP 36 one proof was fairly outlined by Marshall Clagett and the Egyptology community as an ancient multiplication operation. However, meta 2/n table construction details were unexposed until 2006 and published in 2008. A majority of 19th and 20th century Egyptologists correctly identified duplation details of answers without fairly showing the scribal calculations that created unit fraction answers. Missing from academic discussions were the precise meanings of red auxiliary numbers and the context in which the calculations took place. The red number method was implicitly used in 2/n tables, causing scholars to guess, often badly, and muddle the historical record.

In 2008 RMP 36 and related 2/n table methods were explicitly spelled out by solving a critical problem. Rational numbers n/p were scaled to mn/mp by selecting "optimized, but not optimal" m/m LCMs. Ahmes calculated several red number examples in RMP 36. Ahmes considered the divisors of mp, and selected the best set of divisors that added to mn, by denoting his selections in red.

Related Egyptian fraction method also 'healed' an Old Kingdom "Eye of Horus" binary numeration problem that wrote:

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...

by writing

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 10 ro

Middle Kingdom scribes reported the remainder 2/64 as 10/320 in hekat problems The rational number 1/320 was named ro in about 40 RMP problems.

In other texts, scribes scaled 2/64 to 10/320 and (8 + 2)/320 = 1/40 + 1/160 writing a complete statement

1 = 1/2 + 1/4+ 1/16 + 1/32 + 1/40 + 1/160

and in RMP 36 by:

one hekat(unity)=28/53+15/53+5/53+3/53+2/53=53/53

as Middle Kingdom scribes wrote arithmetic statements in unity statements, solving a once impossible "Eye of Horus" problem.

Summary: RMP 36 exposed Ahmes' 2/n table conversion method. Ahmes converted 2/53* by LCM 30 to a unit fraction series by red numbers 4 + 2 + 1. The red numbers, divisors of common denominator 1060, were used to solve 30/53 by 28/53 + 2/53, as Ahmes in RMP 31 solved 28/97 by 26/97 + 2/97. Generally, scribes scaled n/p by optimized, but not optimal LCMs. Difficult n/p conversions included 30/53 and 28/97 solved by substituting (n-2)/p + 2/p. "

An interactive Egyptian Math webinar

will take place on July 21, 2010 at 6:30 PM EST time.

For those that have not visited MATH 2.0 and its new approach to discussing mathematics in an interactive manner, take a spin anytime. A year's worth of weekly broadcasts are available for downloading and viewing.

A list of available topics:

All pages A-Z

1. Home
2. Acceleration of Gifted Math Students
3. Art of Problem Solving
4. bibliography
5. BlockFest
6. borogoves
7. Calculation Nation
8. Calls for articles
10. Circle origami axioms
11. ck12
12. Community of communities
13. conversation
14. Curriki
15. Cut the Knot
16. Designing math-rich games
17. disintermediation
18. edupunk
19. Egypt Math Glossary
20. Egyptian math
21. EightFalls
22. email group
23. Ethnomathematics
24. events
25. Freedom to Learn
26. GameGroup
27. General Math 2.0 presentation
28. GeoGebra authoring environment and community
29. Guaranteach
30. Important questions
31. In Education collaborative article on Math 2.0
32. InstaCalc and BetterExplained
33. IntMath
34. July 15th 2009 Math social objects
35. July 22nd 2009 Math language and math accessibility online
36. July 29th 2009 Engaging students in online math courses
37. key words
38. links
39. Living Math
40. Making Curriculum Pop
41. Mangahigh
42. math 2.0
43. Math 2.0 Activities
44. Math 2.0 at NCTM - a recap
45. Math 2.0 Conference
46. Math 2.0 university research grant work
47. Math 2.0 Unplugged
48. Math Games Framework
49. Math in computer science
50. Math on the Level
51. Math Online 2011
52. math social objects
53. Math teachers at play
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55. MathOverflow
56. Nibipedia for math
57. people and networks
58. Plans and dreams 2010
59. Project Euclid
60. QuestBoard
61. ReactionGrid
62. Science Online unConference
63. Scratch platform as a pathway to math and programming
64. Status of Online Support for Mathematics Discussions
65. Straightedge and compass constructions with Geogebra
66. Sugar on a Stick
67. Tapped In
68. Web Search Fun
69. WikiEducator
70. YourMathGal

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A July 21, 2010 "Egyptian Math' webinar is accessed by:

The site requires Elluminate, a JAVA interactive program, to be downloaded to view the one hour broadcast.

This level of the digital world brings together blogs, Wikipedia, Planetmath, and other URLs to a White Board area ... participants can work in the interactive white board area ... when approved by the moderator ... the first half of my webinar did that ... the second half one of three moderators moved the blog info around for all to see ...

Enjoy this new class of digital library.


oops, the url seems not to work .. one more try:

Dear all,

I would like to ask the following question. Please, correct me if you find errors in formulation, or if it is unclear:

Consider the ring arising from the ring of integers in Q(a, b, c...) by adjunction of square roots. Q(a, b, c....) being the field of the rational functions of the indeterminates a, b, c ...

Does this ring lack unique prime factorization? and why?

Despite of its purely mathematical character, this problem has an historical interest, since it allows to clarify Descartes' theory of equations, and in particular, his technique of reducibility.

Thank you everyone,


Key dates are:

* April 21th: abstract submissions
* April 27th: paper submissions

This could be a great venue for the "PlanetMath at 10 years"
paper I've been thinking about, though a tad rushed.

Here are some things I recently sent to the board about that:


"The idea for the project was hatched by Egge, sometime around fall
2000, in response to the unfortunate removal of the great resource,
"Eric Weisstein's World of Math" (or "MathWorld") from the internet.
This incident created a void for a useful, comprehensive math
encyclopedia freely available online, which we wanted to fill as
quickly as possible. So Nathan and [Aaron] began brainstorming the
concept and working out many of the problems inherent in a system that
would do in real-time what had been static on MathWorld."


I think it would be a Really Good Idea for us to sum up our experiences over the past 10 years and our plans for the next 10 in a jointly written paper that we submit to some high-profile journal (e.g. on the math side, Notices of the AMS; or on the computing side Communications of the ACM; or in some more specialized but no-less-well-reputed context). I think we should give ourselves half a year to work on this, so we get it done "sometime around fall 2010". Probably someone will be having a Special Issue on Crowdsourcing
or some such thing.

I personally have a bunch of ideas (both about content and process for writing) to contribute to this. I also am afraid I've earned myself a reputation as the Mr Mxyzptlk of the PlanetMath universe, but I'll try to keep quiet about this paper until the server transit is finished.

Indeed, the more of our tech goals that we can knock out over the next
six months or so, the better this paper will be, I think.

Math 2.0, A webinar based group, maintains an online library of its one hour weekly events. An agenda for July

includes an Egyptian math presentation (co-hosted by myself) and the following week online and other sources for the writing of Euclid will be presented.

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