Re: Webinar Agenda: Milo Gardner and Bruce Friedman, Egyptian Math (upodate)
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milo gardner
Jun 17, 2010, 11:15:12 AM6/17/10
to mathf...@googlegroups.com, Maria Droujkova, Bruce Friedman
|
Edward Cherlin
Jun 17, 2010, 12:04:19 PM6/17/10
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Thank you. Wonderful stuff. I became aware of the Rhind papyrus and the 2/n problem in my teens, and it is fascinating to hear of more discoveries and developments in our understanding.
I am interested in the questions
1. What do we know that we don't know about Egyptian mathematics?
2. What don't we know that we don't know about Egyptian mathematics?
3. Why didn't the Egyptians go further?
You mention some points on 1. below. For Greek math, the Antikythera mechanism demonstrates that far more went on than is documented, and thus that 2. is important. Similarly for 3., Galileo used only Greek math and observations easily available to the Greeks to work out the first approximation to gravity: parabolic paths in a constant field. Why did the Greeks never notice the parabolic shape of a water fountain, and attack this problem themselves? They would surely have solved it. The Kepler and Newton versions required better observations and stronger math. Why didn't the Greeks attack both after coming so close? And why didn't the Egyptians continue on into the realms opened up by Greeks trained in Egypt?
These questions remain relevant. We don't fully understand the development of modern math, and there is far more to come.
I am interested in the questions
1. What do we know that we don't know about Egyptian mathematics?
2. What don't we know that we don't know about Egyptian mathematics?
3. Why didn't the Egyptians go further?
You mention some points on 1. below. For Greek math, the Antikythera mechanism demonstrates that far more went on than is documented, and thus that 2. is important. Similarly for 3., Galileo used only Greek math and observations easily available to the Greeks to work out the first approximation to gravity: parabolic paths in a constant field. Why did the Greeks never notice the parabolic shape of a water fountain, and attack this problem themselves? They would surely have solved it. The Kepler and Newton versions required better observations and stronger math. Why didn't the Greeks attack both after coming so close? And why didn't the Egyptians continue on into the realms opened up by Greeks trained in Egypt?
These questions remain relevant. We don't fully understand the development of modern math, and there is far more to come.
On Thu, Jun 17, 2010 at 11:15, milo gardner <milog...@yahoo.com> wrote:
\Maria, and group members:
Thanks again for this opportunity. google groups seem not
to allow active http links. What needs to be done to achieve
that step?
The links are active in my mail software.
The following introduction has been updated to included
additional ... non working ... links. My apologies to everyone.
I look forward to introducing Egyptian math to your google group. The Webinar should be fun.
I regret that I cannot attend. The Webinar software is not available for Linux.
I'd like to keep the session low key. Of course as the basic
definitions and applications roll in most participants will likely
feel their modern math glasses getting fogged up. The morning mist will clear putting pencil to paper a few times.
There may be a small number of participants that have
experience with number theory and its ancient built blocks.
I would have been one.
For that group the 4,000 year jump back in time will be an
easy experience.Very few pencil and paper calculations
will need to be made.
The google group posts will attempt to lay out the basic
definitions and applications of Egyptian math, including:
An Egyptian Old Kingdom multiplication operation was
consistent with the modern "Russian Peasant" operation.
This is also an important computer algorithm, more often used for exponentiation, using square and multiply rather than multiply and add.
Egyptian scribes in the Old Kingdom and Middle Kingdom,
the period that Egyptian math will be focused upon in the
Webinar, seemed to only double the desire multiplication
elements. This aspect of the subject will not be explored
in detail. A story or two can to told at this point.
For those, like yourself, that understand Russian peasant
multiplication, a point will be made that the ancient algorithm
was modified into a finite numeration system that included finite arithmetic operations. This will be the topic of the Webinar.
The second historical fact that needs to be introduced is
the binary definition of one (1) used in the Old Kingdom
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
A Dedekind cut! Nice.
Egyptians built a mythology, named the "Eye of Horus, around this definition. An algorithmic statement rounded off statements of any rational number to six-terms, rounding off up to 1/64 of a unit. The Egyptian Old Kingdom arithmetic metaphor was active in the entire history of Egyptian base 10 arithmetic and the entire history of Babylonian base 60 mathematics. "Book of the Dead" stories can be told at this point, citing the weighing of everyone's heart (at death) on a balance beam, by a man that wore two hats, one religious and one science (the mythical leader of Egyptian
mathematics).
Weighing the heart against a feather.
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Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://www.earthtreasury.org/
milo gardner
Jun 17, 2010, 12:48:21 PM6/17/10
to mathf...@googlegroups.com
Edward, Thank you for the questions: A brief *** response*** will be given to each question: |
1. What do we know that we don't know about Egyptian mathematics? |
*** We know a great about its ciphered numeration system, arithmetic, algebra, geometry, weights and measures and arithmetic progressions that were consistent with Gauss summing up 1 to 100 by matching 50 pairs of 101, matching the first and last terms in the series, to the 5050 total. The Kahun Papyrus and RMP 39, 40 and 64 discuss Egyptian arithmetic progressions, found on Wikipedia per: http://en.wikipedia.org/wiki/Kahun_Papyrus *** |
2. What don't we know that we don't know about Egyptian mathematics? |
*** Aspects of cubits and other weights and measures units and calculation styles have not been found in the written records. http://planetmath.org/encyclopedia/CubitsEgyptianGeometryAreasCalculatedIn.html The fragmentary status of the hieratic mathematical texts has caused many scholars to give up, suggesting like Otto Neugebaur did in "Science in Antiquity" (1954?) that the 2/n table was a step backward mathematically. The exact opposite was true. |
3. Why didn't the Egyptians go further? |
Egyptians did go farther, so far that Greek, Arabs and medieval added little to the arithmetic notation, modifying the structure of the unit fraction notations, without modifying the number theory foundations. Unit fraction arithmetic continued for 3,600 years, ending in Europe with the 1585 AD creation of our modern base 10 decimal system. Fibonacci's Liber Abaci, a 1202 AD text
http://liberabaci.blogspot.com/ is a case in point. Fibonacci took 124 pages of a 500 page treatise (that was Euope's arithmetic book for 250 years -until Ottomans over ran Byzantine culture) to describe seven rules (distinctions in Sigler's 2002 Latin to English translation). Egyptian fraction arithmetic to Fibonacci, Arabs, Greeks and Egyptians was demonstrated, often without summary algebra statements (info that I have taken the liberty to add ... from time to time). **** |
You mention some points on 1. below. For Greek math, the Antikythera mechanism demonstrates that far more went on than is documented, and thus that |
*** this was a machine that was cranked one day at a time. To my knowledge the modular arithmetic that "Archimedes" employed to create the gears has not been decoded in Greek arithmetic metaphors, writing 1/2 as a ciphered beta'. |
2. is important. Similarly for 3., Galileo used only Greek math and observations easily available to the Greeks to work out the first approximation to gravity: parabolic paths in a constant field. |
*** Here your question jumps near to our algorithmic base 10 decimal era. By the time of Napier (*early 1600s), Stevin's decimal notation was improved near to the one that we use today. *** |
Why did the Greeks never notice the parabolic shape of a water fountain, and attack this problem themselves? They would surely have solved it. The Kepler and Newton versions required better observations and stronger math. Why didn't the Greeks attack both after coming so close? |
*** You are asking questions that are not present in the historical record. As you may know, we have zero Classical Greek mathematical texts.What we have are "copies" and revisions of reported Classical Greek texts that stressed geometry ... without reporting Greek arithmetic. *** |
And why didn't the Egyptians continue on into the realms opened up by Greeks trained in Egypt? |
*** Your history is backwards. The hieratic arithmetic that will be Webinar'ed next month existed 1,000 years before Classical Greek culture. Plato and others were trained in Egyptian schools, and Greeks continued Egyptian arithmetic, algebra ... and so forth ... in Greece ... Alexandria's library burnt down ... had it not .. many of your probing questions would have been answered. Best Regards, Milo --- On Thu, 6/17/10, Edward Cherlin <eche...@gmail.com> wrote: |
kirby urner
Jun 17, 2010, 1:30:15 PM6/17/10
to mathf...@googlegroups.com
I've been reading The Visual Music by Gadalla (Egyptian) on this topic of Egyptian math.
I'm especially interested in the spatial geometry aspects.
His take on the pyramid and the various ratios are cool.
Here's are a couple screen shots from his I'm using in my RadMath circle.
In my heuristics for Quaker school teachers, the four topic areas we cover are:
Neolithic Math -- deep timeline, back to Sumerians and well before <-- ancient Egyptian goes here
Casino Math -- combinatorics, probability, statistics, Indian gaming (bioregional)
Martian Math -- futuristic, deliberately alien
Supermarket Math -- includes eCommerce, SQL, tcp/ip (the digital stuff most math teachers avoid like the plague)
Here's the Wiki:
Kirby
Maria Droujkova
Jun 20, 2010, 6:29:31 AM6/20/10
to mathf...@googlegroups.com
On Thu, Jun 17, 2010 at 12:48 PM, milo gardner <milog...@yahoo.com> wrote:
>>
>> Bruce and I worked with the Hultsch-Bruins method for over 10 years before finding Ahmes ' shorthand notes that reported his actual ancient calculations. In 2009 Bruce and I transliterated and translated Ahmes 87 problems, including the 2/n table, and documented our efforts on:
>>
>> http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html
>>
>> Let me stop at this point, and re-read the basic definitions
>> that may be necessary for a non-specialist to gain access
>> to this 4,000 year old mathematics.
>>
>> A glossary should be placed here in the next week or so.
>> in the interim, please feel free to ask for a particular phrase
>> or word to be placed in the glossary.
>>
>> Best Regards,
>>
>> Milo Gardner
Milo and Bruce,>>
>> Bruce and I worked with the Hultsch-Bruins method for over 10 years before finding Ahmes ' shorthand notes that reported his actual ancient calculations. In 2009 Bruce and I transliterated and translated Ahmes 87 problems, including the 2/n table, and documented our efforts on:
>>
>> http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html
>>
>> Let me stop at this point, and re-read the basic definitions
>> that may be necessary for a non-specialist to gain access
>> to this 4,000 year old mathematics.
>>
>> A glossary should be placed here in the next week or so.
>> in the interim, please feel free to ask for a particular phrase
>> or word to be placed in the glossary.
>>
>> Best Regards,
>>
>> Milo Gardner
Can you identify a few items related to your topic that would make good "little interactives"? In the spirit of "Eye Openers" Alexander Bogomolny makes: http://www.cut-the-knot.org/pythagoras/tricky.shtml or IntMath applets by Murray Bourne: http://www.intmath.com/
Maybe some visualizations? I am working in GeoGebra with some local people, and we can try to tackle some before your webinar.
Cheers,
Maria Droujkova
http://www.naturalmath.com
Make math your own, to make your own math.
milo gardner
Jun 21, 2010, 12:03:54 PM6/21/10
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Kirby, Maria, et al Reading Gadalla's book online: http://books.google.com/books?id=evYSzuCLXt4C&dq=gadalla,+egyptian&printsec=frontcover&source=bn&hl=en&ei=f1gfTPiTG8aLnQeOocnrDQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CC4Q6AEwBA#v=onepage&q&f=false discussed an Old Kingdom (hieroglyphic) list of Egyptian mythical characters placed on a valid ancient unity stage. The story-line that interests Bruce Friedman and myself considers Anibus as recorded in hieratic script. Anibus took the dual head of Egyptian math, an entity that weighed every dead Egyptian's heart to determine each person's worth in life (in the Old Kingdom and Middle Kingdom). Gadalla's book did not focus upon Anibus' math duties recorded in the Middle Kingdom. When Anibus is focused upon in hieratic script, parsed from spiritual duties, the remaining Middle Kingdom mathematical duties describe a range of solutions to the Old Kingdom "Eye of Horus" problem: http://en.wikipedia.org/wiki/Eye_of_Horus The Middle Kingdom solutions ONLY dealt with rational numbers. Before discussing the weights and measures side of writing hieratic volume units in various classes of unities (10/10, 64/64, 320/320 and so forth) the ancient love of Middle Kingdom unities topic will be covered within Gadalla's broader hieroglyphic view, I agree with the general sense of the wholeness of life, and the unity principles that many gods/characters played in the non-mathematical aspects of life. Both were present in the Old Kingdom and Middle Kingdom. The story-line that Bruce and I will be telling begins with the algorithmic version of one, recorded in the Old Kingdom per 1 = 1/2 + 1/4 + 18 + 116 + 1/32 + 1/64 + ..., a cursive notation that generally threw away upto a 1/64 unit. All algorithmic notations, in any era, MUST do the same thing, rounding off by set standards. Babylonians did the same thing in base 60. Scribe wrote inverse tables of fractions as small as 1/91 as 1/90, throwing away the difference in an algorithmic cursive notation. Modern binary computers, use only zero and, record numbers and mathematical statements within algorithms, rounding off by standard based rules.. The IEEE organization sets up moder 'rounding off standards for analogy and digital computer's use of fixed and floating point notations, and other situations, per: http://en.wikipedia.org/wiki/IEEE_Standards_Association No algorithmic notation, from any era, in any culture, exactly records all numbers and mathematical statements. For example, irrational and higher order numbers and algorithmic statements were not allowed in ancient Egypt as proofs. Duplation statements were often seen in the RMP recorded by Ahmes in 1650 BCE. However, proof of the accuracy of implicit 'algorithmic' statements were explicitly proven by finite arithmetic methods (one of the glossary entries that are under development). Jumping to the modern era, and considering the highest use of algorithmic calculations,as supercomputers came online, Gregory and Albert Chudnovsky of Columbia U., created an algorithm notation based on polar coordinates and Little Fermat. ;Little Fermat is a 2^n type number, setting a standard when the next parallel computer is asked to come online. Working with the IBM Watson think tank, a short-term project turned into a five year project, ending about 20 years ago. When completed the "Little Fermat" operating system worked, speeded up the run time of supercomputers 1,000 times, discussed by: http://www.thefreelibrary.com/Little+Fermat:+a+scheme+for+speeding+up+multiplication+leads+to+a...-a08986294 Jumping back to the the time of the Egyptian Middle Kingdom, scribes had tired of the "Eye of Horus" 'algorithmic round-off problem. Scribes limited arithmetic calculations to rational numbers, thereby eliminating algorithmic calculations a core idea. When a higher order number like pi was needed to solve an area of a cylinder, in the definition of a hekat, one value was set for the number, 256/81. This 256/81 value for pi over-estimated the volume of 1/320 of a heat, named ro by using 3/16 rather than 3.14159. Ahmes in RMP 38 multiplied one hekat, written as 320 ro, by 7/22, an inverse of our modern estimate for pi (22/7) and found 101 + 9/11 ro with 9/11 equal to unit fraction series (2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10) Taking a RMP 38 DISCUSSION from: http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html "In RMP 38 two rational numbers, (35/11)/10 = 35/110 = 7/22, were multiplied 320, by a doubling ' method citing: 1. Initial calculation (320 ro)*(35/11) = (320 ro)*(2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10 = 101 + 9/11 ro 2. Proof (101 9/11 ro) was multiplied by 22/7, and returned one hekat , 320 ro. This class of hekat calculation infers that the traditional Old Kingdom pi value of 256/81 was corrected by considering : " ... that (7/22) and (22/7) were shown and proved to be inverses, and that the AE scribes were skilled and aware of the natural inverse operations of multiplication and division. In effect, the AE were adept at finding reciprocals" (Bruce Friedman)! " Inverse calculations of this type were spread across the RMPs 87 problems. Friedman uses the term 'natural; inverse operations of multiplication and division' Literally, Ahmes defined division as an inverse to multiplication ,and defined multiplication as inverse to division, a set of two modern rules that modern children learning both arithmetic must memorize, and learn to apply. This paired set of facts are not often recognized by Peet, Chace, Gillings and the 1920s translators of Middle Kingdom mathmatical texts. Ahmes recorded In RMP 35-38, plus RMP 66, http://planetmath.org/?op=getobj&from=objects&id=11838 and worked with 320 ro problems rather than one hekat units to solve several classes of problems by adding back once missing rounded off calculaitons, by writing exact finite statements. But what was Ahmes doing in a manner that exactly translates ancient arithmetic statements into our modern base 10 decimal algorithm? Peet, Chace, Gillings, et al, guessed at Ahmes' division operation, improperly suggesting that 'single false position' a medieval root finding operation may have been used in the Middle Kingdom. Ahmes only added back missing "Eye of Horus" fragments that would have been present in the Old Kingdom's binary numeration system. A new Middle Kingdom finite numeration system was created that avoided the 'blind eye' of any algorithmic arithmetic notation. In the Middle Kingdom scribes created non-algorithmic weights and measures, geometry, algebra, arithmetic progressions and arithmetic statements that will be introduced in the webinar, and translated into modern arithmetic statements that validates that scribal division was only inverse to scribal multiplication, inverting the rational number divisor, and multipling, without using 'single false position' or an algorithm. A glossary of terms will be posted to allow novice students of Egyptian mathematics to parse mythological terms from the abstract number theory terms used by Ahmes and other Middle Kingdom scribes. The glossary will contain two sections, a small set of Old Kingdom algorithmic rules, with a correcting set of Middle Kingdom finite arithmetic rules. The second section will include modern arithmetic terms (usually number theory based) that describe the Middle Kingdom correcting rules. For example, the Old Kingdom definition of one: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... added back 1 dja, and 5 ro in weights and measures calculations, and defined a hekat unity as 64/64 (as Hana Vymazalova proved in 2002) when the answers to n = 3, 7, 10, 11 and 13 division problems: (64/64)/n = Q/64 + (5R/n)ro (with Q a quotient, R a remainder, scaled to (5/5) to allow 1/320 units named ro to appear with divisor n was in the range 1/64 < n < 64). were multiplied by the initial divisor n, returning (64/64) in the Akhmim Wooden Tablet http://en.wikipedia.org/wiki/Akhmim_wooden_tablet example, the AWT scribe divided one hekat by 3 by completing the following steps: a.. (64/64)/3 = 21/64 + 1/192 = b. (16 + 4 + 1)/64 + (5/3)ro = c. (1/4 + 1/16 + 1/64)hekat + (1 + 2/.3)ro steps that Hana Vymazalova did not decode. Proof: steps that Hana Vymazalova properly decoded in 2002 a. ((1/4 + 1/16 + 1/64)hekat + (1 + 2/3)(ro] times 3 = b. [(21/64 x 3) = 63/64 + (5/3*(1/320)]x 3 = 5/320] c. 63/64 + 1/64 = 64/64 Again, my apologies for the slowness of creating the needed glossary. A draft should be available in one week. Best Regards, Milo Gardner http://en.wikipedia.org/wiki/User:Milogardner --- On Thu, 6/17/10, kirby urner <kirby...@gmail.com> wrote: |
|
kirby urner
Jun 22, 2010, 12:53:40 AM6/22/10
to mathf...@googlegroups.com
On Mon, Jun 21, 2010 at 9:03 AM, milo gardner <milog...@yahoo.com> wrote:
Reading Gadalla's book online:
http://books.google.com/books?id=evYSzuCLXt4C&dq=gadalla,+egyptian&printsec=frontcover&source=bn&hl=en&ei=f1gfTPiTG8aLnQeOocnrDQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CC4Q6AEwBA#v=onepage&q&f=false
discussed an Old Kingdom (hieroglyphic) list of Egyptian mythical characters placed on a valid ancient unity stage.
Interesting stuff. You're way ahead of me on reading Gadalla. Another Egyptian I read is Midhat Gazale, his Gnomon and Number especially.
The story-line that Bruce and I will be telling begins with the algorithmic version of one, recorded in the Old Kingdom per
1 = 1/2 + 1/4 + 18 + 116 + 1/32 + 1/64 + ...,
a cursive notation that generally threw away upto a 1/64 unit.
All algorithmic notations, in any era, MUST do the same thing, rounding off by set standards.
Babylonians did the same thing in base 60. Scribe wrote inverse tables of fractions as small as 1/91 as 1/90, throwing away the difference in an algorithmic cursive notation.
Modern binary computers, use only zero and, record numbers and mathematical statements within algorithms, rounding off by standard based rules.. The IEEE organization sets up moder 'rounding off standards for analogy and digital computer's use of fixed and floating point notations, and other situations, per:
http://en.wikipedia.org/wiki/IEEE_Standards_Association
Rounding is an important topic. You'll find me bringing up IEEE floating point standards back at the 1995 Oregon Math Forum, with Sir Roger Penrose, Keith Devlin et al.
This very topic has come up recently on another list, where a guy with a calculator is thinking he's discovered some special cube of volume 2.99999... something, all because his calculator is rounding, and the 3rd root of 3 to the 3rd power is not 3. Such is life in the big city.
Extended precision Integer and Decimal types are of rather recent vintage and post date the invention of the electronic computer. The concept of "real number" is useful of course, but tends to have less of a foothold in computer science (a branch of mathematics by some lights), where any "type" needs to be machine-implemented. It's a murky area, with no philosophy of mathematics really nailing it down.
My approach, experimental, is to use pre-college math classes as a venue for introducing "math objects" (generic concept) in the context of algorithmic recipes (programs) in some state of the art, industrial strength computer language, i.e. lets get rid of those calculators and use a real interactive shell for a change (not giving up plotting, just adding more capability, bigger displays etc.).
Along these lines, we might implement a Rational Number type (as distinct from real) that deals with fractions as fractions, i.e. without forcing them to become floating point decimals. When it comes to Egyptian Math, this looks like just what the doctor ordered.
Screen shots:
http://www.flickr.com/photos/17157315@N00/3308514585/sizes/o/ (addable fraction class)
http://www.flickr.com/photos/17157315@N00/4704388291/sizes/l/ (says its Java, but it ain't)
Jumping to the modern era, and considering the highest use of algorithmic calculations,as supercomputers came online, Gregory and Albert Chudnovsky of Columbia U., created an algorithm notation based on polar coordinates and Little Fermat. ;Little Fermat is a 2^n type number, setting a standard when the next parallel computer is asked to come online. Working with the IBM Watson think tank, a short-term project turned into a five year project, ending about 20 years ago.
I also like doing more around primes and composites than is customary, eating into time for calculus. Once you have Euclid's Algorithm to play with, and the extended version thereof, you might also get into Diophantine Equations. Continued fractions are also a mature topic.
One might object that these ideas belong in some advance college course in number theory, post calculus. However, given the use of "math objects" there's no requirement to delay this long.
http://www.thefreelibrary.com/Little+Fermat:+a+scheme+for+speeding+up+multiplication+leads+to+a...-a08986294
Jumping back to the the time of the Egyptian Middle Kingdom, scribes had tired of the "Eye of Horus" 'algorithmic round-off problem. Scribes limited arithmetic calculations to rational numbers, thereby eliminating algorithmic calculations a core idea.
When a higher order number like pi was needed to solve an area of a cylinder, in the definition of a hekat, one value was set for the number, 256/81.
This 256/81 value for pi over-estimated the volume of 1/320 of a heat, named ro by using 3/16 rather than 3.14159. Ahmes in RMP 38 multiplied one hekat, written as 320 ro, by 7/22, an inverse of our modern estimate for pi (22/7) and found
101 + 9/11 ro with 9/11 equal to unit fraction series (2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10)
Approximations to PI remain an important topic as well.
However, I think we should be doing a lot more with Phi. What polynomials solve to Phi? What polyhedra contain Phi in their ratios? These are important topics for classroom investigation (OK to use Google, that's what it's there for).
1. Initial calculation
(320 ro)*(35/11) = (320 ro)*(2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10 = 101 + 9/11 ro
2. Proof
(101 9/11 ro) was multiplied by 22/7, and returned one hekat , 320 ro. This class of hekat calculation infers that the traditional Old Kingdom pi value of 256/81 was corrected by considering : " ... that (7/22) and (22/7) were shown and proved to be inverses, and that the AE scribes were skilled and aware of the natural inverse operations of multiplication and division. In effect, the AE were adept at finding reciprocals" (Bruce Friedman)! "
Inverse calculations of this type were spread across the RMPs 87 problems. Friedman uses the term 'natural; inverse operations of multiplication and division' Literally, Ahmes defined division as an inverse to multiplication ,and defined multiplication as inverse to division, a set of two modern rules that modern children learning both arithmetic must memorize, and learn to apply. This paired set of facts are not often recognized by Peet, Chace, Gillings and the 1920s translators of Middle Kingdom mathmatical texts.
Ahmes recorded In RMP 35-38, plus RMP 66,
http://planetmath.org/?op=getobj&from=objects&id=11838
and worked with 320 ro problems rather than one hekat units to solve several classes of problems by adding back once missing rounded off calculaitons, by writing exact finite statements.
But what was Ahmes doing in a manner that exactly translates ancient arithmetic statements into our modern base 10 decimal algorithm? Peet, Chace, Gillings, et al, guessed at Ahmes' division operation, improperly suggesting that 'single false position' a medieval root finding operation may have been used in the Middle Kingdom.
Ahmes only added back missing "Eye of Horus" fragments that would have been present in the Old Kingdom's binary numeration system. A new Middle Kingdom finite numeration system was created that avoided the 'blind eye' of any algorithmic arithmetic notation. In the Middle Kingdom scribes created non-algorithmic weights and measures, geometry, algebra, arithmetic progressions and arithmetic statements that will be introduced in the webinar, and translated into modern arithmetic statements that validates that scribal division was only inverse to scribal multiplication, inverting the rational number divisor, and multipling, without using 'single false position' or an algorithm.
"Eye of Horus" might be introduced in USA classrooms using the dollar bill. I know some classrooms might not be open to discussing iconography in that way, as it might open the door to weird conspiracy theories. However, here in Portland that's probably not a problem. http://en.wikipedia.org/wiki/Information_Awareness_Office (another example).
The glossary will contain two sections, a small set of Old Kingdom algorithmic rules, with a correcting set of Middle Kingdom finite arithmetic rules. The second section will include modern arithmetic terms (usually number theory based) that describe the Middle Kingdom correcting rules.
For example, the Old Kingdom definition of one: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
added back 1 dja, and 5 ro in weights and measures calculations, and defined a hekat unity as 64/64 (as Hana Vymazalova proved in 2002) when the answers to n = 3, 7, 10, 11 and 13 division problems:
(64/64)/n = Q/64 + (5R/n)ro
(with Q a quotient, R a remainder, scaled to (5/5) to allow 1/320 units named ro to appear with divisor n was in the range 1/64 < n < 64).
were multiplied by the initial divisor n, returning (64/64) in the Akhmim Wooden Tabletexample, the AWT scribe divided one hekat by 3 by completing the following steps:
a.. (64/64)/3 = 21/64 + 1/192 =
b. (16 + 4 + 1)/64 + (5/3)ro =
c. (1/4 + 1/16 + 1/64)hekat + (1 + 2/.3)ro
steps that Hana Vymazalova did not decode.
Proof: steps that Hana Vymazalova properly decoded in 2002
a. ((1/4 + 1/16 + 1/64)hekat + (1 + 2/3)(ro] times 3 =
b. [(21/64 x 3) = 63/64 + (5/3*(1/320)]x 3 = 5/320]
c. 63/64 + 1/64 = 64/64
Again, my apologies for the slowness of creating the needed glossary. A draft should be available in one week.
Best Regards,
Milo Gardner
http://en.wikipedia.org/wiki/User:Milogardner
Amazing stuff.
I think you'd enjoy meeting my friend Glenn Stockton who shares some of your background in military code breaking, went to Presidio Language School.
Glad to see you're using Stellarium, which I likewise admire and endorse using in the classroom, per this web page:
Kirby
milo gardner
Jun 22, 2010, 12:10:55 PM6/22/10
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Kirby, and Math 2.0 members: Thank you Kirby for your thoughtful comments. Our post-WW I classrooms are very good at introducing the algorithmic side of number, and the various classes of mathematics, including calculus, that relies on "the limit theorem" and other classes of algorithms. Prior to WWI per: http://books.google.com/books?id=-oQXAAAAYAAJ&pg=PT155&lpg=PT155&dq=college+arithmetic,+1900&source=bl&ots=mCk68C4lz0&sig=h9hbpRyh3vaD_dEh4JPoC1o2yxs&hl=en&ei=os0gTP7CC5rrnQfzroFy&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBUQ6AEwAQ#v=onepage&q=college%20arithmetic%2C%201900&f=false college arithmetic began with finite arithmetic. In 1910 an international conference was chaired by a Gottingen University math person: http://en.wikipedia.org/wiki/University_of_G%C3%B6ttingen and concluded that K -12 math education should also begin with finite arithmetic. Sadly, after WW I, President Woodrow Wilson and others acted to remove all things German from USA, British and other European K- 12 and college classrooms, by always beginning with algorithms. The modern classrooms in the USA does not include topics that were German connected prior to WW I. How many on this forum are aware of those 1920s actions? In 1963 I ran across the pre-WW I body of knowledge in "Number Theory and its History" by Oystein Ore, a classic book still in publication (thanks to Dover). Problems were written and solved within finite math, the largest body of knowledge, in my view, was the Chinese Remainder Theorem, and congruences (modular arithmetic). Gauss' 1801 treatise, "Discussions on Arithmetic" provided the first rigorous set of arithmetic foundations presented in the Western Tradition, another of my views. NCTM also sells the 1910 International conference notes that that extolled the virtues of college arithmetic and finite arithmetic. Jumping back 4,000 years, Egyptians began with finite arithmetic. The older "Russian Peasant" type multiplication http://www.google.com/search?hl=&q=russian+peasant+multiplication&sourceid=navclient-ff&rlz=1B3GGGL_en___US345&ie=UTF-8&aq=0&oq=russian+peas was an Old Kingdom algorithmic fragment. Its numbers and arithmetic statements were written in, an awkward 6-term rounded off notation (yes, "Eye of Horus" type numbers). By 2050 BCE, a 6-term or less finite notation scaled rational numbers to solvable unit fraction series by looking for a least common multiple m, and a set of red numbers, to write out optimized unit fraction series, a topic that will be discussed in detail in the Webinar. Egyptian finite math slowly adapted to create our modern multiplication operation, first used within mental arithmetic and mental algebra applications, two topics that will be introduced in the Webinar by citing Ahmes 87 problems, and ways to fairly decode the raw hieratic data. Greeks followed Egyptian ciphered numeration and finite math statements. Eudoxus has been given credit for developing the 1/4 geometric series, an algorithmic statement for finding areas. Archimedes used it to find the area of a parabola, first writing 4A/3 = A + A/4 + A/16 + A/64 + ... and proving the hypothesis by taking one phase of the Horus-Eye series by writing out a finite Egyptian fraction series (as Heiberg saw in 1906, and Dijksterhuis reported in "Archimedes", in 1987, consider pre-WW I views on arithmetic) per 4A/3 = A + A/4 + A/12 Note that Archimedes' proof, reported by Heiberg, did not use the "Method of Exhaustion" an algorithm that we all learned in college calculus classes. At this level, it can be said that Egyptians and Greeks both learned to begin with algorithmic infinite series statements, and proved that an exact unit series could be found that solved 'traditional' round off problems. This is the power of finite mathematics, converting most infinite series into a finite series. Sadly, Stanford's project on reading Archimedes' calculus only mentioned the post WW I ":method of exhaustion" side of the topic --on Nova, its web site, and other internet sites. Happily, mathematics is an abstract subject that speaks for itself, calling out the foundational issues to those that read the ancient texts. Over the last few years, Bruce Friedman has taken the language side of hieratic symbols.This is one topic that the US Army's Language School in Monterrey, California (which I qualified to enter -- but chose cryptanalytics) teaches, by offering the oral and written languages. Cyrillic script and Russian are phonetic, hence easily taught. Chinese Mandarin and Arabic are the toughest, the first due to its symbols and syntax and the second due to its regional idioms. Bruce and I tend to confirm the work of Peet,Chace, Gillings, Clagett and others that reported complete transliterations of Middle Kingdom mathematical texts -- WITHOUT fully translating the ancient data into modern arithmetic. More on this topic in the Webinar. Code breaking of ancient mathematical texts requires at least two team members, a linguist and a person that studies the language patterns, a mathematician that also tracks the numerical patterns in the text. The second person (my role) should be skilled in chess and others games of strategy. Ancient texts offer many surprises. For those that have read about Linear B, a Minoan script, it was broken by Ventris and Chadwick, after WW II working in a math -linguist team http://en.wikipedia.org/wiki/Linear_B Chadwick was the cryptanalyst on the team. There are insufficient fragments of Linear A to decode its language side,though the Linear A does include hieroglyphic number symbols, likely used in trade. Continuing with ancient Egypt, due to the poor 20th century translations, wider interdisciplinary teams are needed to be established. In 2009 a mini-conference was chaired by two German PhDs in Egyptology, Tanja Pemmerening and Annette Imhausen per: http://mathforum.org/kb/thread.jspa?threadID=1916972&tstart=15 The usual suspects were invited to the conference, inviting no military code breakers, or serious math historians. It is hoped that future conferences will bring together the broader range of specialists with open invitations. At this point in time, the language side of Egyptology does not formally speak to the numeration and the mathematical side of hieratic mathematical texts. Tanja Pemmerening and I have informally discussed our common interests, the definition of the 1/64 dja unit, recorded with hekat weights and measures (the AWT). Tanja's work clearly shows the 'healing'' side of the "Eye of Horus" round-off problem in hieroglyphic symbols. At this time we do not agree on the two sides of Egyptian math (theoretical and practical statements that scribes used to state and prove every problem). Tanja concludes that our two bodies of work are consistent with one another, but goes no further. Annette Imhausen takes an algorithmic view of hieratic texts, follows the 1990's work of Jim Ritter. To date this group has published no new discoveries that open up previously unread sections of the hieratic math texts. For a reason I do not understand this group parses the literary symbols and subtle meanings of the mythical texts (Book of the Dead +, looking for clues to decode the hieratic texts. Annette and I have shared a small number of emails, but have not moderated our differences, and may not for several years. There is more to this story, info that can be discussed in the Webinar, for those that are interested. Thanks again for this opportunity to lay out a few historical threads. It is hoped that interesting aspects of ancient Egyptian math are being exposed. Reasons why the ancient mathematical styles are not often discussed as ancient scribes recorded their mathematics offers other interesting background info, Hopefully times are changing, thanks to Maria, Kirby, and others. Best Regards, Milo Gardner . |
kirby urner
Jun 22, 2010, 5:01:23 PM6/22/10
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This is fascinating material Milo.
The 1997 Oregon Math Summit I was mentioning (not 1995 -- my
mistake) also featured Ralph Abraham, who used the occasion
to stump for a time-line-based approach to K-12 math learning,
actually studying the mathematics of a civilization while learning
about it in historical terms, proceeding chronologically, grade
by grade. Why not try it? Not every school needs to switch
tracks. Pioneering is OK (as we say in Oregon).
This chrono-sequence wouldn't have to be too ploddingly adhered
to (could have flash-forwards and flash-backs), would be a lot
more like the game 'Civilization' which many students have already
tasted and enjoyed (another current thread with the Devlin group
is a need for more simulations in math class).
Related post to Math Forum (recent vintage):
A large section on Egyptian finite mathematics would make perfect
sense in this context, linked to Sumerian and other sources.
Geographic information is embedded in some of the geometrical
artifacts, so we look at that too (natural periodicities).
Rolling the tape forward a bit, the etymology of "algorithm" in
the name Al Khwarizmi, Wisdom School, Baghdad, suggests
segments on "Iraqi math" (creatively anachronistic name)
aka "algorithmics" ala Liber Abacci (Ed Cherlin and I were
discussing this here, in contra-distinction to Roman numerals,
also abacus-based according to his research).
Europeans were liberated by the phase-in of "cyphers", which
enabled a merchant class to engage in its own double-entry
bookkeeping systems, not over-relying on church-minded
authorities. Enter the Renaissance (not unlike today's open
source revolution in some ways).
As we role forward, we might be able to pick up some of
that lost German content you say got purged by the Wilson
camp. I think more linking should occur at the "lore" level,
i.e. in the context of exploring the history of concepts
(as you are doing below). I'm for going back through
Ada Byron, for example, into the realm of computing
machinery, another way to pick up Pascal and Leibniz.
Through Leibniz, we reach Descartes, paranoid about
some of his results, keeping a secret notebook.... you
can see where students might sit up and listen, if the
development of mathematics were presented more like
the soap opera that it was. We don't skimp on the
concepts, but nor do we omit the lore. "Why was this
concept important?" (not just "what it is").
Example storytelling: Neal Stephenson covers a lot
of this territory (from a "lore" point of view), in the
popular 'Cryptonomicon' a work of historical fiction.
Today's high schoolers should expect to learn about
the Enigma Machine and Alan Turing, as the story
of modern computing unfurls, right down to the
OLPC's XO (where maybe they're reading this), MIT
Media Lab... ILM, Pixar etc.
Like when we get to RSA (say per the Litvins' text
'Mathematics for the Digital Age'), we're prepared to share
stories about how GHCQ dutifully shelved the idea
(of public key crypto), while commerical interests in
North America managed to push it through at MIT,
though not without help from Phil Zimmerman and
company (per the cypher-punk movement, as some
called it then).
Anyway, that's a really face-paced trajectory along a
time line, skipping just about everything, but then this
isn't the time for exhaustive retrospectives. Just doodling
a mathfuture possibility, for schools interested in a more
chronologically integrated curriculum. I'd recommend
'The Book of Numbers' by Conway and Guy for more
cues, along with the Midhat Gazale books. Thanks to
Glenn Stockton for the pointer to the book about
Descarte's secret notebook, decoded by Leibniz but
then lost to history once again:
Kirby
Notes:
More on RSA / crypto here:
Litvin & Litvin:
1997 Oregon Math Summit w/ Ralph Abraham, Sir Roger Penrose etc.
milo gardner
Jun 23, 2010, 9:59:31 AM6/23/10
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Kirby, Thank you again for your very thoughtful comments: |
"This is fascinating material Milo. The 1997 Oregon Math Summit I was mentioning (not 1995 -- my mistake) also featured Ralph Abraham, who used the occasion to stump for a time-line-based approach to K-12 math learning,
actually studying the mathematics of a civilization while learning about it in historical terms, proceeding chronologically, grade by grade. Why not try it? Not every school needs to switch
tracks. Pioneering is OK (as we say in Oregon). This chrono-sequence wouldn't have to be too ploddingly adhered to (could have flash-forwards and flash-backs), would be a lot
more like the game 'Civilization' which many students have already tasted and enjoyed (another current thread with the Devlin group |
is a need for more simulations in math class)." |
The Portland and Oregon approaches to teaching children mathematics are enlightened compared to California's sterile approach. In 1990 the State of California was required to publish a K-12 Math Framework. The draft copy was written up by a kindergarten teacher from Northridge, LA County. Basic math concepts like function were muddled, as were the social learning sides of math history story lines. All of the ethnic story-lines, once traditional math framework topics, had been removed. State Board of Education, Sacramento, testimony included five to six ethnic community groups. My story line cited Gauss' child story of adding 1-100 by matching the first and last numbers in the series: 101, 50 times, that mentally were summed to 5050 by an algebraic concept (that Ahmes discussed in RMP 39. 40 and 64 following a 200 year older document, the Kahun Papyrus: http://en.wikipedia.org/wiki/Kahun_Papyrus ) Many numerate children, over the years, seem to have independently developed this basic arithmetic progression knowledge. Sadly, the California State Board of Education did not agree with our testimonies. A professional writer was hired to fix the muddled math concepts, a process that took an additional two years. In the end, California formally excised ancient and modern ethnic math story lines from its federally mandated K-12 Mathematical Framework. Luckily, knowledgeable college math professors were present (one from UC Davis, Henry Alder author of "Probability and Statistics" a text that I happily studied as an undergrad) shook their collective heads on the socially sterile final report. College math professors and average high school math teachers are aware that arithmetic remains as the main bridge to algebra. There are many classes of historical story lines that can be taken from many cultures. Reporting the human sides of how and why children can be introduced to the power of numbers, and the fundamental concepts of arithmetic and algebra, is fun. Thank you all, for not following California's 1990 lead, and showing the fun side of the long and rich history of mathematics to our children. Best Regards, Milo Gardner |
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