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- Berlin Papyrus and second degree equations | planetmath.org

x2 y2 100 4x 3y 0 what are x and y Clagett s single false position suggestion was borrowed from 1920s attempts to read closely related Rhind Mathematical Papyrus problems and methods an approach that not involved in the Berlin Papyrus either The 1900 BCE Berlin Papyrus solution was reported by Schack Schackenberg in 1900 AD as Ahmes reported his 1650 BCE solution in RMP 69 Assume the square of the first side y to be 1 cubit Then the other side x will be 1 2 1 4 Then y2 1 and using Egyptian multiplication we determine x2 1 2 1 4 1 2 1 4 1 8 1 4 1 8 1 16 1 2 1 4 1 4 1 8 1 8 1 16 1 2 1 16 Thus x2 y2 1 1 2 1 16 Now 1 1 2 1 16 1 2 1 1 4 and 100 1 2 10 Divide 10 by 1 1 4 and you get 8 Ahmes pesu an inverse proportional valuation of commodities was finitely informed by Egyptian square root The pesu used in the RMP 69 shed light on the Berlin Papyrus in other ways The Middle Kingdom method was not consistently parsed by scholars in the 20th century related to confusion over single false position and other issues Gillings and Clagett missed MK scribal arithmetic details by reporting personalized versions of the text s math By reading RMP 69 the 10 by 10 cubit was broken into two squares in the ratio of 1 3 4 to one another in the Berlin Papyrus Following Schack Schackenburg properly footnoted by Clagett the pesu method offered a direct proof that abstract mathematics solved two second 1900 BCE second degree equations within hekat and loaf conversions to Pesu units Clagett was not alone in reporting the Berlin Papyrus method contained single false position division operation Raw transliterated hieratic data shows that Ahmes obtained 5 4 from the pesu step and not from single false position The simplest version of the data says that 10 was divided by 5 4 and solved by 10 times 4 5 8 as we do today So we get x 8 The Berlin Papyrus reported y2 100 64 y 6 obtained y 6 using modern arithmetic steps such that 64 36 100 was proven Berlin Papyrus Problem 2 You are told the area of a square of 400 square cubits was equal to that of two smaller squares the side of one square is 1 2 1 4 of the other reported by the scribe as 2 3 2 of one another What are the sides of the two unknown squares This is analogous to problem 1 except that the Berlin Papyrus scribe s pesu calculation used 2 3 2 an analysis that placed the pesu calculation in a different logical step than the first problem Subtle differences are important as historians report scribal shorthand notes as the mathematics was recorded The second Berlin Papyrus problem

Original URL path: http://www.planetmath.org/berlinpapyrusandseconddegreeequations (2016-04-25)

Open archived version from archive - milogardner | planetmath.org

PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A01A16 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module History Member for 10 years 1 month My articles Acano a lunar calendar method Ahmes bird feeding rate method Akhmim Wooden Tablet An over view of Ahmes Papyrus Arabic numerals Archimedes calculus Berlin Papyrus and second degree equations economic context of Egyptian fractions Egyptian and Greek square root Egyptian fraction Egyptian geometry areas calculated in cubits khets and setats Egyptian Mathematical Leather Roll Egyptian multiplication and division Egyptian weights and measures hekat divisions Hibeh Papyrus Hultsch Bruins method Kahun Papyrus and Arithmetic Progressions Liber Abaci Mathematics in Egypt Mathematical Leather Roll 2014 Update Mayan math Mayan Seasonal Almanac Mayan Super number arithmetic Moscow Mathematical Papyrus Plato s mathematics Red Auxiliary numbers the first LCM method Reisner Papyrus remainder arithmetic remainder arithmetic vs Egyptian fractions Rhind Mathematical Papyrus RMP 35 to 38 plus RMP 66 RMP 36 and the 2 n table RMP 47 and the hekat RMP 53 54 55 RMP 69 and the Berlin Paprus proportion method square root of 3 5 6 7 and 29 Why Study Egyptian Fraction Mathematics Buddy list actions Create Buddy List Designate Existing Team Member of Egyptian Fractions Hultsch Bruins Method editors Wikipedia Group World Writable Forename Milo Surname Gardner City sacramento State CA Country USA Homepage http en wikipedia org wiki User Milogardner Preamble this is the default PlanetMath preamble as your knowledge of TeX increases you will probably want to edit this but it should be fine

Original URL path: http://www.planetmath.org/users/milogardner (2016-04-25)

Open archived version from archive - Acano a lunar calendar method | planetmath.org

Quoting Barrios As a matter of fact to record a date on the acano you only need to write a number from 1 to 30 on one of its squares The selected square fixes the moon while the number fixes the day of the moon counted let us say from new to new Accordingly it is possible to record unambiguously on a single acano the 33 successive dates fixing a whole round of the summer solstice through the lunar year What is of the utmost importance is that this can be accomplished either through the years by actual observation either at any desired moment by performing an easy arithmetical exercise on the acano Indeed once recorded on the acano the date of a particular summer solstice we obtain the dates of the next summer solstices simply adding 11 days by year to the previous number Each time the accumulated shift is greater than 29 or 30 days we jump to the next square reduce the shift by 29 or 30 days write the new date on the square and continue the count Actually this exercise can be done even mentally for a number of years DISCUSSIONS Aaboe argues for several classes of lunar eclipse calendars in ancient cultures One was the acano Several scholars argue the Egyptian and Babylonian 135 moon case including red to denote life and black to denote death following an acano like cycle Jose Barrios Garcia argues for Canarian 520 day eclipse and 270 moon calendars in alternating red and black day colors F Lounsbury and A Lebeuf argue the Mayan 260 day calendar a Canarian 520 day calendar and a 405 moon calendar that include red to denote east good and black to denote west almost bad followed acano cycles Aztecs and Mesoamericans used black to denote north and death Acano luni solar eclipse calendars were well known to ancient cultures such as 9 moon 260 day and 99 moon calendars cycles that may relate to ancients predicting eclipses Two recent acano type uses aligned civil and scientific calendars connected to eclipses tables and validated longitudes in land map making and ocean navigation The far ranging Phoenicians circumnavigated Africa in 600 BCE made later trips as did Columbus in 1492 assisted by lunar eclipse tables and the Armed Guards of Polaris calibrated daily by 1 2 hour glasses for longitude and Polaris for latitude Acano type lunar eclipse prediction tables referenced to Spain port city longitudes were carried by Columbus on four New World trips Columbus like the Phoenicians Libyans had used the 135 moon Egyptian eclipse tables Phoenicians Libyans reached the Canary Islands in 900 BCE as deposed Egyptian pharaohs denoted by mummification and calendars and stayed there until 400 AD as a stop over on long voyages Columbus debarked from the Canary Islands and observed two eclipses on his voyages determining longitudes more accurate than historians have credited him by only reading Bishop de Landa s edited data Columbus had based his longitude

Original URL path: http://www.planetmath.org/acanoalunarcalendarmethod (2016-04-25)

Open archived version from archive - conchoid of Nicomedes | planetmath.org

FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit conchoid of Nicomedes The conchoid of Nicomedes is the locus of the endpoints of a line segment length 2 b 2 b 2b the midpoint of which is on a given line l l l and which lies on a line through a given point O O O with distance a a a from l l l The curve was invented by the Greek matematician Nicomedes in about 200 BCE With his conchoid he solved two classical problems of constructibility viz doubling the cube and trisecting the angle The name of the curve is derived from Greek ϰ o γ χ η ϰ normal o γ χ η varkappa acute o gamma chi eta mussel ε ι δ o ς ε ι δ o ς varepsilon iota delta o varsigma form kind type psaxes Dx 9 Dy 9 0 0 1 2 5 4 8 5 x x x y y y l l l O O O a a a b b b b b b a h a h a h Choosing for O O O the origin and l l l vertical we have in the polar coordinates r φ r φ r varphi for the conchoid of Nicomedes the expression r a cos φ b r plus or minus a φ b displaystyle r frac a cos varphi pm b 1 or r a cos φ 2 b 2 superscript r a φ 2 superscript b 2 left r frac a cos varphi right 2 b 2 Her we may substitute cos φ x r φ x r cos varphi frac x r and then r 2 x 2 y 2 superscript r 2 superscript x 2 superscript y 2 r 2 x 2 y 2 when the equation of the curve can be simplified to x 2 y 2 x a 2 b 2 x 2 superscript x 2 superscript y 2 superscript x a 2 superscript b 2 superscript x 2 displaystyle x 2 y 2 x a 2 b 2 x 2 2 Hence the conchoid is an algebraic curve The form y x x a b 2 x a 2 y plus or minus x x a superscript b 2 superscript x a 2 displaystyle y pm frac x x a sqrt b 2 x a 2 3 of the equation tells that the curve has as an asymptote the line x a x a x a It s not hard to derive the following parametric presentation of the conchoid of Nicomedes x a a b a 2 h 2 y h b h a 2 h 2 formulae sequence x plus or minus a a b superscript a 2 superscript h 2 y plus or minus h b h superscript a

Original URL path: http://www.planetmath.org/conchoidofnicomedes (2016-04-25)

Open archived version from archive - compass and straightedge construction | planetmath.org

compass was opened exactly π 𝜋 times wider than the length of the original line segment Compass and straightedge constructions are of historical significance The ancient Greeks are the most well known civilization for investigating these constructions on an elementary level It should be pointed out that the compasses that they used were collapsible That is you could open the compass and draw an arc but immediately after you removed a point of the compass from the plane where you drew the arc the compass would close completely It turns out that whether a collapsible compass or a modern day compass is used to perform these constructions makes no difference This statement is justified by the fact that one can use a collapsible compass to construct a circle with a given radius at any point as shown by this entry One of the greatest applications of abstract algebra is being able to determine which constructions are possible and which are not The connection between constructions and abstract algebra is that the set of all constructible points is in one to one correspondence with the elements of the field of constructible numbers Without abstract algebra one would be hard pressed to prove statements about constructibility such as A 20 superscript 20 angle is not constructible with compass and straightedge Defines compass straightedge ruler constructible collapsible compass collapsible Related ConstructibleNumbers TheoremOnConstructibleAngles MotivationOfDefinitionOfConstructibleNumbers ClassicalProblemsOfConstructibility Synonym straightedge and compass construction ruler and compass construction compass and ruler construction Major Section Reference Type of Math Object Definition Groups audience Chi and me Mathematics Subject Classification 01A20 no label found 51M15 no label found Add a correction Attach a problem Ask a question Comments Minimal group axiom set Permalink Submitted by dh2718 on Wed 06 27 2007 22 13 This is an attempt to reduce the standard axiom set for group Let S be a set with an associative law defined everywhere for any a and b in S there exists an unique c such that ab c This associative law satisfies the following axioms 1 There exists at least one right neutral element e may be more f g such that ae a for all a in S af a ag a 2 Every a in S has at least one right inverse a maybe more with respect to one of the right elements e such that aa e Axiom 2 differs from the standard definition for a and b in S there are right inverse a and b not necessarily unique such that aa e and bb f but e and f could be different right neutral elements Is such a set still a group Log in to post comments Re Minimal group axiom set Permalink Submitted by stevecheng on Wed 06 27 2007 22 51 Here s what I think 1 Under the proposed axiom set there exists at most one right inverse but there could exist none for you only say the right hand side of ab c exists If I ask

Original URL path: http://www.planetmath.org/compassandstraightedgeconstruction (2016-04-25)

Open archived version from archive - ruled surface | planetmath.org

PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M15 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M15 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit ruled surface A straight line g 𝑔 moving continuously in space sweeps a ruled surface Formally A surface S 𝑆 in ℝ 3 superscript ℝ 3 is a ruled surface if it is connected and if for any point p 𝑝 of S 𝑆 there is a line g 𝑔 such that p g S 𝑝 𝑔 𝑆 Such a surface may be formed by using two auxiliary curves given e g in the parametric forms r a t r b t formulae sequence 𝑟 𝑎 𝑡 𝑟 𝑏 𝑡 Using two parameters s 𝑠 and t 𝑡 we express the position vector of an arbitrary point of the ruled surface as r a t s b t 𝑟 𝑎 𝑡 𝑠 𝑏 𝑡 Here r a t 𝑟 𝑎 𝑡 is a curve on the ruled surface and is called directrix or the base curve of the surface while r b t 𝑟 𝑏 𝑡 is the director curve of the surface Every position of g 𝑔 is a generatrix or ruling of the ruled surface Examples 1 Choosing the z 𝑧 axis r c t k 𝑟 𝑐 𝑡 𝑘 c 0 𝑐 0 as the directrix and the unit circle r i cos t j sin t 𝑟 𝑖 𝑡 𝑗 𝑡 as the director curve we get the helicoid screw surface cf the circular helix r c t k s i cos t j sin t s cos t s sin t c t 𝑟 𝑐 𝑡 𝑘 𝑠 𝑖 𝑡 𝑗

Original URL path: http://www.planetmath.org/ruledsurface (2016-04-25)

Open archived version from archive - Recent revisions | planetmath.org

3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Add new Article View sorted by MSC navigate by MSC Map experimental View sorted by creation date A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 0 9 other symbols Major Section field section Any Collaboration Reference Research Recreation Education Non Newtonian Calculus Last updated by smithpith on Monday April 25 2016 10 42 field arising from special relativity Last updated by pahio on Wednesday April 20

Original URL path: http://www.planetmath.org/revisions?section=All (2016-04-25)

Open archived version from archive - planetmath.org | Math for the people, by the people.

can t understand what you are trying to do Greetings p Approximate values of real by RKJCHENNAI Feb 11 http rkmath yolasite com resources Roots 20sum 20and 20multiplication pdf Finding approximate values of real roots equation by RKJCHENNAI Feb 11 p search engine by akdevaraj Feb 4 Search engine still not functioning p Legendre s symbol by akdevaraj Jan 29 a p is the Legendre symbol for quadratic residues and non residues If a p 1 it means a is not a quadratic residue of p Another way of expressing this the p under ref is an impossible prime factor of x 2 a For further examples of this concept see A123239 A 072936 and A 119691 on OEIS p Book by akdevaraj Jan 26 Currently reading prime numbers a computational perspective by Richard Crandall and Carl Pomerance This is a monumental book on prime numbers primality testing and factorisation p Book by akdevaraj Jan 26 Currently reading prime numbers a computational perspective by Richard Crandall and Carl Pomerance This is a monumental book on prime numbers primality testing and factorisation p Book by akdevaraj Jan 26 Currently reading prime numbers a computational perspective by Richard Crandall and Carl Pomerance This is a monumental book on prime numbers primality testing and factorisation p Indirect primality testing by akdevaraj Jan 24 Dear dh2718 thanks for the compliment p Indirect primality testing by akdevaraj Jan 23 Dear Permalink you are right about bio age I am 85 p Indirect primality testing by akdevaraj Jan 13 Dear Peruchio f x k f x star means multiply k belongs to Z Thus in this case since f x x 2 x 1 f x k f x means 1 3k Trust this is clear more Latest Messages p Another good book by akdevaraj Mar 27 Prime obsession by John Derbyshire it is on Riemann Hypothesis extremely readable pstricks by Wkbj79 Mar 15 p Deleting by akdevaraj Feb 15 How do we delete a message I clicked on edit then tried to delete but did not succeed p Indirect primality testing by perucho Feb 15 Thanks akdevaraj for clarify what means k confused me since here in PM one simply writes f x kf x Your example is quite complex so f x kf x is too long But let s suppose f x x 2 so f x kf x f x kx 2 x kx 2 2 k 2x 4 2kx 3 x 2 Let s prove a couple of examples If k 0 f x 0f x 0 2x 4 2 0 x 3 x 2 x 2 f x as it should be If k 1 f x 1f x 1 2x 4 2 1 x 3 x 2 x 2 2x 3 x 4 x x 2 2 f x x 2 again as it should be Thus I can t understand what you are trying to do Greetings p Approximate values of real by RKJCHENNAI Feb 11 http rkmath

Original URL path: http://www.planetmath.org/node?destination=node (2016-04-25)

Open archived version from archive