Rhind Mathematical Papyrus

Rhind Mathematical Papyrus
manuscript (mul)
Bayanai
Suna saboda	Alexander Henry Rhind (en)
Muhimmin darasi	Lissafi
Mawallafi	Ahmes (en)
Ranar wallafa	16 century "BCE"
Catalog code (en)	17
Ƙasa da aka fara	Tsohuwar Masar
Harshen aiki ko suna	Harshen Misira
Kayan haɗi	papyrus (en)
Collection (en)	British Library (en)
Copyright status (en)	public domain (en) da public domain (en)

Rhind Mathematical Papyrus (RMP; wanda aka fi sani da papyrus British Museum 10057, pBM 10058, da Brooklyn Museum 37.1784Ea-b) yana daya daga cikin sanannun misalai na Lissafin Masar na dā.

Yana daya daga cikin sanannun takardun lissafi guda biyu, tare da Moscow Mathematical Papyrus . Rhind Papyrus shine mafi girma, amma ƙarami, daga cikin biyun.

A cikin sakin layi na farko na papyrus Ahmes ya gabatar da papyrus kamar yadda yake ba da "Dubbin lissafi na gaskiya don yin bincike a cikin abubuwa, da kuma ilimin dukkan abubuwa, asiri ... duk asirin".  Ya ci gaba da cewa:

An kwafe wannan littafin a cikin shekara ta 33, watan 4 na Akhet, a ƙarƙashin ubangijin Sarkin Upper da Lower Egypt, Awserre, wanda aka ba da rai, daga tsohuwar kwafin da aka yi a lokacin Sarkin Uppar da Lower Egypt Nimaatre. Marubucin Ahmose ya rubuta wannan kwafin.

An buga littattafai da labarai da yawa game da Rhind Mathematical Papyrus, kuma kaɗan daga cikin waɗannan sun fito fili. An buga Rhind Papyrus a cikin 1923 ta masanin kimiyyar Masar na Ingilishi T. Eric Peet kuma ya ƙunshi tattaunawa game da rubutun da ya biyo bayan littafin Francis Llewellyn Griffith na I, II da III. Chace ya wallafa wani tarin a cikin 1927-29 wanda ya haɗa da hotuna na rubutun. An buga wani bayyani na baya-bayan nan na Rhind Papyrus a cikin 1987 ta Robins da Shute.

Tarihi

Rhind Mathematical Papyrus ya kasance a cikin Lokaci na Biyu na Masar. Marubucin Ahmes ne ya kwafe shi (watau, Ahmose; Ahmes tsohuwar rubutun da masana tarihi na lissafi suka fi so) daga rubutun da ya ɓace yanzu daga mulkin sarki na 12 Amenemhat III.

Ya kasance a kusa da 1550 BC. Takardar ta kasance a Shekara ta 33 na Sarkin Hyksos Apophis kuma ta ƙunshi wani baya tarihi na baya-bayan nan a kan bayanta mai yiwuwa daga "Shekara 11" na magajinsa, Khamudi.

Alexander Henry Rhind, masanin tarihi na Scotland, ya sayi sassa biyu na papyrus a 1858 a Luxor, Misira; an bayyana cewa an samo shi a "ɗaya daga cikin ƙananan gine-gine kusa da Ramesseum", kusa da Luxor.

Gidan kayan gargajiya na Burtaniya, inda aka adana mafi yawan papyrus yanzu, ya samo shi a 1865 tare da Rubutun Leather na Masar, wanda Henry Rhind ya mallaka.

An sayi wasu sassan rubutun da kansu a Luxor ta hanyar masanin kimiyyar Masar na Amurka Edwin Smith a tsakiyar shekarun 1860, 'yarsa ta ba da gudummawa a 1906 ga New York Historical Society, kuma yanzu Gidan Tarihi na Brooklyn ne ke gudanar da su. Wani sashi na tsakiya na 18 centimetres (7.1 in) in) ya ɓace.

An fara fassara papyrus kuma an fassara shi da lissafi a ƙarshen karni na 19. Al'amarin fassarar lissafi ya kasance ba cikakke ba a fannoni da yawa.

Littattafai

Littafin I - Arithmetic da Algebra

Sashe na farko na Rhind papyrus ya ƙunshi teburin bincike da tarin 21 arithmetic da 20 algebraic matsaloli. Matsalolin suna farawa ne tare da maganganu masu sauƙi, sannan kuma matsalolin kammalawa (sekem) da kuma ƙarin ƙididdigar layi (matsalaraha).

Sashe na farko na papyrus an ɗauke shi da teburin 2 / n. Kashi 2 / n don n mai ban mamaki daga 3 zuwa 101 ana bayyana su azaman jimlar sassan raka'a. Misali,

2 15

=

1 10

+

1 30

{\displaystyle {\frac {2}{15}}={\frac {1}{10}}+{\frach {1} {30}}}

. Rarrabawar 2 / n cikin kashi na raka'a ba ta wuce kalmomi 4 ba kamar misali:

${\frac {2}{101}}={\frac {1}{101}}+{\frac {1}{202}}+{\frac {1}{303}}+{\frac {1}{606}}$

Wannan teburin yana biye da ƙarami, ƙaramin tebur na maganganu masu raguwa don lambobi 1 zuwa 9 da aka raba da 10. Misali an rubuta rabuwa da 7 da 10 kamar haka:

7 raba da 10 yana samar da 2/3 + 1/30

Bayan wadannan teburin biyu, papyrus ya rubuta matsaloli 91 gaba ɗaya, waɗanda zamani suka sanya su a matsayin matsaloli (ko lambobi) 1-87, gami da wasu abubuwa huɗu waɗanda aka sanya su a cikin matsalolin 7B, 59B, 61B da 82B. Matsalolin 1-7, 7B da 8-40 sun damu da lissafi da algebra na farko.

Matsalolin 1-6 suna lissafin rarraba wasu gurasar gurasa ta maza 10 kuma suna yin rikodin sakamakon a cikin sassan raka'a. Matsalolin 7-20 suna nuna yadda za a ninka maganganun 1 + 1/2 + 1/4 = 7/4, da kuma 1 + 2/3 + 1/3 = 2 ta bangarori daban-daban.Matsalolin 21-23 matsaloli ne a kammala, wanda a cikin ƙididdigar zamani kawai matsalolin cirewa ne. Matsalolin 24-34 sune matsalolin "aha"; waɗannan ma'auni ne na layi. Matsalar 32 misali ta dace (a cikin bayanin zamani) don warware x + 1/3 x + 1/4 x = 2 don x. Matsalar 35-38 ta haɗa da rarrabuwa na heqat, wanda shine tsohuwar ƙungiyar Masar ta girma. Farawa a wannan lokacin, nau'ikan ma'auni iri-iri sun zama mafi mahimmanci a duk sauran papyrus, kuma hakika babban la'akari a duk sauran Papyrus shine nazarin girma. Matsalolin 39 da 40 suna lissafin rarraba gurasa kuma suna amfani da ci gaban lissafi.

Littafin II - Geometry

Wani bangare na Rhind Papyrus

Sashe na biyu na Rhind papyrus, kasancewar matsalolin 41-59, 59B da 60, ya ƙunshi matsalolin lissafi. Peet ya kira wadannan matsalolin a matsayin "matsalar ma'auni".

Matsalolin 41-46 suna nuna yadda za a sami ƙarar ɗakunan ajiya na cylindrical da rectangular. A cikin matsala 41 Ahmes ya lissafa girman hatsi na cylindrical. Idan aka ba da diamita d da tsawo h, an ba da ƙarar V ta hanyar:

$V=\left[\right(1-1/9\left)d\right]^{2}h$

A cikin lissafin lissafi na zamani (kuma ta amfani da d = 2r) wannan yana ba da

V = (8

/

9

)

2

d

2

h = (256

/

81 )

r

2

h

{\displaystyle V= (8/9) ^{2}d^{2}h= (256/81) r^{2}.

. Kalmar raguwa 256/81 tana kusa da darajar π kamar yadda yake 3.1605..., kuskuren kasa da kashi daya cikin dari.

Matsalar 47 tebur ne tare da daidaito na raguwa wanda ke wakiltar yanayi goma inda yawan jiki na "100 na hudu" ya kasu kashi ɗaya daga cikin ninki goma, daga goma zuwa ɗari. Ana bayyana ƙididdigar ta hanyar ɓangarorin Ido na Horus, wani lokacin kuma ta amfani da ƙaramin ɗayan ƙarar da aka sani da "rashin sau huɗu". Heqat sau huɗu da ro sau huɗu sune raka'a na ƙarar da aka samo daga heqat da ro mai sauƙi, don waɗannan raka'a huɗu na ƙarar su gamsar da alaƙa masu zuwa: 1 heqat sau huɗar = 4 heqat = 1280 ro = 320 ro sau huɗara. Don haka,

100/10 sau huɗu heqat = 10 sau huɗu

100/20 heqat sau huɗu = 5 heqat sau hudu

100/30 heqat sau huɗu = (3 + 1/4 + 1/16 + 1/64) heqat sau hudu + (1 + 2/3) matsayi sau huɗu

100/40 heqat sau huɗu = (2 + 1/2) heqat sau hudu

100/50 heqat sau huɗu = 2 heqat sau hudu

100/60 sau huɗu heqat = (1 + 1/2 + 1/8 + 1/32) sau huɗu Heqat + (3 + 1/3) sau huɗu

100/70 heqat sau huɗu = (1 + 1/4 + 1/8 + 1/32 + 1/64) heqat sau hudu + (2 + 1/14 + 1/21 + 1/42) ro sau huɗu

100/80 sau huɗu heqat = (1 + 1/4) sau huɗu

100/90 sau huɗu heqat = (1 + 1/16 + 1/32 + 1/64) sau huɗu Heqat + (1/2 + 1/18) sau huɗu

100/100 heqat sau huɗu = 1 heqat sau hudu

Yankuna

Matsalolin 48-55 suna nuna yadda za a lissafa yankuna iri-iri. Matsalar 48 sananne ce saboda taƙaice tana ƙididdige Yankin da'irar ta hanyar kimanta π. Musamman, matsala ta 48 a bayyane ta karfafa yarjejeniyar (wanda aka yi amfani da shi a ko'ina cikin sashin lissafi) cewa "yanki na da'irar yana tsaye da na murabba'in murabba'insa a cikin rabo 64/81." Daidai, papyrus ya kusan π kamar 256/81, kamar yadda aka riga aka lura a sama a cikin bayanin matsalar 41.

Sauran matsaloli suna nuna yadda za a sami yankin rectangles, triangles da trapezoids.

Pyramids

Matsalolin karshe guda shida suna da alaƙa da gangaren pyramids. An bayar da rahoton matsala kamar haka:

Idan dala tana da tsayi 250 kuma gefen tushe 360 yana da tsayi, menene ya sa?"

An ba da mafita ga matsalar a matsayin rabo na rabin gefen tushe na dala zuwa tsayinsa, ko kuma rabo na gudu zuwa tashi na fuskarsa. A wasu kalmomi, adadin da aka samo don seked shine cotangent na kusurwa zuwa tushe na dala da fuskarsa.

Littafin III - Miscellany

Sashe na uku na Rhind papyrus ya ƙunshi sauran matsalolin 91, kasancewar 61, 61B, 62-82, 82B, 83-84, da "lambobi" 85-87, waɗanda abubuwa ne waɗanda ba na lissafi ba ne a yanayi. Wannan sashi na ƙarshe ya ƙunshi teburin bayanai masu rikitarwa (wanda sau da yawa ya haɗa da ɓangarorin ido na Horus), matsaloli da yawa na pefsu waɗanda ke da matsalolin algebraic na asali game da shirye-shiryen abinci, har ma da matsala mai ban sha'awa (79) wanda ke nuna ci gaban lissafi, jerin lissafi, da wasu matsalolin da suka biyo baya da ma'anar tarihi. Matsalar 79 ta bayyana a bayyane, "gidaje bakwai, cats 49, beraye 343, kunnuwa 2401 na spelt, hekats 16807. " A cikin matsala 79 ya shafi yanayin da gidaje 7 kowannensu yana dauke da beraye bakwai, wanda duk suna cinye beraye bakwai na hatsi, kowannensu zai samar da matakai bakwai na hatimi. Sashe na uku na Rhind papyrus sabili da haka wani nau'i ne na miscellany, gina a kan abin da aka riga aka gabatar.Matsalar 61 ta shafi ninkawar raguwa. Matsalar 61B, a halin yanzu, tana ba da cikakkiyar magana don lissafin 2/3 na 1/n, inda n ba daidai ba ne. A cikin bayanin zamani tsarin da aka bayar shine

${\frac {2}{3n}}={\frac {1}{2n}}+{\frac {1}{6n}}$

Hanyar da aka bayar a cikin 61B tana da alaƙa da samo teburin 2 / n.

Matsalolin 62-68 sune matsalolin algebraic. Matsalolin 69-78 duk matsalolin pefsu ne a wani nau'i ko wani. Sun haɗa da lissafi game da ƙarfin burodi da giya, dangane da wasu albarkatun da aka yi amfani da su wajen samar da su.

Matsalar 79 ta tara kalmomi biyar a cikin Ci gaban lissafi. Harshensa yana da karfi yana nuna ma'anar zamani da kuma yarinya "Yayin da nake zuwa St Ives".Matsalolin 80 da 81 suna lissafin ɓangarorin Ido na Horus na hinu (ko nlats). Abubuwa huɗu na lissafi na ƙarshe, matsalolin 82, 82B da 83-84, suna lissafin adadin abinci da ake buƙata ga dabbobi daban-daban, kamar tsuntsaye da shanu. Koyaya, waɗannan matsalolin, musamman 84, suna fama da rashin tabbas, rikice-rikice, da rashin daidaito mai sauƙi.

Abubuwa uku na ƙarshe a kan Rhind papyrus an sanya su a matsayin "lambobi" 85-87, sabanin "matsala", kuma an warwatsa su ko'ina a gefen baya na papyrus, ko verso. Su ne, bi da bi, karamin magana wanda ke kawo karshen takardar (kuma yana da 'yan damar fassara, wanda aka bayar a ƙasa), wani takarda da ba shi da alaƙa da jikin takardar, wanda aka yi amfani da shi don riƙe shi tare (duk da haka yana dauke da kalmomi da ɓangarorin Masar waɗanda yanzu sun saba da mai karatu na takardar), da kuma karamin bayanin tarihi wanda ake zaton an rubuta shi wani lokaci bayan kammala jikin rubutun papyrus. Ana tunanin wannan bayanin don bayyana abubuwan da suka faru a lokacin "mulkin Hyksos", lokacin katsewa na waje a cikin al'ummar Masar ta dā wanda ke da alaƙa da lokacin tsakiya na biyu. Tare da waɗannan ba-mathematical ba duk da haka a tarihi da kuma ilimin harshe mai ban sha'awa, rubutun papyrus ya zo ƙarshe.

Haɗin haɗin kai

Yawancin kayan Rhind Papyrus sun damu da ma'aunin Masar na dā kuma musamman nazarin girman da aka yi amfani da shi don canzawa tsakanin su. An ba da daidaituwa na ma'auni da aka yi amfani da shi a cikin papyrus a cikin hoton.

Rukunin ma'auni da aka yi amfani da su a cikin Rhind Papyrus .

Abun ciki

Wannan teburin ya taƙaita abubuwan da ke cikin Rhind Papyrus ta hanyar taƙaitaccen fassarar zamani. Ya dogara ne akan bayyanar kundi biyu na papyrus wanda Arnold Buffum Chace ya buga a 1927, kuma a 1929. [1] Gabaɗaya, papyrus ya ƙunshi sashe huɗu: shafi na taken, teburin 2 / n, ƙaramin tebur "1-9/10", da matsaloli 91, ko "lambar". An ƙidaya waɗannan daga 1 zuwa 87 kuma sun haɗa da abubuwa huɗu na lissafi waɗanda zamani suka sanya su a matsayin matsalolin 7B, 59B, 61B, da 82B. Lamba 85-87, a halin yanzu, ba abubuwa ne na lissafi waɗanda suka zama wani ɓangare na jikin takardar ba, amma a maimakon haka bi da bi: ƙaramin magana ce da ke ƙare takardar, wani yanki na "takardar" da aka yi amfani da shi don riƙe takardar tare (wanda ya riga ya ƙunshi rubuce-rubucen da ba su da alaƙa), da bayanin tarihi wanda ake zaton ya bayyana wani lokaci jim kadan bayan kammala jikin papyrus. Wadannan abubuwa uku na ƙarshe an rubuta su a wurare daban-daban na baya papyrus (gefen baya), nesa da abubuwan lissafi. Sabili da haka Chace ya bambanta su ta hanyar tsara su a matsayin lambobi ba tare da Matsalolin ba, kamar sauran abubuwa 88 da aka ƙidaya.

Bibiyar Tarihi

• Chace, Arnold Buffum; et al. (1927). The Rhind Mathematical Papyrus. 1. Oberlin, Ohio: Mathematical Association of America – via Internet Archive.
• Chace, Arnold Buffum; et al. (1929). The Rhind Mathematical Papyrus. 2. Oberlin, Ohio: Mathematical Association of America – via Internet Archive.
• Gillings, Richard J. (1972). Mathematics in the Time of the Pharaohs (Dover reprint ed.). MIT Press. ISBN 0-486-24315-X.
• Robins, Gay; Shute, Charles (1987). The Rhind Mathematical Papyrus: an Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4.

Mahaɗa

• British Museum webpage on the first section of the Papyrus
• British Museum webpage on the second section of the Papyrus
• British Museum webpage on the Papyrus at the Wayback Machine (archived ga Yuni, 29, 2015).
• "Mathematics in Egyptian Papyri". MacTutor History of Mathematics archive.
• Samfuri:MathWorld
• Williams, Scott W. Mathematicians of the African Diaspora, containing a page on Egyptian Mathematics Papyri.