Rhind Mathematical Papyrus 2/n tebur

Rhind Mathematical Papyrus 2/n table
mathematical table (en)
Bayanai
Bangare na	Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus, wani tsohon aikin lissafi na Masar, ya haɗa da tebur na lissafi don canza lambobi masu ma'ana na nau'i 2/ n zuwa ɓangarorin Masarawa ( jimlar juzu'i na raka'a daban-daban ), nau'i da Masarawa suka yi amfani da su don rubuta lambobi. Rubutun ya bayyana wakilcin lambobi 50 masu hankali. An rubuta shi a lokacin tsaka-tsakin lokaci na biyu na Masar (kimanin 1650-1550). BCE) na Ahmes, marubucin farko na ilimin lissafi wanda aka san sunansa. Wataƙila an kwafi ɓangarori na takaddar daga 1850 da ba a san su ba Rubutun BCE.

Tebur

Tebur mai zuwa yana ba da fadadawa da aka jera a cikin papyrus.

Tebur 2/n daga Rhind Mathematical Papyrus

2/3  = 1/2 + 1/6Samfuri:Figure space	2/5   = 1/3 + 1/15Samfuri:Figure spaceSamfuri:Figure space	2/7  = 1/4 + 1/28Samfuri:Figure space
2/9  = 1/6 + 1/18Samfuri:Figure space	2/11  = 1/6 + 1/66Samfuri:Figure space	2/13 = 1/8 + 1/52 + 1/104
2/15 = 1/10 + 1/30	2/17  = 1/12 + 1/51 + 1/68Samfuri:Figure space	2/19 = 1/12 + 1/76 + 1/114
2/21 = 1/14 + 1/42	2/23  = 1/12 + 1/276Samfuri:Figure space	2/25 = 1/15 + 1/75
2/27 = 1/18 + 1/54	2/29  = 1/24 + 1/58 + 1/174 + 1/232Samfuri:Figure space	2/31 = 1/20 + 1/124 + 1/155
2/33 = 1/22 + 1/66	2/35  = 1/30 + 1/42Samfuri:Figure space	2/37 = 1/24 + 1/111 + 1/296
2/39 = 1/26 + 1/78	2/41  = 1/24 + 1/246 + 1/328Samfuri:Figure space	2/43 = 1/42 + 1/86 + 1/129 + 1/301
2/45 = 1/30 + 1/90	2/47  = 1/30 + 1/141 + 1/470Samfuri:Figure space	2/49 = 1/28 + 1/196
2/51 = 1/34 + 1/102	2/53  = 1/30 + 1/318 + 1/795Samfuri:Figure space	2/55 = 1/30 + 1/330
2/57 = 1/38 + 1/114	2/59  = 1/36 + 1/236 + 1/531Samfuri:Figure space	2/61 = 1/40 + 1/244 + 1/488 + 1/610
2/63 = 1/42 + 1/126	2/65  = 1/39 + 1/195Samfuri:Figure space	2/67 = 1/40 + 1/335 + 1/536
2/69 = 1/46 + 1/138	2/71  = 1/40 + 1/568 + 1/710Samfuri:Figure space	2/73 = 1/60 + 1/219 + 1/292 + 1/365
2/75 = 1/50 + 1/150	2/77  = 1/44 + 1/308Samfuri:Figure space	2/79 = 1/60 + 1/237 + 1/316 + 1/790
2/81 = 1/54 + 1/162	2/83  = 1/60 + 1/332 + 1/415 + 1/498Samfuri:Figure space	2/85 = 1/51 + 1/255
2/87 = 1/58 + 1/174	2/89  = 1/60 + 1/356 + 1/534 + 1/890Samfuri:Figure space	2/91 = 1/70 + 1/130
2/93 = 1/62 + 1/186	2/95  = 1/60 + 1/380 + 1/570Samfuri:Figure space	2/97 = 1/56 + 1/679 + 1/776
2/99 = 1/66 + 1/198	2/101 = 1/101 + 1/202 + 1/303 + 1/606

Wannan bangare na Rhind Mathematical Papyrus an yada shi a kan takardu tara na papyrus.

Bayani

Duk wani lamari mai ma'ana yana da fadadawa daban-daban a matsayin jimlar raguwa, kuma tun lokacin da aka gano Rhind Mathematical Papyrus masu lissafi sun yi ƙoƙari su fahimci yadda tsoffin Masarawa zasu iya lissafin takamaiman fadadawa da aka nuna a cikin wannan tebur.

Shawarwarin da Gillings ya bayar sun hada da dabaru daban-daban guda biyar. Matsala ta 61 a cikin Rhind Mathematical Papyrus ya ba da tsari ɗaya:

${\frac {2}{3n}}={\frac {1}{2n}}+{\frac {1}{6n}}$, wanda za a iya bayyana daidai da ${\frac {2}{n}}={\frac {1}{2}}{\frac {1}{n}}+{\frac {3}{2}}{\frac {1}{n}}$ ( n raba ta 3 a cikin ma'auni na ƙarshe).

Sauran hanyoyin da za a iya amfani da su sune:

$\,{\frac {2}{n}}\;=\;{\frac {1}{3}}{\frac {1}{n}}+{\frac {5}{3}}{\frac {1}{n}}$ ( n raba ta 5)

$\!{\frac {2}{mn}}\!={\frac {1}{m}}\!{\frac {1}{k}}+{\frac {1}{n}}{\frac {1}{k}}$ (inda k shine matsakaicin m da n )

$\,{\frac {2}{n}}\;=\,{\frac {1}{n}}+{\frac {1}{2n}}+{\frac {1}{3n}}+{\frac {1}{6n}}$ . Wannan dabara yana haifar da bazuwar ga n = 101 a cikin tebur.

An ba da shawarar Ahmes ya canza 2 / p (inda p shine lambar farko) ta hanyoyi biyu, da hanyoyi uku don canza 2 / q. Sauran sun ba da shawarar cewa Ahmes ya yi amfani da hanya ɗaya kawai wanda ya yi amfani le dalilai masu yawa kamar mafi ƙarancin yawa. Abdulrahman Abdulaziz ya ba da cikakken bayani mai sauƙi game da yadda za a iya lalata teburin 2 / p.

Kwatanta da sauran rubutun tebur

Wani tsohuwar papyrus na Masar ya ƙunshi irin wannan tebur na ɓangarorin Masar; Lahun Mathematical Papyri, wanda aka rubuta a kusa da 1850 KZ, game da shekarun wani tushen da ba a sani ba ne na Rhind papyrus.  Kashi n Kahun 2 / n sun kasance daidai da raguwar da aka bayar a cikin teburin Rhind Papyrus 2 / n .

Rubutun fatar lissafi na Masar (EMLR), a kusa da 1900 KZ, ya lissafa rushewar ɓangarorin nau'in 1/n zuwa wasu ɓangarorin raka'a.  Teburin ya kunshi jerin sassan raka'a 26 na nau'in 1/n da aka rubuta a matsayin jimlar wasu lambobi masu ma'ana.

Akhmim katako ya rubuta ɓan masu wahala na nau'in 1/n (musamman, 1/3, 1/7, 1/10, 1/11 da 1/13) dangane da ɓangaro Eye of Horus waɗanda suka kasance ɓangarori na nau'ikan ⁠1/2k⁠ da sauran da aka bayyana dangane da ɗayan da ake kira ro. Samfuri:SfracAn bincika amsoshin ta hanyar ninka mai rarraba na farko ta hanyar mafita da aka gabatar da kuma bincika cewa amsar da ta samo asali ita ce 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 5 ro, wanda ya yi daidai da 1.