A cikin wannan labarin za mu gwada ilimin lissafi na ChatGPT. Za mu yi ƙoƙari mu yi amfani da basirar ɗan adam don nemo misalan abin da ya dace da Muhimmin Theorem na Algebra, gano cewa ba shakka zai ƙaddamar da mu zuwa lambar yabo ta Filaye.
Idan muka yi tambaya game da tushen tushen digiri na 3 mai yawa, a wannan yanayin duk na gaske, ChatGPT yana jayayya cewa ƙudurin nazari na iya dogara da tsarin da aka tsara, don haka muna ba da shawarar yin amfani da hanyar ƙididdigewa kamar hanyar Newton-Raphson.
Kuskure a cikin lissafin abin da aka samo asali
Ya zuwa yanzu, ba za mu iya shakkar ikon ilimin lissafi na AI ba, don haka mun yi ƙoƙari mu magance matsalar gano tushen polynomial p (x) = x3 - 3 × 2 + 4 kuma ga mamakinmu ya yi lissafin kuskure. na abin da aka samo asali , don haka samun tushen ba daidai ba ne. Ya dawo x = 0 a matsayin tushen polynomial kuma mun tambaye shi ya duba shi. A zahiri, yana sane da kasancewar kuskure amma bai san inda ya faru ba. Mun ga cewa kuskuren yana cikin asalin polynomial kuma muna tambayar cewa an ƙididdige shi daga tushen ta hanyar Newton-Raphson. Abin mamaki, yana sake yin kuskuren lissafi, wannan lokacin a cikin aiki mai sauƙi, kamar yadda muke iya gani a cikin hoto mai zuwa:
Kuskure
Ganin kuskuren da ke cikin lissafin, sai mu sake tambayarsa, ya sake yin wani kuskure, don haka muka ba shi tsarin farko na hanyar Newton-Raphson, wato, x₁ = 5/3 kuma muna neman ci gaba da maimaitawa, wanda ya haifar da x₁ = 5. /3 shine tushen polynomial. Muna tabbatarwa ta hanyar sake tambaya ko ƙimar 5/3 shine tushen polynomial, kuma mun sami amsa mai gamsarwa. Muna neman yin lissafin ƙimar polynomial a wannan ƙimar, kuma, tun da sakamakon ya bambanta da sifili, muna nuna cewa ba zai iya zama tushen ba. Ya fahimce shi kuma ya ba da hakuri kamar yadda muke gani a kasa:
Mun kammala cewa ka'idar Newton-Raphson hanya ce daidai, amma aikace-aikacensa ba daidai ba ne, don haka muna ƙoƙarin nemo tushen ta hanyar amfani da wata hanya, kamar haɓakar polynomial.
A wannan yanayin, mun gano cewa tushen polynomial p (x) sune x = r da x = 1 ± 2i.
Lokacin da aka tambaye shi don tabbatar da cewa ƙimar p(1+2i) ba sifili ba ce kuma don haka ba zai iya zama tushen tushen yawan adadin mu ba, sake yarda da kuskuren. Lokacin da muka isa wannan yanayin, zamu tafi tare da alama, kuma muna gaya masa cewa x = - 1 shine ainihin tushen polynomial kuma sauran tushen suna lissafin. Amsarsa ta farko ba za ta iya zama abin mamaki ba, yana gaya mana cewa ban da x = - 1, sauran tushen tushen polynomial p(x) = 4 - 3 × 2 + x3 sune x = 1 + 2i da x = 1 - 2i . Har sau hudu yana sake ba da sakamakon da ba daidai ba, don haka ba mu da wani zaɓi face mu samar masa da sabon tushe. A wannan yanayin, maimakon mu ba shi, muna tambaya idan x = 2 shine tushen yawan adadin mu. Ku hukunta kanku amsar, ko kuma, lissafin da ChatGPT ke yi don duba cewa x = 2 ba tushe bane:
Bayan sake duba lissafin ku, gama ta hanyar bayyana cewa tushen asalin mu shine x = - 1, x = 1, da x = 2.
Za mu nuna muku cewa duk tushen uku ne, mayar da daidai, wato, x = 1 ba tushe ba ne yayin da sauran dabi'u biyu suke. Ba mu yi kasa a gwiwa ba, mu yi kokarin nemo tushen tushe na uku na polynomial, kuma a nan ne muka sami babbar fassara. Bayanin mu: “Asalin Theorem na Algebra ya tabbatar da cewa kowane nau'in digiri yana da ainihin tushen n, na gaske da kuma hadaddun. Don haka, idan polynomial na digiri n yana da tushen tushe na nk, to dole ne ya kasance yana da tushen hadaddun nk. Ya zuwa yanzu mun yarda. Mu ci gaba da: “A cikin yanayin p(x) = 4 – 3×2 + x3, mun gano cewa yana da tushen asali guda biyu, x = – 1 da x = 2. Tunda p(x) ne a polynomial na digiri 3, to dole ne ya kasance yana da ƙarin hadadden tushen. Wannan hadadden tushen shine x = 1 ± 2i." Ba za mu iya fita daga cikin mamakinmu ba kuma muna tsammanin ya so ya nuna mana daya daga cikin tushen biyu ne kawai, don haka muka sake ba shi wata dama, wanda ya haifar da:
Don haka idan mun yi gaskiya, mun sami wani misali ga Muhimmin Theorem na Algebra, babban digiri na 3 mai tushe 4. Shin muna neman lambar yabo ta Filaye?
AI ta sake tabbatar da cewa amsarta daidai ne har sau biyu, yana nuna cewa digiri na 3 polynomial na iya samun tushen 4. Har ma mun tashi nemo su ta hanyar amfani da Hanyar Bisection. Yanzu eh, mun daina neman tushen digiri mai sauƙi 3 polynomial. Muna bankwana da kwaya ɗaya ta ƙarshe:
A matsayin taƙaitaccen taƙaitaccen bayani, ba muna cewa ChatGPT ba shi da hankali na Artificial Intelligence, nesa da shi, idan ba kawai akasin haka ba, yana da kyau AI sosai, amma a cikin kansa, a cikin Harshen Halitta, kodayake a cikin Lissafi har yanzu yana da dogon hanya don tafiya. koyi. Dole ne mu yi suka game da sakamakon da injuna suka dawo mana: ba gaskiya ba ne ko ta yaya aka yi bayanin su, kuma da alama mutum ya ɓace wanda zai iya tabbatar da gaskiyar su.
GAME DA MARUBUCI
Sarría Martínez De Mendivil
An fara buga wannan labarin akan Tattaunawar.
