Nginike ubufakazi bokuthi impande eyisikwele ka-2 ayinangqondo
Nansi ubufakazi bokuphikisana bokuthi impande eyisikwele ka-2 ayinangqondo. Okokuqala, khumbula ukuthi izinombolo ezingenakubalwa ziyizinombolo ezingenakubhalwa njenge - p / q ngezinombolo eziphelele u - p no - q. Ake sithi impande eyisikwele ka-2 inengqondo, okusho ukuthi sqrt(2) = p / q kwezinye izinombolo eziphelele p no q. Ngaphezu kwalokho, ngaphandle kokulahleka kokujwayelekile, cabanga ukuthi le ngxenyana isesimweni esincishisiwe, okusho ukuthi u-p no-q abanabo abahlanganisi abafanayo. Singakusho lokhu ngoba noma iyiphi inqwaba yezinombolo ephelele ingabhalwa ngendlela encishisiwe, ngakho ukuba khona kwanoma yikuphi ukumelwa okunengqondo kwe-sqrt ((2) kungasho ukuba khona kokunye ngendlela encishisiwe. Manje sikhulisa izinhlangothi zombili zesibalo esingenhla, sithola  2 = p^2 / q^2  Kwezinye izinombolo eziphelele p no-q ezingenabo bonke abahlanganisi abafanayo. Manje phindaphinda ngo q^2 ukuze uthole  2 * q^2 = p^2  Phawula ukuthi uhlangothi lwesobunxele luyi-pair ngenxa ye-factor ka-2, ngakho uhlangothi lwesokudla p^2 luyi-pair futhi. Lokhu kusho ukuthi u-p kumelwe futhi abe u-pair, kungenjalo u-p^2 ubengeke abe u-pair. Ngakho p ingabhalwa ngokuthi 2 * x ngenombolo ephelele x. Uma sifaka lokhu kule equation engenhla, sithola  2 * q ^ 2 = (2 * x) ^ 2 = 2 ^ 2 * x ^ 2  Qaphela ukuthi zombili izinhlangothi ze-equation zinomlingani ongu-2 esingawuqeda. Ngokwenza lokhu, sithola  q^2 = 2 * x^2  Ngombono ofana nalona ongenhla, singaphetha ngokuthi u-q uyisibalo esilinganayo. Nokho lokhu kuphikisana nokucabanga kwethu kokuthi u-p no-q babengenaso isici esifanayo. Ngakho-ke ukucabanga kwethu kokuqala kokuthi sqrt(2) kunengqondo kwakungelona iqiniso, okuyilokho esazama ukukufakazela.