Al-Khwarizmi yaranditse ibijanye n’ingene abantu bashobora guharura bafatiye ku macumi, arasigura neza yongera aramenyekanisha cane uburyo bwo gukora ibiharuro bimwebimwe vy’ubupfindo. Ubwo buryo yabusiguye mu gitabu yanditse (The Book of Restoring and Balancing). Ijambo al-jabr riboneka mw’izina ry’icarabu ry’ico gitabu (Kitab al-jabr wa’l-muqabala) ni ryo ryakomotseko ijambo ry’igifaransa algèbre. Umwanditsi w’ivya siyansi yitwa Ehsan Masood avuga yuko ibiharuro vyitwa algèbre ari “bwo buryo bwo gukora ibiharuro buhambaye gusumba ubundi bwose bumaze kwiyumvirwa, bukaba bufasha mu mice yose y’ubuhinga bwa siyansi.” *

Umwanditsi umwe yavuze atera inkuru ati: “Abanyeshure benshi bo mu mashure yisumbuye bari gushima ko [al-Khwarizmi] atazana ivyo biharuro.” Ariko al-Khwarizmi yavuze ko intumbero yiwe yari iyo gusigurira abantu uburyo  bworoshe bwo guharura bokoresha mu rudandaza, mu vyo kugabura intoranwa, mu vyigwa bijanye n’amatohoza, no mu bindi.

Mu binjana vyakurikiye, abahinga mu biharuro b’Abanyaburaya, ushizemwo Galilée na Fibonacci, baremera cane al-Khwarizmi kubera ko yasiguye neza ibijanye n’ibiharuro vy’ubupfindo. Ivyo Al-Khwarizmi yasiguye vyaraciriye inzira ibindi vyigwa vyagizwe muri vya biharuro vyitwa algèbre, mu vy’uguharura ibitigiri (arithmétique) no mu biharuro bijanye n’ingero z’inyabutatu (trigonométrie). Ibiharuro bijanye n’ingero z’inyabutatu vyarafashije abahinga bo mu Karere k’ibihugu vy’Abarabu gupima imfuruka z’inyabutatu n’impande zazo, biranabafasha gutera imbere mu vyigwa bijanye n’ibisyo vyo mu kirere. *

Abisunze ivyo al-Khwarizmi yanditse baratanguje ubuhinga bushasha bwo gukoresha ibice vy’ibitigiri, baranateza imbere uburyo bushasha bwo gupima uburinganire bw’ikintu n’ubwuzure bwaco (uburiba). Abafundi n’abahinga mu vy’ubwubatsi bo mu bihugu vy’Abarabu barakoresheje ubwo buhinga buteye imbere, bagenzi babo b’Abanyaburaya bataranabumenya. Abanyaburaya bagiye kubumenya mu gihe ca za ntambara z’abakirisu n’abisilamu zabera muri Palestina. Basubiye iwabo bajanye ubwo buhinga, bafashijwe n’abisilamu bize bari bagizwe imbohe. Barafashijwe kandi n’abimukira bize.

Ivyo al-Khwarizmi yanditse vyahavuye bishirwa mu kilatini. Umuhinga mu biharuro w’Umutaliyano yitwa Fibonacci (uwitwa kandi Léonard de Pise, akaba yabayeho mu 1170-1250), ni we ahanini avugwa ko yakwiragije ibiharuro vy’abahindu n’Abarabu i Buraya. Yamenye ivyo biharuro igihe yagira ingendo mu karere k’Ikiyaga Mediterane, araheza yandika igitabu yise Igitabu c’ibiharuro (mu kilatini).

Vyasavye imyaka amajana kugira ivyo al-Khwarizmi yasiguye bimenyekane cane. Ariko ubuhinga yakoresha be n’ibiharuro bijanye na bwo, bisigaye ari vyo nkingi y’ubuhinga bwa siyansi n’ubundi buhinga bwa none, tutibagiye ivy’ubudandaji n’amahinguriro.