Use of AL-KARAJI method in calculating sum

Аннотация

This article provides several ways to find the sum of some number series. Al-Karaji's contribution to the science of mathematics lies in the method that he used to calculate the sums, and finding of the sum of a series of cubes of natural numbers is proved in this way. To explain al-Karaji's method more widely, other sums were calculated in the same way, and a general result was obtained. Keywords: Number series, Newton binomial, binomial coefficient, arithmetic progression, short multiplication formula, rectangle, square. Introduction: The development of Arab mathematics of mathematicians and astronomers who worked in the began in the 7th century AD, just in the era of the House of Wisdom, a kind of academy founded in emergence of the religion of Islam. It grew out of the Baghdad during the reign of al-Mammun (813-833). many challenges posed by trade, architecture, Five works by al-Khorezmi have survived, partially astronomy, geography, optics, and deeply combined revised, of which two treatises on arithmetic and the desire to solve these practical problems and algebra had a decisive impact on the further intense theoretical work. Arab mathematicians development of mathematics. His treatise on achieved significant achievements and made a number arithmetic is known only in the Latin version of the 13th of undeniable discoveries in the development of century, which, no doubt, is not an accurate algebraic calculus, both abstract and practical, the translation. It could be titled "A Book on Addition and formation of the theory of equations, algorithmic Subtraction Based on Indian Calculus." This is, in any methods at the junction of algebra and arithmetic. In case, the first book that sets out the decimal number the development of Arab mathematics, two stages can system and the operations performed in this system, be distinguished: first of all, the assimilation in the 7th including multiplication and division. In particular, a and 8th centuries of the Greek and Eastern heritage. small circle was used there, which served as a zero. Al- Baghdad was the first major scientific center during the Khorezmi explained how to pronounce numbers using reign of al-Mansur (754-775) and Harun al-Rashid (786- the concepts of one, ten, hundreds, thousands, 809). There were a large number of libraries, and many thousands of thousands ... that he defined. But the copies of scientific works were made. The works of form of the numbers used by al-Khorezmi is unknown, ancient Greece (Euclid, Archimedes, Apollonius, Heron, perhaps they were the letters of the Arabic alphabet or Ptolemy, Diophantus) were translated, and works from the Arabic numerals of the East. In fact, a purely India, Persia and Mesopotamia were also studied. But alphabetic number system existed for a very long time, by the 9th century, a real Arab mathematical culture of as evidenced by the "Book of arithmetic for scribes and its own had formed, and new work went beyond the traders", written by Abu-l-Wafa between 961 and 976, limits defined by the Hellenic mathematical heritage. and the famous "A sufficient book on the science of The first famous scientist of the Baghdad school was arithmetic," written by al-Karadzhi at the end of the X - Muhammad al-Khwarizmi, whose activity took place in beginning of the XI century. [1-3] the first half of the 9th century. He was part of a group American Journal of Applied Science and Technology 42 https://theusajournals.com/index.php/ajast American Journal of Applied Science and Technology (ISSN: 2771-2745) Al-Karaji (late X - early XI centuries), a native of the city He applied arithmetic operations to monomials, and of Karaj, located between Tehran and Qazvin, is the then to expressions composed of monomials, that is, to author of many very important works, namely, "A polynomials, considering addition and subtraction on Sufficient Book on the Science of Arithmetic", "Al- equal terms. As for division, he limited himself to Fakhri", an extensive algebraic treatise, dedicated to division into monomials. We get acquainted with the the vizier of Baghdad Fakhr al-Mulk, as well as the book results of the al-Karaji school concerning division and "Al-Badi" devoted to the study of indefinite equations. square root extraction from the work of his follower al- The Sufficient Book is a textbook of practical arithmetic, Samawal, who continued his research. much like another book he wrote between 961 and But al-Karaji has already managed to describe what 976. Abu-l-Wafa and which bears the name "Book of could now be called the algebra of polynomials. These arithmetic for scribes and merchants". The numbers methods of "arithmetization of algebra", according to were written there verbally and nowhere was the Rashed, are based, on the one hand, on the initial decimal positional number system used, which was elements of the algebra of al-Khorezmi and Abu-Kamil, more in line with the habits of traders. Abu-l-Wafa and on the other hand, on the t