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Abu Abdallah Muhammad ibn Musa al-Khwarizmi, earlier transliterated as Algoritmi or Algaurizin, (c. 780 – c. 850) was a Persian mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad.
The word al-Khwarizmi is pronounced in classical Arabic as Al-Khwarithmi hence the Latin transliteration is Algoritmi. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal positional number system to the Western world. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic. In Renaissance Europe, he was considered the original inventor of algebra, although we now know that his work is based on older Indian or Greek sources. He revised Ptolemy's Geography and wrote on astronomy and astrology.
Some words reflect the importance of al-Khwarizmi's contributions to mathematics. "Algebra" is derived from "al-jabr", one of the two operations he used to solve quadratic equations. "Algorism" and "algorithm" stem from "Algoritmi" – the Latin form of his name.
He was born in a Persian family, and his birthplace is given as Chorasmia by Ibn al-Nadim. Few details of al-Khwarizmi's life are known with certainty. His name may indicate that he came from Khwarezm (Khiva), then in Greater Khorasan, which occupied the eastern part of the Greater Iran, now Xorazm Province in Uzbekistan. Abu Rayhan Biruni calls the people of Khwarizm "a branch of the Persian tree".
Al-Khwarizmi's contributions to mathematics, geography, astronomy and cartography established the basis for innovation in algebra and trigonometry. His systematic approach to solving linear and quadratic equations led to algebra, a word derived from the title of his 830 book on the subject, "The Compendious Book on Calculation by Completion and Balancing".
On the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Indian system of numeration throughout the Middle East and Europe. It was translated into Latin as "Algoritmi de numero Indorum". Al-Khwarizmi, rendered as (Latin) Algoritmi, led to the term "algorithm". One of his work was based on Persian and Babylonian astronomy, Indian numbers, and Greek mathematics.
Al-Khwarizmi systematized and corrected Ptolemy's data for Africa and the Middle east. Another major book was Kitab surat al-ard ("The Image of the Earth"; translated as Geography), presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea, Asia, and Africa. He also wrote on mechanical devices like the astrolabe and sundial.
He assisted a project to determine the circumference of the Earth and in making a world map for al-Ma'mun, the caliph, overseeing 70 geographers. When, in the 12th century, his works spread to Europe through Latin translations, it had a profound impact on the advance of mathematics in Europe. He introduced Arabic numerals into the Latin West, based on a place-value decimal system developed from Indian sources.
Al-Khwarizmi’s method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where a, b and c are positive integers):
– squares equal roots (ax2 = bx),
– squares equal number (ax2 = c),
– roots equal number (bx = c),
– squares and roots equal number (ax2 + bx = c),
– squares and number equal roots (ax2 + c = bx),
– roots and number equal squares (bx + c = ax2),
by dividing out the coefficient of the square and using the two operations al-jabr ("restoring" or "completion") and al-muqabala ("balancing"). Al-jabr is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. Al-muqabala is the process of bringing quantities of the same type to the same side of the equation.
The above discussion uses modern mathematical notation for the types of problems which the book discusses. However, in al- Khwarizmi’s day, most of this notation had not yet been invented, so he had to use ordinary text to present problems and their solutions. For example, for one problem he writes, (from an 1831 translation): "If someone say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times". Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts".
J.O'Conner and E.Robertson wrote in the "MacTutor History of Mathematics archive": "Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". He gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before".
Al-Khwarizmi’s second major work was on the subject of arithmetic, which survived in a Latin translation but was lost in the original Arabic. The translation was most likely done in the 12th century by Adelard of Bath, who had also translated the astronomical tables in 1126.
Al-Khwarizmi’s work on arithmetic was responsible for introducing the Arabic numerals, based on the Hindu-Arabic numeral system developed in Indian mathematics, to the Western world. The term "algorithm" is derived from the algorism, the technique of performing arithmetic with Hindu-Arabic numerals developed by al-Khwarizmi. Both "algorithm" and "algorism" are derived from the Latinized forms of al-Khwarizmi's name, Algoritmi and Algorismi, respectively.
"Astronomical tables of Sind and Hind" is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine values. This is the first of many Arabic Zijes based on the Indian astronomical methods known as the sindhind. The work contains tables for the movements of the sun, the moon and the five planets known at the time. This work marked the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge.
"Astronomical tables of Sind and Hind" also contained tables for the trigonometric functions of sine and cosine. A related treatise on spherical trigonometry is also attributed to him.

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