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The history of mathematics during the Golden Age of Islam, especially during the 9th and the 10th centuries, building on Greek predecessors such as Euclid, Archimedes, and Apollonius as well as incorporating Indian sources such as Aryabhata, saw some important developments, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra (named for the work of scholar al-Kwarizmi) and certain advances in geometry and trigonometry.^{[1]} Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries.^{[2]}
The study of algebra, which itself is an Arabic word meaning "reunion of broken parts",^{[3]} flourished during the Islamic golden age. Al-Khwarizmi is, along with the Greek mathematician Diophantus, known as the father of algebra.^{[4]} In his book The Compendious Book on Calculation by Completion and Balancing Al-Khwarizmi deals with ways to solve for the positive roots of first and second degree (linear and quadratic) polynomial equations.^{[5]} He also introduces the method of reduction, and unlike Diophantus, gives general solutions for the equations he deals with.^{[4]}
Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, which was syncopated, where some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn al-Banna' al-Marrakushi and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī.^{[6]}
On the work done by Al-Khwarizmi, J. J. O'Connor and Edmund F. Robertson said:^{[7]}
"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". It 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." — MacTutor History of Mathematics archive
Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. Omar Khayyam, along with Sharaf al-Dīn al-Tūsī, found several solutions of the cubic equation. Omar Khayyam found the general geometric solution of a cubic equation.
The Greeks had discovered Irrational numbers, but were not happy with them and only able to cope by drawing a distinction between magnitude and number. In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically; and indeed Greek mathematics was mainly geometrical. Islamic mathematicians including Abū Kāmil Shujāʿ ibn Aslam slowly removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations. They worked freely with irrationals as objects, but they did not examine closely their nature.^{[8]}
In the twelfth century, Latin translations of Al-Khwarizmi's Arithmetic on the Indian numerals introduced the decimal positional number system to the Western world.^{[9]} His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations. In Renaissance Europe, he was considered the original inventor of algebra, although it is now known that his work is based on older Indian or Greek sources.^{[10]} He revised Ptolemy's Geography and wrote on astronomy and astrology.
The earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite (c. 300 BCE). The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665).
In between, implicit proof by induction for arithmetic sequences was introduced by al-Karaji (c. 1000) and continued by al-Samaw'al, who used it for special cases of the binomial theorem and properties of Pascal's triangle.
Omar Khayyám (c. 1038/48 in Iran – 1123/24)^{[11]} wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of third-degree equations, going beyond the Algebra of Khwārazmī.^{[12]} Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections. This method had been used by the Greeks,^{[13]} but they did not generalize the method to cover all equations with positive roots.^{[14]}
Sharaf al-Dīn al-Ṭūsī (? in Tus, Iran – 1213/4) developed a novel approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. For example, to solve the equation \ x^3 + a = b x, with a and b positive, he would note that the maximum point of the curve \ y = b x - x^3 occurs at x = \textstyle\sqrt{\frac{b}{3}}, and that the equation would have no solutions, one solution or two solutions, depending on whether the height of the curve at that point was less than, equal to, or greater than a. His surviving works give no indication of how he discovered his formulae for the maxima of these curves. Various conjectures have been proposed to account for his discovery of them.^{[15]}
Engraving of Abū Sahl al-Qūhī's perfect compass to draw conic sections.
The theorem of Ibn Haytham.
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