Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as full development of the decimal placevalue system to include decimal fractions, the first systematised study of algebra, and advances in geometry and trigonometry.^{[1]}
Arabic works played an important role in the transmission of mathematics to Europe during the 10th—12th centuries.^{[2]}
Further information: History of algebra 
The study of algebra, the name of which is derived from the Arabic word meaning completion or "reunion of broken parts",^{[3]} flourished during the Islamic golden age. Muhammad ibn Musa alKhwarizmi, a Persian scholar in the House of Wisdom in Baghdad was the founder of algebra, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing, AlKhwarizmi deals with ways to solve for the positive roots of first and second degree (linear and quadratic) polynomial equations. He introduces the method of reduction, and unlike Diophantus, also gives general solutions for the equations he deals with.^{[4]}^{[5]}^{[6]}
AlKhwarizmi'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, meaning that some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn alBanna' alMarrakushi and Abū alḤasan ibn ʿAlī alQalaṣādī.^{[7]}^{[6]}
On the work done by AlKhwarizmi, J. J. O'Connor and Edmund F. Robertson said:^{[8]}
"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of alKhwarizmi, 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 the 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."
Several other mathematicians during this time period expanded on the algebra of AlKhwarizmi. Abu Kamil Shuja' wrote a book of algebra accompanied with geometrical illustrations and proofs. He also enumerated all the possible solutions to some of his problems. Abu alJud, Omar Khayyam, along with Sharaf alDīn alTūsī, found several solutions of the cubic equation. Omar Khayyam found the general geometric solution of a cubic equation.^{[citation needed]}
Further information: Cubic equation 
Omar Khayyam (c. 1038/48 in Iran – 1123/24)^{[9]} wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of cubic or thirdorder equations, going beyond the Algebra of alKhwārizmī.^{[10]} Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections. This method had been used by the Greeks,^{[11]} but they did not generalize the method to cover all equations with positive roots.^{[10]}
Sharaf alDīn alṬūsī (? 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 , with a and b positive, he would note that the maximum point of the curve occurs at , 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.^{[12]}
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See also: Mathematical induction § History 
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 alKaraji (c. 1000) and continued by alSamaw'al, who used it for special cases of the binomial theorem and properties of Pascal's triangle.
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 and Ibn Tahir alBaghdadi slowly removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations.^{[13]}^{[14]} They worked freely with irrationals as mathematical objects, but they did not examine closely their nature.^{[15]}
In the twelfth century, Latin translations of AlKhwarizmi's Arithmetic on the Indian numerals introduced the decimal positional number system to the Western world.^{[16]} 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.^{[17]}^{[18]} He revised Ptolemy's Geography and wrote on astronomy and astrology. However, C.A. Nallino suggests that alKhwarizmi's original work was not based on Ptolemy but on a derivative world map,^{[19]} presumably in Syriac or Arabic.
Further information: Law of sines and History of trigonometry 
The spherical law of sines was discovered in the 10th century: it has been attributed variously to AbuMahmud Khojandi, Nasir alDin alTusi and Abu Nasr Mansur, with Abu alWafa' Buzjani as a contributor.^{[13]} Ibn Muʿādh alJayyānī's The book of unknown arcs of a sphere in the 11th century introduced the general law of sines.^{[20]} The plane law of sines was described in the 13th century by Nasīr alDīn alTūsī. In his On the Sector Figure, he stated the law of sines for plane and spherical triangles and provided proofs for this law.^{[21]}
Further information: Negative numbers 
In the 9th century, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid.^{[22]} AlKhwarizmi did not use negative numbers or negative coefficients.^{[22]} But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication .^{[23]} AlKaraji wrote in his book alFakhrī that "negative quantities must be counted as terms".^{[22]} In the 10th century, Abū alWafā' alBūzjānī considered debts as negative numbers in A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen.^{[23]}
By the 12th century, alKaraji's successors were to state the general rules of signs and use them to solve polynomial divisions.^{[22]} As alSamaw'al writes:
the product of a negative number — alnāqiṣ — by a positive number — alzāʾid — is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (martaba khāliyya), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.^{[22]}
Further information: False position method 
Between the 9th and 10th centuries, the Egyptian mathematician Abu Kamil wrote a nowlost treatise on the use of double false position, known as the Book of the Two Errors (Kitāb alkhaṭāʾayn). The oldest surviving writing on double false position from the Middle East is that of Qusta ibn Luqa (10th century), an Arab mathematician from Baalbek, Lebanon. He justified the technique by a formal, Euclideanstyle geometric proof. Within the tradition of Golden Age Muslim mathematics, double false position was known as hisāb alkhaṭāʾayn ("reckoning by two errors"). It was used for centuries to solve practical problems such as commercial and juridical questions (estate partitions according to rules of Quranic inheritance), as well as purely recreational problems. The algorithm was often memorized with the aid of mnemonics, such as a verse attributed to Ibn alYasamin and balancescale diagrams explained by alHassar and Ibn alBanna, who were each mathematicians of Moroccan origin.^{[24]}
Sally P. Ragep, a historian of science in Islam, estimated in 2019 that "tens of thousands" of Arabic manuscripts in mathematical sciences and philosophy remain unread, which give studies which "reflect individual biases and a limited focus on a relatively few texts and scholars".^{[25]}^{[full citation needed]}