's "Cubic equations and intersections of conic sections" the first page of the two-chaptered manuscript kept in Tehran University
Algebra The study of
algebra, the name of which is derived from the
Arabic word meaning completion or "reunion of broken parts", flourished during the
Islamic golden age.
Muhammad ibn Musa al-Khwarizmi, a Arab 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, Al-Khwarizmi 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. 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, meaning that 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. Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi.
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 al-Jud,
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.
Cubic equations x2 =
ay, a
circle with diameter
b/
a2, and a vertical line through the intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the
x-axis.
Omar Khayyam (c. 1038/48 in
Iran – 1123/24) wrote the
Treatise on Demonstration of Problems of Algebra containing the systematic solution of
cubic or third-order equations, going beyond the
Algebra of al-Khwārizmī. Khayyám obtained the solutions of these equations by finding the intersection points of two
conic sections. This method had been used by the Greeks, but they did not generalize the method to cover all equations with positive
roots.
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.
Induction 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 (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.
Irrational numbers 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 al-Baghdadi 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 mathematical objects, but they did not examine closely their nature. 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. 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. He revised
Ptolemy's
Geography and wrote on astronomy and astrology. However,
C.A. Nallino suggests that al-Khwarizmi's original work was not based on Ptolemy but on a derivative world map, presumably in
Syriac or
Arabic.
Spherical trigonometry The spherical
law of sines was discovered in the 10th century: it has been attributed variously to
Abu-Mahmud Khojandi,
Nasir al-Din al-Tusi and
Abu Nasr Mansur, with
Abu al-Wafa' Buzjani as a contributor.
Ibn Muʿādh al-Jayyānī's
The book of unknown arcs of a sphere in the 11th century introduced the general law of sines. The plane law of sines was described in the 13th century by
Nasīr al-Dīn al-Tūsī. In his
On the Sector Figure, he stated the law of sines for plane and spherical triangles and provided proofs for this law.
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.
Al-Khwarizmi did not use negative numbers or negative coefficients.
Al-Karaji wrote in his book
al-Fakhrī that "negative quantities must be counted as terms". == Influences ==