Mathematics in medieval Islam

To solve the third-degree equation x3 + a2x = b Khayyám constructed the parabola 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.

In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed by the Islamic civilization between circa 622 and 1600.[1] Islamic science and mathematics flourished under the Islamic caliphate established across the Middle East, extending from the Iberian Peninsula in the west to the Indus in the east.

Katz, in A history of mathematics says that:[2]

"A complete history of mathematics of medieval Islam cannot yet be written, since so many of these Arabic manuscripts lie unstudied... Still, the general outline... is known. In particular, Islamic mathematicians fully developed the decimal place-value number system to include decimal fractions, systematised the study of algebra and began to consider the relationship between algebra and geometry, studied and made advances on the major Greek geometrical treatises of Euclid, Archimedes, and Apollonius, and made significant improvements in plane and spherical geometry."

A page from the The Compendious Book on Calculation by Completion and Balancing by Al-Khwarizmi.

An important role was played by the translation and study of Greek mathematics, which was the principal route of transmission of these texts to Western Europe. Smith notes that:[3]

"the world owes a great debt to Arab scholars for preserving and transmitting to posterity the classics of Greek mathematics... their work was chiefly that of transmission, although they developed considerable ingenuity in algebra and showed some genius in their work in trigonometry."

There is some controversy as to the relative weight of transmission versus original work in the value of the medieval Islamic contribution.

History
[edit] Algebra

The most important contribution of the Islamic mathematicians was the development of algebra; combining the Babylonian material with the Greek geometry to produce a new algebra.
[edit] Irrational numbers

The Greeks had discovered Irrational numbers, but were not happy with them and only able to cope by drawing a distinction between between magnitude and number. In the Greek view, magnitudes varied continuously and could be used for entities such as line segements, 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.
[edit] Induction
See also: Mathematical induction#History

The earliest implicit traces of mathematical induction can be found in Euclid's [4] proof that the number of primes is infinite. 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 around 1000 AD and continued by al-Samaw'al, who used it for special cases of the binomial theorem and properties of Pascal's triangle.
[edit] Major figures and developments
[edit] Omar Khayyám
To solve the third-degree equation x3 + a2x = b Khayyám constructed the parabola 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 Khayyám (c. 1038/48–1123/24)[5] wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of third-degree equations, going beyond the Algebra of al-Khwārizmī.[6] Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections. This method had been used by the Greeks,[7] but they did not generalize the method to cover all equations with positive roots.[8]
[edit] Sharaf al-Dīn al-Ṭūsī

Sharaf al-Dīn al-Ṭūsī (? in Tus, Iran – 1213/4 in Iran) 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, in order 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.[9]