'' The first century of the
Islamic
Arab Empire saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and research in other parts of the world had faded. In the second half of the 8th century, Islam had a cultural awakening, and research in mathematics and the sciences increased. The Muslim
Abbasid caliph al-Mamun (809–833) is said to have had a dream where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translation be made of as many Greek works as possible, including Ptolemy's
Almagest and Euclid's
Elements. Greek works would be given to the Muslims by the
Byzantine Empire in exchange for treaties, as the two empires held an uneasy peace. Arabic mathematicians established algebra as an independent discipline, and gave it the name "algebra" (
al-jabr). They were the first to teach algebra in an
elementary form and for its own sake. Throughout their time in power, the Arabs used a fully rhetorical algebra, where often even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (e.g. twenty-two) with
Arabic numerals (e.g. 22), but the Arabs did not adopt or develop a syncopated or symbolic algebra
Persian mathematician
Muhammad ibn Mūsā al-Khwārizmī, described as the father or founder of
algebra, was a faculty member of the "
House of Wisdom" (
Bait al-Hikma) in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 AD, wrote more than half a dozen mathematical and
astronomical works. The book also introduced the fundamental concept of "
reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as
al-jabr. The name "algebra" comes from the "
al-jabr" in the title of his book. R. Rashed and Angela Armstrong write:
Al-Jabr is divided into six chapters, each of which deals with a different type of formula. The first chapter of
Al-Jabr deals with equations whose squares equal its roots \left(ax^2 = bx\right), the second chapter deals with squares equal to number \left(ax^2 = c\right), the third chapter deals with roots equal to a number \left(bx = c\right), the fourth chapter deals with squares and roots equal a number \left(ax^2 + bx = c\right), the fifth chapter deals with squares and number equal roots \left(ax^2 + c = bx\right), and the sixth and final chapter deals with roots and number equal to squares \left(bx + c = ax^2\right). In
Al-Jabr, al-Khwarizmi uses geometric proofs, He also recognizes that the
discriminant must be positive and described the method of
completing the square, though he does not justify the procedure. The Greek influence is shown by
Al-Jabr's geometric foundations and by one problem taken from Heron. He makes use of lettered diagrams but all of the coefficients in all of his equations are specific numbers since he had no way of expressing with parameters what he could express geometrically; although generality of method is intended. which became known to the Arabs sometime before the 10th century. :\Delta^{\Upsilon} \overline{\alpha} \varsigma \overline{\iota} \,\;\sigma\;\, \Mu \lambda \overline{\theta} And al-Khwarizmi would have written as The manuscript gives exactly the same geometric demonstration as is found in
Al-Jabr, and in one case the same example as found in
Al-Jabr, and even goes beyond
Al-Jabr by giving a geometric proof that if the discriminant is negative then the quadratic equation has no solution. The
Egyptian mathematician
Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers in the form of a
square root or
fourth root as solutions to quadratic equations or as
coefficients in an equation. He was also the first to solve three non-linear
simultaneous equations with three unknown
variables.
Al-Karaji (953–1029), also known as Al-Karkhi, was the successor of
Abū al-Wafā' al-Būzjānī (940–998) and he discovered the first numerical solution to equations of the form ax^{2n} + bx^n = c. Al-Karaji only considered positive roots.
Omar Khayyám, Sharaf al-Dīn al-Tusi, and al-Kashi x^2 = a y,, a
circle with diameter b/a^2, 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. 1050 – 1123) wrote a book on Algebra that went beyond
Al-Jabr to include equations of the third degree. Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general
cubic equations since he mistakenly believed that arithmetic solutions were impossible. He understood the importance of the
discriminant of the cubic equation and used an early version of
Cardano's formula to find algebraic solutions to certain types of cubic equations. Some scholars, such as Roshdi Rashed, argue that Sharaf al-Din discovered the
derivative of cubic polynomials and realized its significance, while other scholars connect his solution to the ideas of Euclid and Archimedes. Sharaf al-Din also developed the concept of a
function. In his analysis of the equation x^3 + d = bx^2 for example, he begins by changing the equation's form to x^2(b - x) = d. He then states that the question of whether the equation has a solution depends on whether or not the "function" on the left side reaches the value d. To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when \textstyle x = \frac{2b}{3}, which gives the functional value \textstyle \frac{4b^3}{27}. Sharaf al-Din then states that if this value is less than d, there are no positive solutions; if it is equal to d, then there is one solution at \textstyle x = \frac{2b}{3}; and if it is greater than d, then there are two solutions, one between 0 and \textstyle \frac{2b}{3} and one between \textstyle \frac{2b}{3} and b. In the early 15th century,
Jamshīd al-Kāshī developed an early form of
Newton's method to numerically solve the equation x^P - N = 0 to find roots of N. Al-Kāshī also developed
decimal fractions and claimed to have discovered it himself. However, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by the
Baghdadi mathematician
Abu'l-Hasan al-Uqlidisi as early as the 10th century.
Abū al-Hasan ibn Alī al-Qalasādī (1412–1486) was the last major medieval
Arab algebraist, who made the first attempt at creating an
algebraic notation since
Ibn al-Banna two centuries earlier, who was himself the first to make such an attempt since
Diophantus and
Brahmagupta in ancient times. The syncopated notations of his predecessors, however, lacked symbols for
mathematical operations. Al-Qalasadi "took the first steps toward the introduction of algebraic symbolism by using letters in place of numbers" and by "using short Arabic words, or just their initial letters, as mathematical symbols." ==Europe and the Mediterranean region==