الرياضيات في العصر الذهبي للحضارة الإسلامية

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

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 place-value system to include decimal fractions, the first systematised study of algebra, and advances in geometry and trigonometry.

Arabic works played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 المفاهيم

 الحساب

 تبني ونشر النظام العشري الهندي

تطور الأعداد الهندية-العربية

The essential element of the decimal place value representation of numbers is a symbol for the zero , which indicates that the corresponding step number does not occur at this point: The number 207 contains two 100s, no 10 and seven times 1; in contrast to 27, which contains two 10s and seven 1s. This important idea of ​​zero dates back to Indian mathematics, where it was used at least as early as the 7th century AD and was described by the Indian astronomer and mathematician Brahmagupta . [12] Indic numerals also spread to Syria and Mesopotamia by the 8th centuryand were adopted by Islamic mathematics in the 9th century. Previously, the Abjad numeral script was used by the Arabs , [13] in which, similar to Greek numeral script, the letters of the alphabet stand for specific numerical values. [14] With the Arabic translation of the Siddhānta by the Indian mathematician Aryabhata in the 8th century, the number zero found its way into Arabic-language literature. [15] The zero was called sifr ('empty', 'nothing') in Arabic; From this designation developed, among other things, the German word "Ziffer" and the English "zero" for zero.[13]

The first known description of this new number system in Arabic comes from the polymath al-Khwarizmi , one of Islam's most important mathematicians. He was probably of Khorezm descent, was born around 780, worked in the House of Wisdom in Baghdad and died between 835 and 850. [16] His work kitāb al-ḥisāb al-hindī (Book of Calculations with Indian Numerals) or kitab al-jam ' wa'l-tafriq al-ḥisāb al-hindī ('Addition and Subtraction in Indic Arithmetic'), translated into Latin in the 12th century, introduced Indo-Arabic numerals and the decimal system to Europe. [17]The work has survived in only one Latin manuscript, the Arabic original has been lost. [18] The Latin translation begins with the words: " Dixit Algorizmi " ("Al-Chwarizmi said"). [19] From this developed the word " algorithm ", which is used today for systematic calculation methods. [20] Contrary to its title, al-Chwarizmi's introduction to the Indian number system included not only procedures for written addition and subtraction , but also for multiplying , dividing and taking square roots. One of the earliest surviving works on arithmetic in the original Arabic, the book Fundamentals of Indian Arithmetic by Kushyar ibn Labban (fl. 971–1029), was very influential in Islamic countries and played an important role in the final spread of the decimal system.

جمع 5625 و 839 على سبورة كوشيار بن لبان

The written calculation techniques introduced by al-Khwarizmi and Kuschyar ibn Labban differed significantly from those used today. The reason for this was that they were optimized for the calculations that were common at the time on a so-called dust table , a flat tray sprinkled with fine sand. In contrast to calculating with pen and paper, only a relatively small number of digits could be written on a dust tablet at the same time, but it offered the advantage that digits could be erased very quickly and others could write over them. [23] However, dust tables as arithmetic aids soon fell out of use in favor of ink and paper. This is what Abu l-Hasan al-Uqlidisi wrote in his around 953Book of Chapters on Indian Arithmetic , that the use of the dust table "is not proper" because otherwise one only sees it with "good-for-nothings" who "deny their living in the streets with astrology". Accordingly, in his book, al-Uqlidisi gave written computational techniques optimized for covering letters on paper.

 اختراع الكسور العشرية

In al-Uqlidisi's book on Indian arithmetic, in addition to arithmetic with natural numbers in decimal representation, there is also the oldest known treatment of decimal fractions . Previously, it was customary to specify non-integer parts in the sexagesimal system. [25] Al-Uqlidisi introduced decimal fractions in connection with divisions by 2 and by 10 and showed the usefulness of this new form of representation with examples: He halved the number 19 five times and got 0.59375 or increased the number 135 five times by a tenth, which results in a decimal fraction of 217.41885. However, Al-Uqlidisi did not yet use today's notation with a decimal separator, but marked the ones place by putting a small vertical line over it. [26]

Al-Uqlidisi's use of decimal fractions still appeared largely as a technical device and arithmetic aids; [26] it is unclear whether he fully recognized their mathematical significance. [27] The full mathematical understanding of decimal fractions for the approximate representation of real numbers, however, can only be found more than 200 years later in a treatise on arithmetic by as-Samaw'al (around 1130 to around 1180) from the year 1172. As-Samaw'al carefully introduced it as a method to approximate numbers with (in principle) arbitrary precision, and demonstrated this with examples by, among other things, developing decimal fractions of{\displaystyle {\tfrac {210}{13}}}{\displaystyle {\tfrac {210}{13}}}and from{\displaystyle {\sqrt {10}}}\sqrt{10}certain. In order to calculate higher roots, as-Samaw'al also used numerical iteration methods, in which the idea of ​​"convergence" of the calculated approximations against the value sought becomes clear. [25] [28] The last great mathematician in the countries of Islam during the European Middle Ages, Jamshid Masʿud al-Kashi (around 1389 to 1429), wrote the work Key to Arithmetic in 1427 in which, based on the binomial theorem , described a general method for calculating nth roots.

 الجبر

للمزيد من المعلومات: تاريخ الجبر

The study of algebra, the name of which is derived from the Arabic word meaning completion or "reunion of broken parts", flourished during the العصر الذهبي للإسلام. محمد بن موسى الخوارزمي، 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. في كتابه The Compendious Book on Calculation by Completion and Balancing، يتعامل الخوارزمي مع طرق حل 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. one of six standard shapes can be formed. In modern notation with the unknown $x$ and with coefficients $p$ و {$q$ , denoting given positive numbers, are:

1) $x^{2}=px$,	2) $x^{2}=p$,	3) $x=p$,
4) $x^{2}+px=q$,	5) $x^{2}+q=px$,	6) $px+q=x^{2}$.

Zwei Fälle quadratischer Gleichungen bei الخوارزمي (نسخة عربية من القرن 14)

In the first three cases, the solution can be determined directly, for cases 4, 5 and 6 al-Chwarizmi gave rules for the solution and proved them geometrically by completing the square . Although he always used concrete numerical examples, he emphasized the general validity of the considerations. [45] [46]

The procedure is to be explained in the example of case 5, in which al-Chwarizmi established that it is the only one of the six cases in which no, exactly one or exactly two (positive) solutions can exist. All other cases, on the other hand, always have a uniquely determined solution. [47] The equation is given $2x^{2}+100-20x=58$. This is first transformed by al-dschabr , which means that terms that are subtracted (in this case $20x$), are added on both sides of the equation, so that finally only additions appear in the equation; in the example results $2x^{2}+100=58+20x$. The second transformation step al-muqabala is to combine like terms on the left and right sides of the equation on one page; in the example you get $2x^{2}+42=20x$. Finally, dividing the equation by 2 gives the normal form $x^{2}+21=10x$. [48] ​​With the rule given by al-Chwarizmi for case 5, the two solutions can now be determined:

${\frac {10}{2}}-{\sqrt {\left({\frac {10}{2}}\right)^{2}-21}}=3\quad$ und $\quad {\frac {10}{2}}+{\sqrt {\left({\frac {10}{2}}\right)^{2}-21}}=7$.

 التطور اللاحق للجبر في الإسلام

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.

On the work done by Al-Khwarizmi, J. J. O'Connor and Edmund F. Robertson said:

"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 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."

— MacTutor History of Mathematics archive

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.

The scholar Abu Kamil (c. 850 to c. 930), believed to be of Egyptian origin, published a highly influential book entitled Algebra . The collection of problems contained therein was, for example , taken up intensively towards the end of the 12th century by the Italian mathematician Leonardo of Pisa . [53] Abu Kamil's Algebra , intended as a commentary on al-Khwarizmi's work, contains numerous advances in algebraic transformations. Among other things, he showed arithmetic rules for multiplying out expressions that contain the unknown, or arithmetic rules for roots, such as ${\sqrt {a}}\cdot {\sqrt {b}}={\sqrt {a\cdot b}}$. He led careful proofs for elementary transformations such as{\displaystyle ax\cdot bx=ab\cdot x^{2}}{\displaystyle ax\cdot bx=ab\cdot x^{2}}. [54] The second part of Abu Kamil's algebra contains numerous problems that illustrate the theoretical first part. According to John Lennart Berggren , one of the most interesting problems shows his “virtuoso” handling of the rules of algebra: Abu Kamil considered the nonlinear system of equations in it $10=x+y+z$, $z^{2}=x^{2}+y^{2}$ ، $xz=y^{2}$ with three unknowns and detailed the calculation steps that eventually led to the solution $x=5-{\sqrt {{\sqrt {3125}}-50}}$ to lead.

In the period that followed, algebra was further arithmeticized, which means that its geometric origins receded into the background and the purely algebraic calculation laws were further developed. [56] The Persian mathematician al- Karaj (953–1029) considered arbitrary powers of the unknown{\displaystyle x}xas well as sums and differences formed from them. In doing so, he took an important step in the direction of arithmetic for polynomials , but still failed to come up with a generally valid formulation of polynomial division , since he – like all Islamic mathematicians before him – lacked the concept of negative numbers. [57] Not until as-Samaw'al , about 70 years later, does the power law appear, among other things $x^{m}\cdot x^{n}=x^{m+n}$ for any positive and negative exponent{\displaystyle m}mand{\displaystyle n}n. [58] As-Samaw'al was thus able to provide an efficient tabular procedure with which any polynomial division can be carried out; for example he used it to calculate

${\frac {20x^{6}+2x^{5}+58x^{4}+75x^{3}+125x^{2}+96x+94+140x^{-1}+50x^{-2}+90x^{-3}+20x^{-4}}{2x^{3}+5x+5+10x^{-1}}}$.

Omar Khayyám's "Cubic equations and intersections of conic sections" the first page of the two-chaptered manuscript kept in Tehran University

In the field of solving algebraic equations, the Persian scientist and poet Omar Chayyam (1048–1131) took al-Khwarizmi's classification of quadratic equations and extended it to include cubic equations , that is, equations involving the cube of the unknown. [60] He showed that these can be reduced to one of 25 standard forms, 11 of which can be reduced to quadratic equations. For the remaining 14 types, Omar Chayyam gave methods with which the solutions can be constructed geometrically as the intersections of conics . [61]In his treatise, he also expressed the "wish" to be able to calculate the solution algebraically by root expressions, as with the quadratic equations, also with the cubic ones. However, according to Omar Chayyam, neither he nor any other algebraist was successful. [62] Chayyam's wish was only to be fulfilled in 1545 with the publication of solution formulas for third-degree equations by the Italian scholar Gerolamo Cardano .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 المعادلات التكعيبية

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.

للمزيد من المعلومات: Cubic equation

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 $\ x^{3}+a=bx$, with a and b positive, he would note that the maximum point of the curve $\ y=bx-x^{3}$ 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

انظر أيضاً: 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 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.

 حساب المثلثات الكري

للمزيد من المعلومات: قانون الجيب and تاريخ حساب المثلثات

الثلاث نقاط A و B و C على سطح كرة يشكلون مثلثاً كرياً بأضلع a, b, c وزوايا α, β و γ.

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.

An important advance in Islamic mathematics that greatly simplified calculations over Menelaus' theorem was the law of sines for spherical triangles . It was formulated and proved by Abu'l-Wafa and, presumably independently, by al-Biruni and one of his teachers. [77] Thus, for the first time, a possibility was available to directly calculate angles (and not only sides) of spherical triangles. [78] The theorem says: In a spherical triangle with angles $\alpha$ , $\beta$, $\gamma$ والأضلاع $a$, $b$, $c$ on the opposite sides applies:

${\frac {\sin a}{\sin \alpha }}={\frac {\sin b}{\sin \beta }}={\frac {\sin c}{\sin \gamma }}$

In particular, a spherical triangle can be calculated from three given quantities if one side and an opposite angle are given.

Sphärisches Dreieck zur Bestimmung der Gebetsrichtung

Spherical trigonometry is of great importance not only in astronomy, but also in geography , when measurements and calculations take into account the spherical shape of the earth. An important application for the Islamic religion is found in al-Biruni: the determination of the qibla , the direction of prayer towards Mecca . Al-Biruni addressed this problem in a work on mathematical geography entitled Determination of the Coordinates of Cities . He assumed that the longitude and latitude of a city $\mathrm {B}$ as well as the longitude and latitude of Mecca $\mathrm {A}$ given are. In the spherical triangle $\mathrm {BAP}$ with the North Pole $\mathrm {P}$ are then the two sides $\mathrm {PA}$ and $\mathrm {PB}$ as well as their intermediate angles $\mathrm {P}$ famous. Since the side opposite the given angle is unknown, the law of sines cannot be applied directly. This problem would be solved today, for example, with the law of cosines , which, however, was not yet available to al-Biruni. Instead, he used auxiliary triangles and repeated application of the law of sines to find the angle at the point $\mathrm {B}$, i.e. the Qibla , to be calculated.

 الهندسة الإقليدية

The Elements , which the Greek mathematician Euclid used around 300 B.C. that systematically summarized the geometry of his day were available in Arabic translation by the late 8th century and had a very large influence on Islamic mathematicians. [81] Archimedes ' treatise On the Sphere and Cylinder and Apollonius' work Konika on conic sections were also pillars on which geometry in Islamic countries was based. [82] A popular subject of study was the construction of regular polygons using ruler and compass. For regular triangles, quadrilaterals, pentagons and 15-gons and the regular polygons resulting from them by doubling sides, construction was only known with compasses and straightedges; on the other hand, regular heptagons and nine- corners can only be constructed by using additional tools. Abu al-Wafa, in his work On those parts of geometry that craftsmen need, gave, among other things, various constructions of these two cases with the help of conic sections or by so-called insertion (neusis) . [83]

Another important mathematician who systematically dealt with geometric constructions was Abu Sahl al-Quhi (around 940 to around 1000). In particular, he wrote a treatise on the "perfect compass," an instrument used to draw conic sections. [84] In addition to theoretical considerations on the construction of geometric figures, conic sections were also of great importance for practical applications such as sundials or burning mirrors. Ibrahim ibn Sinan (908-946), a grandson of Thabit ibn Qurra, in his work On the Drawing of the Three Conics, gave various methods for constructing the three types of conics, ellipse, parabola and hyperbola. [85]Also of theoretical and practical interest in Islamic mathematics were geometric constructions that result from the limitation of the classical Euclidean tools. For example, Abu'l-Wafa wrote a paper dealing with constructions using a ruler and a compass with a fixed opening, also called "rusty compass". For example, he showed how these tools can be used to divide a line into any number of sections of equal size, or to construct squares and regular pentagons.

Euclid's Axiom of Parallels: If the sum α+β of the interior angles is less than 180°, then the lines h and k intersect at a point S that lies on the same side of g as the two angles lie on.

A purely theoretical problem with which several Islamic mathematicians dealt intensively was the question of what role the parallels postulate plays in the axiomatic structure of Euclidean geometry. In his Elements , Euclid used the "modern" structure of a mathematical theory by proving theorems based on definitions and axioms , i.e. statements that are assumed to be true without proof. A special role was played by the parallel axiom, which was considered non-obvious from the start because of its relative complexity. Accordingly, there were already numerous attempts in antiquity to prove this statement using the other axioms. [87] [88]For example, Alhazen (around 965 to after 1040) tried to approach this problem by reformulating the concept of parallel straight lines. Omar Chayyam later objected to this, considering Alhazen's use of a "moving line" to be non-obvious, and formulated a new postulate himself, replacing the Euclidean one. In the 13th century, Nasir ad-Din at-Tusi took up the proof attempts of his predecessors and added more to them. [89] It has been known since the 19th century that the parallel axiom is independent of the other axioms, i.e. it cannot be proven. All attempts that had been made since antiquity were therefore incorrect or circular reasoning.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 التوافيق ونظرية الأعداد

The ancient Indian results in combinatorics were adopted by Islamic mathematicians. There were also individual further developments in this area. [91] Statements about numbers or natural numbers in general can often be proved by the principle of complete induction . In the works of Islamic mathematicians there are some considerations that contain all the important components of this method of proof. So al-Karaj showed the formula in connection with sums of powers

$\sum _{i=1}^{n}i^{3}=\left(\sum _{i=1}^{n}i\right)^{2}$.

He led the inductive step with the concrete example $n=10$ out, but his approach was independent of his choice for $n$. In al-Karaj and even more clearly in as-Samaw'al there are reflections on the essential steps towards a proof for the binomial theorem

$(x+y)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{n-k}y^{k}$

Pascalsches Dreieck zur Bestimmung der Binomialkoeffizienten

contained by complete induction - even if the mathematical expression possibilities of the time were not sufficient to even formulate such a general statement. For calculating the binomial coefficients ${\tbinom {n}{k}}$ al-Karaj and al-Samaw'al used Pascal's triangle long before Blaise Pascal . [94] [95]

The mathematician Ibn Munim (died 1228) from al-Andalus made significant contributions to combinatorics. In his book Fiqh al-Hisab (“Arithmetic Laws”) he started with the task of determining the number of all possible words with a maximum of 10 letters in the Arabic language. He approached this quite demanding problem - among other things, the rules of how consonants and vowels must follow each other in word formation must be observed - via various individual problems. So he first determined the number of different colored tassels that arise when one out $n$ possible colors $k$ choose different colors. About the relationships of the occurring binomial coefficients (see also combination (combinatorics) ) he finally succeeded in recursively determining the number of possible words of fixed length from the numbers of shorter words. [96] [97]

In addition to magic squares [98] and figured numbers [99] , Islamic number theory also dealt with perfect numbers and their generalization, the friendly numbers . Two numbers are called friendly if each equals the sum of the proper divisors of the other. Only one example, the pair 220 and 284, has been known since antiquity, but no general mathematical statement about friendly numbers. In the 9th century, Thabit ibn Qurra was able to state and prove a law of formation (see theorem of Thabit ibn Qurra ). [100] With the help of which al-Farisi in the late 13th century added another pair, namely 17,296 and 18,416.

 الأعداد السالبة

للمزيد من المعلومات: الأعداد السالبة

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. But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication $(a\pm b)(c\pm d)$. Al-Karaji wrote in his book al-Fakhrī that "negative quantities must be counted as terms". In the 10th century, Abū al-Wafā' al-Būzjānī considered debts as negative numbers in A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen.

By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions. As al-Samaw'al writes:

the product of a negative number — al-nāqiṣ — by a positive number — al-zāʾid — 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.

 Double false position

للمزيد من المعلومات: False position method

Between the 9th and 10th centuries, the Egyptian mathematician Abu Kamil wrote a now-lost treatise on the use of double false position, known as the Book of the Two Errors (Kitāb al-khaṭāʾayn). 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, Euclidean-style geometric proof. Within the tradition of medieval Muslim mathematics, double false position was known as hisāb al-khaṭāʾayn ("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 al-Yasamin and balance-scale diagrams explained by al-Hassar and Ibn al-Banna, who were each mathematicians of Moroccan origin.

 Other major figures

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".

• 'Abd al-Hamīd ibn Turk (fl. 830) (quadratics)
• Thabit ibn Qurra (826–901)
• Sind ibn Ali (d. after 864)
• Ismail al-Jazari (1136–1206)
• Abū Sahl al-Qūhī (c. 940–1000) (centers of gravity)
• Abu'l-Hasan al-Uqlidisi (952–953) (arithmetic)
• 'Abd al-'Aziz al-Qabisi (d. 967)
• Ibn al-Haytham (c. 965–1040)
• Abū al-Rayḥān al-Bīrūnī (973–1048) (trigonometry)
• Ibn Maḍāʾ (c. 1116–1196)
• Jamshīd al-Kāshī (c. 1380–1429) (decimals and estimation of the circle constant)

 Gallery

• 

  Engraving of Abū Sahl al-Qūhī's perfect compass to draw conic sections.

• 

  The theorem of Ibn Haytham.

 See also

• Arabic numerals
• Indian influence on Islamic mathematics in medieval Islam
• History of calculus
• History of geometry
• Science in the medieval Islamic world
• Timeline of science and engineering in the Muslim world

 Sources

• Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Victor J. Katz (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-11485-9.
• Boyer, Carl B. (1991), "Greek Trigonometry and Mensuration, and The Arabic Hegemony", A History of Mathematics (2nd ed.), New York City: John Wiley & Sons, ISBN 0-471-54397-7, https://archive.org/details/historyofmathema00boye 
• Katz, Victor J. (1993). A History of Mathematics: An Introduction. HarperCollins college publishers. ISBN 0-673-38039-4.
• Nallino, C.A. (1939), "Al-Ḥuwārismī e il suo rifacimento della Geografia di Tolomeo" (in it), Raccolta di scritti editi e inediti, V, Rome: Istituto per l'Oriente, pp. 458–532 
• Rosen, Fredrick (1831). The Algebra of Mohammed Ben Musa. Kessinger Publishing. ISBN 1-4179-4914-7.
• Smith, David E. (1958). History of Mathematics. Dover Publications. ISBN 0-486-20429-4.
• Struik, Dirk J. (1987), A Concise History of Mathematics (4th rev. ed.), Dover Publications, ISBN 0-486-60255-9, https://archive.org/details/concisehistoryof0000stru_m6j1 

 Further reading

Books on Islamic mathematics

• Berggren, J. Lennart (1986). Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag. ISBN 0-387-96318-9.
  • Review: Toomer, Gerald J.; Berggren, J. L. (1988). "Episodes in the Mathematics of Medieval Islam". American Mathematical Monthly. Mathematical Association of America. 95 (6): 567. doi:10.2307/2322777. JSTOR 2322777.
  • Review: Hogendijk, Jan P.; Berggren, J. L. (1989). "Episodes in the Mathematics of Medieval Islam by J. Lennart Berggren". Journal of the American Oriental Society. American Oriental Society. 109 (4): 697–698. doi:10.2307/604119. JSTOR 604119.
• Daffa', Ali Abdullah al- (1977). The Muslim contribution to mathematics. London: Croom Helm. ISBN 0-85664-464-1.
• Ronan, Colin A. (1983). The Cambridge Illustrated History of the World's Science. Cambridge University Press. ISBN 0-521-25844-8.
• Rashed, Roshdi (2001). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Translated by A. F. W. Armstrong. Springer. ISBN 0-7923-2565-6.
• Youschkevitch, Adolf P.; Rozenfeld, Boris A. (1960). Die Mathematik der Länder des Ostens im Mittelalter. Berlin.{{cite book}}: CS1 maint: location missing publisher (link) Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62–160.
• Youschkevitch, Adolf P. (1976). Les mathématiques arabes: VIII^e–XV^e siècles. translated by M. Cazenave and K. Jaouiche. Paris: Vrin. ISBN 978-2-7116-0734-1.

Book chapters on Islamic mathematics

• Cooke, Roger (1997). "Islamic Mathematics". The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0-471-18082-3.

Books on Islamic science

• Daffa, Ali Abdullah al-; Stroyls, J.J. (1984). Studies in the exact sciences in medieval Islam. New York: Wiley. ISBN 0-471-90320-5.
• Kennedy, E. S. (1984). Studies in the Islamic Exact Sciences. Syracuse Univ Press. ISBN 0-8156-6067-7.

Books on the history of mathematics

• Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd ed.). Princeton University Press. ISBN 0-691-00659-8. (Reviewed: Katz, Victor J.; Joseph, George Gheverghese (1992). "The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph". The College Mathematics Journal. Mathematical Association of America. 23 (1): 82–84. doi:10.2307/2686206. JSTOR 2686206.)
• Youschkevitch, Adolf P. (1964). Gesichte der Mathematik im Mittelalter. Leipzig: BG Teubner Verlagsgesellschaft.

Journal articles on Islamic mathematics

• Høyrup, Jens. “The Formation of «Islamic Mathematics»: Sources and Conditions”. Filosofi og Videnskabsteori på Roskilde Universitetscenter. 3. Række: Preprints og Reprints 1987 Nr. 1.

Bibliographies and biographies

• Brockelmann, Carl. Geschichte der Arabischen Litteratur. 1.–2. Band, 1.–3. Supplementband. Berlin: Emil Fischer, 1898, 1902; Leiden: Brill, 1937, 1938, 1942.
• Sánchez Pérez, José A. (1921). Biografías de Matemáticos Árabes que florecieron en España. Madrid: Estanislao Maestre.
• Sezgin, Fuat (1997). Geschichte Des Arabischen Schrifttums (in الألمانية). Brill Academic Publishers. ISBN 90-04-02007-1.
• Suter, Heinrich (1900). Die Mathematiker und Astronomen der Araber und ihre Werke. Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft. Leipzig.{{cite book}}: CS1 maint: location missing publisher (link)

Television documentaries

• Marcus du Sautoy (presenter) (2008). "The Genius of the East". The Story of Maths. BBC.
• Jim Al-Khalili (presenter) (2010). Science and Islam. BBC.

 وصلات خارجية

• Hogendijk, Jan P. (January 1999). "Bibliography of Mathematics in Medieval Islamic Civilization".
• O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews .
• Richard Covington, Rediscovering Arabic Science, 2007, Saudi Aramco World
• List of Inventions and Discoveries in Mathematics During the Islamic Golden Age