Iraqi mathematics, or Mesopotamian mathematics, refers to the history of mathematics in Iraq, also known as Mesopotamia, from ancient Sumerian and Babylonian mathematics through through to medieval Islamic mathematics.

## Babylonian mathematics Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, which is about six decimal figures.1 + 24/60 + 51/602 + 10/603 = 1.41421296...

Babylonian mathematics (also known as Assyro-Babylonian mathematics) refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. In respect of time they fall in two distinct groups: one from the Old Babylonian period (1830-1531 BC), the other mainly Seleucid from the last three or four centuries BC. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia. In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, and baked hard in an oven or by the heat of the sun. The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC 7289 gives an approximation to accurate to five decimal places.

### Babylonian numerals

The Babylonian system of mathematics was sexagesimal (base-60) numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a Highly composite number, having divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60, facilitating calculations with fractions. Additionally, unlike the Egyptians and Romans, the Babylonians and Indians had a true place-value system, where digits written in the left column represented larger values (much as in our base ten system: 734 = 7×100 + 3×10 + 4×1). The Sumerians and Babylonians were pioneers in this respect.

### Sumerian mathematics (3000–2000 BC) A temple mosaic from the ancient Sumerian city of Uruk IV (3400–3100 BC) showing a tessellation pattern in the tile colours.

The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.

Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.

### Old Babylonian mathematics (2000–1000 BC)

The Old Babylonian period is the period to which most of the clay tablets on Babylonian mathematics belong, which is why the mathematics of Mesopotamia is commonly known as Babylonian mathematics. Some clay tablets contain mathematical lists and tables, others contain problems and worked solutions.

#### Arithmetic

The Babylonians made extensive use of pre-calculated tables to assist with arithmetic. For example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. The Babylonians used the lists of squares together with the formulas  to simplify multiplication.

The Babylonians did not have an algorithm for long division. Instead they based their method on the fact that together with a table of reciprocals. Numbers whose only prime factors are 2, 3 or 5 (known as 5-smooth or regular numbers) have finite reciprocals in sexagesimal notation, and tables with extensive lists of these reciprocals have been found.

Reciprocals such as 1/7, 1/11, 1/13, etc. do not have finite representations in sexagesimal notation. To compute 1/13 or to divide a number by 13 the Babylonians would use an approximation such as #### Algebra

As well as arithmetical calculations, Babylonian mathematicians also developed algebraic methods of solving equations. Once again, these were based on pre-calculated tables.

To solve a quadratic equation, the Babylonians essentially used the standard quadratic formula. They considered quadratic equations of the form where here b and c were not necessarily integers, but c was always positive. They knew that a solution to this form of equation is and they would use their tables of squares in reverse to find square roots. They always used the positive root because this made sense when solving "real" problems. Problems of this type included finding the dimensions of a rectangle given its area and the amount by which the length exceeds the width.

Tables of values of n3 + n2 were used to solve certain cubic equations. For example, consider the equation Multiplying the equation by a2 and dividing by b3 gives Substituting y = ax/b gives which could now be solved by looking up the n3 + n2 table to find the value closest to the right hand side. The Babylonians accomplished this without algebraic notation, showing a remarkable depth of understanding. However, they did not have a method for solving the general cubic equation.

#### Growth

Babylonians modeled exponential growth, constrained growth (via a form of sigmoid functions), and doubling time, the latter in the context of interest on loans.

Clay tablets from c. 2000 BCE include the exercise "Given an interest rate of 1/60 per month (no compounding), compute the doubling time." This yields an annual interest rate of 12/60 = 20%, and hence a doubling time of 100% growth/20% growth per year = 5 years.

#### Plimpton 322

The Plimpton 322 tablet describes a method for solving what we would nowadays describe as quadratic equations of the form, ,

by steps (described in geometric terms) in which the solver calculates a sequence of intermediate values v1 = c/2, v2 = v12, v3 = 1 + v2, and v4 = v31/2, from which one can calculate x = v4 + v1 and 1/x = v4 - v1.

Robson's research (2001, 2002), published by the Mathematical Association of America, notes that Plimpton 322 can be interpreted as the following values, for regular number values of x and 1/x in numerical order:

v3 in the first column,
v1 = (x - 1/x)/2 in the second column, and
v4 = (x + 1/x)/2 in the third column.

In this interpretation, x and 1/x would have appeared on the tablet in the broken-off portion to the left of the first column. For instance, row 11 of Plimpton 322 can be generated in this way for x = 2.

Robson points out that Plimpton 322 reveals mathematical "methods -— reciprocal pairs, cut-and-paste geometry, completing the square, dividing by regular common factors -— [which] were all simple techniques taught in scribal schools" of that time period.

Though the table was formerly popularly interpreted by leading mathematicians as a listing of Pythagorean triples and trigonometric functions, in 2002 the Mathematical Association of America published Robson's research and (in 2003) awarded her with the Lester R. Ford Award for a modern day interpretation formally rejecting prior mathematical misconceptions.

Based on one interpretation of the Plimpton 322 cuneiform tablet (circa 1900 BC), some have even asserted that the ancient Babylonians had a trigonometric table of secants. There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table.

#### Geometry

Babylonians have known the common rules for measuring volumes and areas. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a time-mile used for measuring the travel of the Sun, therefore, representing time.

The ancient Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries, but they lacked the concept of an angle measure and consequently, studied the sides of triangles instead.

The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere.

They also used a form of Fourier analysis to compute ephemeris (tables of astronomical positions), which was discovered in the 1950s by Otto Neugebauer.

### Neo-Babylonian mathematics (700 BC–400 AD)

In the Hellenistic world, Babylonian astronomy and mathematics exerted a great influence on the mathematicians of Alexandria, in Ptolemaic Egypt and Roman Egypt. This is particularly apparent in the astronomical and mathematical works of Hipparchus, Ptolemy, Hero of Alexandria, and Diophantus. In Diophantus' case, the Babylonian influence is so strong in his Arithmetica that some scholars have argued that he himself may have been a Hellenized Babylonian. The strong Babylonian influence on Hero's work had also led to speculation that he may have been a Phoenician.

Some of the Neo-Babylonian (also known as Chaldean) mathematicians and/or mathematical astronomers during this period include Naburimannu (fl. 6th–3rd century BC), Kidinnu (d. 330 BC), Berossus (3rd century BC), Sudines (fl. 240 BC), Seleucus of Seleucia (b. 190 BC), and Diophantus (fl. 3rd century AD).

#### Influence in Hellenistic world

Since the rediscovery of the Babylonian civilization, it has become apparent that Greek and Hellenistic mathematicians and astronomers, and in particular Hipparchus, borrowed greatly from the Babylonians.

Franz Xaver Kugler demonstrated in his book Die Babylonische Mondrechnung ("The Babylonian lunar computation", Freiburg im Breisgau, 1900) the following: Ptolemy had stated in his Almagest IV.2 that Hipparchus improved the values for the Moon's periods known to him from "even more ancient astronomers" by comparing eclipse observations made earlier by "the Chaldeans", and by himself. However Kugler found that the periods that Ptolemy attributes to Hipparchus had already been used in Babylonian ephemerides, specifically the collection of texts nowadays called "System B" (sometimes attributed to Kidinnu). Apparently Hipparchus only confirmed the validity of the periods he learned from the Chaldeans by his newer observations.

It is clear that Hipparchus (and Ptolemy after him) had an essentially complete list of eclipse observations covering many centuries. Most likely these had been compiled from the "diary" tablets: these are clay tablets recording all relevant observations that the Chaldeans routinely made. Preserved examples date from 652 BC to AD 130, but probably the records went back as far as the reign of the Babylonian king Nabonassar: Ptolemy starts his chronology with the first day in the Egyptian calendar of the first year of Nabonassar, i.e., 26 February 747 BC.

This raw material by itself must have been hard to use, and no doubt the Chaldeans themselves compiled extracts of e.g., all observed eclipses (some tablets with a list of all eclipses in a period of time covering a saros have been found). This allowed them to recognise periodic recurrences of events. Among others they used in System B (cf. Almagest IV.2):

• 223 synodic months = 239 returns in anomaly (anomalistic month) = 242 returns in latitude (draconic month). This is now known as the saros period which is very useful for predicting eclipses.
• 251 (synodic) months = 269 returns in anomaly
• 5458 (synodic) months = 5923 returns in latitude
• 1 synodic month = 29;31:50:08:20 days (sexagesimal; 29.53059413… days in decimals = 29 days 12 hours 44 min 3⅓ s)

The Babylonians expressed all periods in synodic months, probably because they used a lunisolar calendar. Various relations with yearly phenomena led to different values for the length of the year.

Similarly various relations between the periods of the planets were known. The relations that Ptolemy attributes to Hipparchus in Almagest IX.3 had all already been used in predictions found on Babylonian clay tablets.

All this knowledge was transferred to the Greeks probably shortly after the conquest by Alexander the Great (331 BC). According to the late classical philosopher Simplicius (early 6th century AD), Alexander ordered the translation of the historical astronomical records under supervision of his chronicler Callisthenes of Olynthus, who sent it to his uncle Aristotle. It is worth mentioning here that although Simplicius is a very late source, his account may be reliable. He spent some time in exile at the Sassanid (Persian) court, and may have accessed sources otherwise lost in the West. It is striking that he mentions the title tèresis (Greek: guard) which is an odd name for a historical work, but is in fact an adequate translation of the Babylonian title massartu meaning "guarding" but also "observing". Anyway, Aristotle's pupil Callippus of Cyzicus introduced his 76-year cycle, which improved upon the 19-year Metonic cycle, about that time. He had the first year of his first cycle start at the summer solstice of 28 June 330 BC (Proleptic Julian calendar date), but later he seems to have counted lunar months from the first month after Alexander's decisive battle at Gaugamela in fall 331 BC. So Callippus may have obtained his data from Babylonian sources and his calendar may have been anticipated by Kidinnu. Also it is known that the Babylonian priest known as Berossus wrote around 281 BC a book in Greek on the (rather mythological) history of Babylonia, the Babyloniaca, for the new ruler Antiochus I; it is said that later he founded a school of astrology on the Greek island of Kos. Another candidate for teaching the Greeks about Babylonian astronomy/astrology was Sudines who was at the court of Attalus I Soter late in the 3rd century BC.

In any case, the translation of the astronomical records required profound knowledge of the cuneiform script, the language, and the procedures, so it seems likely that it was done by some unidentified Chaldeans. Now, the Babylonians dated their observations in their lunisolar calendar, in which months and years have varying lengths (29 or 30 days; 12 or 13 months respectively). At the time they did not use a regular calendar (such as based on the Metonic cycle like they did later), but started a new month based on observations of the New Moon. This made it very tedious to compute the time interval between events.

What Hipparchus may have done is transform these records to the Egyptian calendar, which uses a fixed year of always 365 days (consisting of 12 months of 30 days and 5 extra days): this makes computing time intervals much easier. Ptolemy dated all observations in this calendar. He also writes that "All that he (=Hipparchus) did was to make a compilation of the planetary observations arranged in a more useful way" (Almagest IX.2). Pliny states (Naturalis Historia II.IX(53)) on eclipse predictions: "After their time (=Thales) the courses of both stars (=Sun and Moon) for 600 years were prophesied by Hipparchus, …". This seems to imply that Hipparchus predicted eclipses for a period of 600 years, but considering the enormous amount of computation required, this is very unlikely. Rather, Hipparchus would have made a list of all eclipses from Nabonasser's time to his own. Problem II.8 in Diophantus' Arithmetica (1670 edition), annotated with Fermat's comment which became Fermat's Last Theorem.

Other traces of Babylonian practice in Hipparchus' work are:

• first Greek known to divide the circle in 360 degrees of 60 arc minutes.
• first consistent use of the sexagesimal number system.
• the use of the unit pechus ("cubit") of about 2° or 2½°.
• use of a short period of 248 days = 9 anomalistic months.

#### Diophantine mathematics

Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Hellenistic mathematics. During this period, the Hellenized Babylonian mathematician Diophantus made advances in Babylonian algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis". The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations.

The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares). Diophantus also made significant advances in notation, the Arithmetica being the earliest known instance of algebraic syncopation.

## Islamic mathematics

After the Islamic conquest of Persian Mesopotamia, the region of Mesopotamia came to be known as "Iraq" in the Arabic language. Under the Abbasid caliphate, the capital city of the Arab Empire was Baghdad, which was built in Iraq during the 8th century. From the 8th to 13th centuries, often known as the "Islamic Golden Age", Iraq/Mesopotamia once again became the centre of mathematical activity. Many of the greatest mathematicians at the time were active in Iraq, including Muḥammad ibn Mūsā al-Khwārizmī (Algoritmi), Al-Abbās ibn Said al-Jawharī, 'Abd al-Hamīd ibn Turk, Al-Kindi (Alkindus), Hunayn ibn Ishaq (Johannitius), the Banū Mūsā brothers, the Thābit ibn Qurra family, Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius), the Brethren of Purity, Al-Saghani, Abū Sahl al-Qūhī, Ibn Sahl, Abu Nasr Mansur ibn Iraq, Ibn al-Haytham (Alhazen), Ibn Tahir al-Baghdadi, and Ibn Yahyā al-Maghribī al-Samaw'al. Mathematical activity came to an end in Iraq/Mesopotamia after sack of Baghdad in 1258.

### Origins and influences

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 eighth 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 Hellenistic works as possible, including Ptolemy's Almagest and Euclid's Elements. Hellenistic works would be given to the Muslims by the Byzantine Empire in exchange for treaties, as the two empires held an uneasy peace. Many of these Hellenistic works were translated by Thabit ibn Qurra (826-901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius. Historians are in debt to many Islamic translators, for it is through their work that many ancient Greek texts have survived only through Arabic translations.

Hellenistic/Egyptian, Indian and Babylonian mathematics all played an important role in the development of early Islamic mathematics. The works of mathematicians such as Euclid, Apollonius, Archimedes, Diophantus, Aryabhata and Brahmagupta were all acquired by the Islamic world and incorporated into their mathematics. Perhaps the most influential mathematical contribution from India was the decimal place-value Indo-Arabic numeral system, also known as the Hindu numerals. The Persian historian al-Biruni (c. 1050) in his book Tariq al-Hind states that al-Ma'mun had an embassy in India from which was brought a book to Baghdad that was translated into Arabic as Sindhind. It is generally assumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.

Besides the Hellenistic and Indian traditions, a third tradition which had a significant influence on mathematics in medieval Islam was the "mathematics of practitioners", which included the applied mathematics of "surveyors, builders, artisans, in geometric design, tax and treasury officials, and some merchants." This applied form of mathematics transcended ethnic divisions and was a common heritage of the lands incorporated into the Islamic world. This tradition also includes the religious observances specific to Islam, which served as a major impetus for the development of mathematics as well as astronomy.

#### Islam and mathematics

A major impetus for the flowering of mathematics as well as astronomy in medieval Islam came from religious observances, which presented an assortment of problems in astronomy and mathematics, specifically in trigonometry, spherical geometry, algebra and arithmetic.

The Islamic law of inheritance served as an impetus behind the development of algebra (derived from the Arabic al-jabr) by Muhammad ibn Mūsā al-Khwārizmī and other medieval Islamic mathematicians. Al-Khwārizmī's Hisab al-jabr w’al-muqabala devoted a chapter on the solution to the Islamic law of inheritance using algebra. He formulated the rules of inheritance as linear equations, hence his knowledge of quadratic equations was not required. Later mathematicians who specialized in the Islamic law of inheritance included Al-Hassār, who developed the modern symbolic mathematical notation for fractions in the 12th century, and Abū al-Hasan ibn Alī al-Qalasādī, who developed an algebraic notation which took "the first steps toward the introduction of algebraic symbolism" in the 15th century.

### Importance

J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:

"Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the 16th, 17th, and 18th centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of Greek mathematics."

R. Rashed wrote in The development of Arabic mathematics: between arithmetic and algebra:

"Al-Khwarizmi's successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose."

### Biographies

Al-Ḥajjāj ibn Yūsuf ibn Maṭar (786 – 833)
Al-Ḥajjāj translated Euclid's Elements into Arabic.
Al-Khwārizmī was a Persian mathematician, astronomer, astrologer and geographer. He worked most of his life as a scholar in the House of Wisdom in Baghdad. His Algebra was the first book on the systematic solution of linear and quadratic equations. Latin translations of his Arithmetic, on the Indian numerals, introduced the decimal positional number system to the Western world in the 12th century. He revised and updated Ptolemy's Geography as well as writing several works on astronomy and astrology.
Al-Jawharī was a mathematician who worked at the House of Wisdom in Baghdad. His most important work was his Commentary on Euclid's Elements which contained nearly 50 additional propositions and an attempted proof of the parallel postulate.
ʿAbd al-Hamīd ibn Turk (fl. 830 Baghdad)
Ibn Turk wrote a work on algebra of which only a chapter on the solution of quadratic equations has survived.
Yaʿqūb ibn Isḥāq al-Kindī (c. 801 Kufa – 873 Baghdad)
Al-Kindī (or Alkindus) was a philosopher and scientist who worked as the House of Wisdom in Baghdad where he wrote commentaries on many Hellenistic works. His contributions to mathematics include many works on arithmetic and geometry.
Hunayn ibn Ishaq (808 Al-Hirah – 873 Baghdad)
Hunayn (or Johannitus) was a translator who worked at the House of Wisdom in Baghdad. Translated many Hellenistic works including those by Plato, Aristotle, Galen, Hippocrates, and the Neoplatonists.
The Banū Mūsā were three brothers who worked at the House of Wisdom in Baghdad. Their most famous mathematical treatise is The Book of the Measurement of Plane and Spherical Figures, which considered similar problems as Archimedes did in his On the Measurement of the Circle and On the sphere and the cylinder. They contributed individually as well. The eldest, Jaʿfar Muḥammad (c. 800) specialised in geometry and astronomy. He wrote a critical revision on Apollonius' Conics called Premises of the book of conics. Aḥmad (c. 805) specialised in mechanics and wrote a work on pneumatic devices called On mechanics. The youngest, al-Ḥasan (c. 810) specialised in geometry and wrote a work on the ellipse called The elongated circular figure.
Al-Mahani
Ahmed ibn Yusuf
Thabit ibn Qurra (Syria-Iraq, 835-901)
Al-Hashimi (Iraq? ca. 850-900)
Muḥammad ibn Jābir al-Ḥarrānī al-Battānī (c. 853 Harran – 929 Qasr al-Jiss near Samarra)
Sinan ibn Tabit (ca. 880 - 943)
Al-Nayrizi
Ibrahim ibn Sinan (Iraq, 909-946)
Al-Khazin (Iraq-Iran, ca. 920-980)
Al-Karabisi (Iraq? 10th century?)
Ikhwan al-Safa' (Iraq, first half of 10th century)
The Ikhwan al-Safa' ("brethren of purity") were a (mystical?) group in the city of Basra in Irak. The group authored a series of more than 50 letters on science, philosophy and theology. The first letter is on arithmetic and number theory, the second letter on geometry.
Al-Uqlidisi (Iraq-Iran, 10th century)
Al-Saghani (Iraq-Iran, ca. 940-1000)
Abū Sahl al-Qūhī (Iraq-Iran, ca. 940-1000)
Al-Khujandi
Abū al-Wafāʾ al-Būzjānī (Iraq-Iran, ca. 940-998)
Ibn Sahl (Iraq-Iran, ca. 940-1000)
Abu Nasr ibn `Iraq (Iraq-Iran, ca. 950-1030)
Ibn al-Haytham (Iraq-Egypt, ca. 965-1040)
Ibn Yaḥyā al-Maghribī al-Samawʾal (ca. 1130, Baghdad – c. 1180, Maragha)
Naṣīr al-Dīn al-Ṭūsī (18 February 1201 in Tus, Khorasan – 26 June 1274 in Kadhimain near Baghdad)

### Algebra A page from the Al-jabr wa'l muqabalah, written circa 820 by Al-Khwarizmi. This manuscript is dated circa 240 AH (854 AD).

The term algebra is derived from the Arabic term al-jabr in the title of Al-Khwarizmi's Al-jabr wa'l muqabalah. He originally used the term al-jabr to describe the method 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.

There are three theories about the origins of Islamic algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence, and the third emphasizes Hellenistic influence. Many scholars believe that it is the result of a combination of all three sources.

Throughout their time in power, before the fall of Islamic civilization, the Arabs used a fully rhetorical algebra, where sometimes even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (eg. twenty-two) with Arabic numerals (eg. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra, until the work of Ibn al-Banna al-Marrakushi in the 13th century and Abū al-Hasan ibn Alī al-Qalasādī in the 15th century.

There were four conceptual stages in the development of algebra, three of which either began in, or were significantly advanced in, the Islamic world. These four stages were as follows:

• Geometric stage, where the concepts of algebra are largely geometric. This dates back to the Babylonians and continued with the Hellenistic/Egyptian mathematicians, and was revived by Omar Khayyam.
• Static equation-solving stage, where the objective is to find numbers satisfying certain relationships. The move away from geometric algebra dates back to Diophantus and Brahmagupta, but algebra didn't decisively move to the static equation-solving stage until Al-Khwarizmi's Al-Jabr.
• Dynamic function stage, where motion is an underlying idea. The idea of a function began emerging with Sharaf al-Dīn al-Tūsī, but algebra didn't decisively move to the dynamic function stage until Gottfried Leibniz.
• Abstract stage, where mathematical structure plays a central role. Abstract algebra is largely a product of the 19th and 20th centuries.

#### Static equation-solving algebra

Al-Khwarizmi and Al-jabr wa'l muqabalah

The Muslim Persian mathematician Muhammad ibn Mūsā al-Khwārizmī (c. 780-850) 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 A.D., wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian Sindhind. One of al-Khwarizmi's most famous books is entitled Al-jabr wa'l muqabalah or The Compendious Book on Calculation by Completion and Balancing, and it gives an exhaustive account of solving polynomials up to the second degree. The book also introduced the fundamental method 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.

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 (ax² = bx), the second chapter deals with squares equal to number (ax² = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax² + bx = c), the fifth chapter deals with squares and number equal roots (ax² + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax²).

J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:

"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 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."

Debate exists over whether Al-Khwarizmi or the Hellenistic Babylonian mathematician Diophantus should be called "the father of algebra". Many agree that Al-Khwarizmi deserves this title most. Those who support Diophantus point to the fact that the algebra found in Al-Jabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, introduced the fundamental methods of reduction and balancing, and was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers. The "novelty of Al-Khwarizmi lies in his extremely systematic treatment, aiming at a general classification of linear and quadratic equations, and at general methods of solving them which are established with proofs."  In addition, R. Rashed and Angela Armstrong write:

"Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."

Ibn Turk and Logical Necessities in Mixed Equations

'Abd al-Hamīd ibn Turk (fl. 830) authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr. The manuscript gives the exact 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 determinant is negative then the quadratic equation has no solution. The similarity between these two works has led some historians to conclude that Islamic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.

#### Linear algebra

In linear algebra and recreational mathematics, magic squares were known to Arab mathematicians, possibly as early as the 7th century, when the Arabs got into contact with Indian or South Asian culture, and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics. It has also been suggested that the idea came via China. The first magic squares of order 5 and 6 appear in an encyclopedia from Baghdad circa 983 AD, the Rasa'il Ihkwan al-Safa (Encyclopedia of the Brethren of Purity); simpler magic squares were known to several earlier Arab mathematicians.