Muhammad ibn Musa al-Khwarizmi
Overview
 
There is some confusion in the literature on whether al-Khwārizmī's full name is or . Ibn Khaldun notes in his encyclopedic work: "The first who wrote upon this branch (algebra) was Abu ʿAbdallah al-Khowarizmi, after whom came Abu Kamil Shojaʿ ibn Aslam." (MacGuckin de Slane). (Rosen 1831, pp. xi–xiii) mentions that "[Abu Abdallah Mohammed ben Musa] lived and wrote under the caliphat of Al Mamun, and must therefore be distinguished from Abu Jafar Mohammed ben Musa, likewise a mathematician and astronomer, who flourished under the Caliph Al Motaded (who reigned A.H.
Encyclopedia
There is some confusion in the literature on whether al-Khwārizmī's full name is or . Ibn Khaldun notes in his encyclopedic work: "The first who wrote upon this branch (algebra) was Abu ʿAbdallah al-Khowarizmi, after whom came Abu Kamil Shojaʿ ibn Aslam." (MacGuckin de Slane). (Rosen 1831, pp. xi–xiii) mentions that "[Abu Abdallah Mohammed ben Musa] lived and wrote under the caliphat of Al Mamun, and must therefore be distinguished from Abu Jafar Mohammed ben Musa, likewise a mathematician and astronomer, who flourished under the Caliph Al Motaded (who reigned A.H. 279-289, A.D. 892-902)." In the introduction to his critical commentary on Robert of Chester's Latin translation of al-Khwārizmī's Algebra, L.C. Karpinski notes that Abū Jaʿfar Muḥammad ibn Mūsā refers to the eldest of the Banū Mūsā
Banu Musa
The Banū Mūsā brothers , namely Abū Jaʿfar Muḥammad ibn Mūsā ibn Shākir , Abū al‐Qāsim Aḥmad ibn Mūsā ibn Shākir and Al-Ḥasan ibn Mūsā ibn Shākir , were three 9th-century Persian scholars of Baghdad who are known for their Book of Ingenious Devices on automata and mechanical devices...

 brothers. Karpinski notes in his review on (Ruska 1917) that in (Ruska 1918): "Ruska here inadvertently speaks of the author as Abū Gaʿfar M. b. M., instead of Abū Abdallah M. b. M."
, earlier transliterated as Algoritmi or Algaurizin, (c. 780, Khwārizm
Khwarezm
Khwarezm, or Chorasmia, is a large oasis region on the Amu Darya river delta in western Central Asia, which borders to the north the Aral Sea, to the east the Kyzylkum desert, to the south the Karakum desert and to the west the Ustyurt Plateau...

 – c. 850) was a Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...

 mathematician
Islamic mathematics
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...

, astronomer
Islamic astronomy
Islamic astronomy or Arabic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age , and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa, and...

 and geographer
Islamic geography
Geography and cartography in medieval Islam refers to the advancement of geography, cartography and the earth sciences in the medieval Islamic civilization....

, a scholar in the House of Wisdom
House of Wisdom
The House of Wisdom was a library and translation institute established in Abbassid-era Baghdad, Iraq. It was a key institution in the Translation Movement and considered to have been a major intellectual centre during the Islamic Golden Age...

 in Baghdad
Baghdad
Baghdad is the capital of Iraq, as well as the coterminous Baghdad Governorate. The population of Baghdad in 2011 is approximately 7,216,040...

.

In the twelfth century, Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 translations of his work on the Indian numerals
Indian numerals
Most of the positional base 10 numeral systems in the world have originated from India, where the concept of positional numeration was first developed...

, introduced the decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 positional number system
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...

 to the Western world
Western world
The Western world, also known as the West and the Occident , is a term referring to the countries of Western Europe , the countries of the Americas, as well all countries of Northern and Central Europe, Australia and New Zealand...

. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

 and quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

s in Arabic. In Renaissance Europe, he was considered the original inventor of algebra, although we now know that his work is based on older Indian or Greek sources. He revised Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

's Geography and wrote on astronomy and astrology.

Some words reflect the importance of al-Khwarizmi's contributions to mathematics. "Algebra" is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism
Algorism
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist...

and algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

stem from Algoritmi, the Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 form of his name. His name is also the origin of (Spanish
Spanish language
Spanish , also known as Castilian , is a Romance language in the Ibero-Romance group that evolved from several languages and dialects in central-northern Iberia around the 9th century and gradually spread with the expansion of the Kingdom of Castile into central and southern Iberia during the...

) guarismo and of (Portuguese
Portuguese language
Portuguese is a Romance language that arose in the medieval Kingdom of Galicia, nowadays Galicia and Northern Portugal. The southern part of the Kingdom of Galicia became independent as the County of Portugal in 1095...

) algarismo, both meaning digit
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...

.

Life

He was born in a Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...

 family, and his birthplace is given as Chorasmia by Ibn al-Nadim
Ibn al-Nadim
Abu'l-Faraj Muhammad bin Is'hāq al-Nadim , whose father was known as al-Warrāq was a Shia Muslim scholar and bibliographer. Some scholars regard him as a Persian, but this is not certain. He is famous as the author of the Kitāb al-Fihrist...

.

Few details of al-Khwārizmī's life are known with certainty. His name may indicate that he came from Khwarezm
Khwarezm
Khwarezm, or Chorasmia, is a large oasis region on the Amu Darya river delta in western Central Asia, which borders to the north the Aral Sea, to the east the Kyzylkum desert, to the south the Karakum desert and to the west the Ustyurt Plateau...

 (Khiva), then in Greater Khorasan
Greater Khorasan
Greater Khorasan or Ancient Khorasan is a historical region of Greater Iran mentioned in sources from Sassanid and Islamic eras which "frequently" had a denotation wider than current three provinces of Khorasan in Iran...

, which occupied the eastern part of the Greater Iran
Greater Iran
Greater Iran refers to the regions that have significant Iranian cultural influence. It roughly corresponds to the territory on the Iranian plateau and its bordering plains, stretching from Iraq, the Caucasus, and Turkey in the west to the Indus River in the east...

, now Xorazm Province
Xorazm Province
Xorazm Province or Khorezm Province as it is still more commonly known, is a viloyat of Uzbekistan located in the northwest of the country in the lower reaches of the Amu-Darya River. It borders with Turkmenistan, Karakalpakstan, and Buxoro Province. It covers an area of 6,300 km²...

 in Uzbekistan
Uzbekistan
Uzbekistan , officially the Republic of Uzbekistan is a doubly landlocked country in Central Asia and one of the six independent Turkic states. It shares borders with Kazakhstan to the west and to the north, Kyrgyzstan and Tajikistan to the east, and Afghanistan and Turkmenistan to the south....

. Abu Rayhan Biruni calls the people of Khwarizm "a branch of the Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...

 tree".

Al-Tabari
Muhammad ibn Jarir al-Tabari
Abu Ja'far Muhammad ibn Jarir al-Tabari was a prominent and influential Sunni scholar and exegete of the Qur'an from Persia...

 gave his name as Muhammad ibn Musa al-Khwārizmī al-Majousi al-Katarbali (Arabic: ). The epithet
Epithet
An epithet or byname is a descriptive term accompanying or occurring in place of a name and having entered common usage. It has various shades of meaning when applied to seemingly real or fictitious people, divinities, objects, and binomial nomenclature. It is also a descriptive title...

 al-Qutrubbulli could indicate he might instead have come from Qutrubbul (Qatrabbul), a viticulture
Viticulture
Viticulture is the science, production and study of grapes which deals with the series of events that occur in the vineyard. When the grapes are used for winemaking, it is also known as viniculture...

 district near Baghdad
Baghdad
Baghdad is the capital of Iraq, as well as the coterminous Baghdad Governorate. The population of Baghdad in 2011 is approximately 7,216,040...

. However, Rashed suggests:
Regarding al-Khwārizmī's religion, Toomer writes:
Ibn al-Nadīm
Ibn al-Nadim
Abu'l-Faraj Muhammad bin Is'hāq al-Nadim , whose father was known as al-Warrāq was a Shia Muslim scholar and bibliographer. Some scholars regard him as a Persian, but this is not certain. He is famous as the author of the Kitāb al-Fihrist...

's Kitāb al-Fihrist includes a short biography on al-Khwārizmī, together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833. After the Islamic conquest of Persia
Islamic conquest of Persia
The Muslim conquest of Persia led to the end of the Sassanid Empire in 644, the fall of Sassanid dynasty in 651 and the eventual decline of the Zoroastrian religion in Persia...

, Baghdad became the centre of scientific studies and trade, and many merchants and scientists from as far as China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...

 and India
History of India
The history of India begins with evidence of human activity of Homo sapiens as long as 75,000 years ago, or with earlier hominids including Homo erectus from about 500,000 years ago. The Indus Valley Civilization, which spread and flourished in the northwestern part of the Indian subcontinent from...

 traveled to this city, as did Al-Khwārizmī. He worked in Baghdad as a scholar at the House of Wisdom
House of Wisdom
The House of Wisdom was a library and translation institute established in Abbassid-era Baghdad, Iraq. It was a key institution in the Translation Movement and considered to have been a major intellectual centre during the Islamic Golden Age...

 established by Caliph
Caliph
The Caliph is the head of state in a Caliphate, and the title for the ruler of the Islamic Ummah, an Islamic community ruled by the Shari'ah. It is a transcribed version of the Arabic word   which means "successor" or "representative"...

 , where he studied the sciences and mathematics, which included the translation of Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

 and Sanskrit
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...

 scientific manuscripts.

D. M. Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā
Banu Musa
The Banū Mūsā brothers , namely Abū Jaʿfar Muḥammad ibn Mūsā ibn Shākir , Abū al‐Qāsim Aḥmad ibn Mūsā ibn Shākir and Al-Ḥasan ibn Mūsā ibn Shākir , were three 9th-century Persian scholars of Baghdad who are known for their Book of Ingenious Devices on automata and mechanical devices...

.

Contributions

Al-Khwārizmī's contributions to mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, geography
Geography
Geography is the science that studies the lands, features, inhabitants, and phenomena of Earth. A literal translation would be "to describe or write about the Earth". The first person to use the word "geography" was Eratosthenes...

, astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, and cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

 established the basis for innovation in algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

 and trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

. His systematic approach to solving linear
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

 and quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

s led to algebra, a word derived from the title of his 830 book on the subject, "The Compendious Book on Calculation by Completion and Balancing" (al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabalaالكتاب المختصر في حساب الجبر والمقابلة).

On the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Indian system of numeration
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...

 throughout the Middle East
Middle East
The Middle East is a region that encompasses Western Asia and Northern Africa. It is often used as a synonym for Near East, in opposition to Far East...

 and Europe
Europe
Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...

. It was translated into Latin as Algoritmi de numero Indorum. Al-Khwārizmī, rendered as (Latin) Algoritmi, led to the term "algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

".

Some of his work was based on Persian
Iran
Iran , officially the Islamic Republic of Iran , is a country in Southern and Western Asia. The name "Iran" has been in use natively since the Sassanian era and came into use internationally in 1935, before which the country was known to the Western world as Persia...

 and Babylonian astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, Indian numbers
Indian numerals
Most of the positional base 10 numeral systems in the world have originated from India, where the concept of positional numeration was first developed...

, and Greek
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

 mathematics.

Al-Khwārizmī systematized and corrected Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

's data for Africa
Africa
Africa is the world's second largest and second most populous continent, after Asia. At about 30.2 million km² including adjacent islands, it covers 6% of the Earth's total surface area and 20.4% of the total land area...

 and the Middle east
Middle East
The Middle East is a region that encompasses Western Asia and Northern Africa. It is often used as a synonym for Near East, in opposition to Far East...

. Another major book was Kitab surat al-ard ("The Image of the Earth"; translated as Geography), presenting the coordinates of places based on those in the Geography of Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

 but with improved values for the Mediterranean Sea
Mediterranean Sea
The Mediterranean Sea is a sea connected to the Atlantic Ocean surrounded by the Mediterranean region and almost completely enclosed by land: on the north by Anatolia and Europe, on the south by North Africa, and on the east by the Levant...

, Asia, and Africa.

He also wrote on mechanical devices like the astrolabe
Astrolabe
An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...

 and sundial
Sundial
A sundial is a device that measures time by the position of the Sun. In common designs such as the horizontal sundial, the sun casts a shadow from its style onto a surface marked with lines indicating the hours of the day. The style is the time-telling edge of the gnomon, often a thin rod or a...

.

He assisted a project to determine the circumference of the Earth and in making a world map for al-Ma'mun
Al-Ma'mun
Abū Jaʿfar Abdullāh al-Māʾmūn ibn Harūn was an Abbasid caliph who reigned from 813 until his death in 833...

, the caliph, overseeing 70 geographers.

When, in the 12th century, his works spread to Europe
Europe
Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...

 through Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 translations, it had a profound impact on the advance of mathematics in Europe. He introduced Arabic numerals into the Latin West, based on a place-value decimal system developed from Indian sources.

Algebra


is a mathematical book written approximately 830 CE. The book was written with the encouragement of the Caliph al-Ma'mun
Al-Ma'mun
Abū Jaʿfar Abdullāh al-Māʾmūn ibn Harūn was an Abbasid caliph who reigned from 813 until his death in 833...

 as a popular work on calculation and is replete with examples and applications to a wide range of problems in trade, surveying and legal inheritance. The term algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

is derived from the name of one of the basic operations with equations described in this book. The book was translated in Latin as Liber algebrae et almucabala by Robert of Chester
Robert of Chester
Robert of Chester was an English arabist of the 12th century. He translated several historically important books from Arabic to Latin, by authors such as Abu Musa Jabir Ibn Hayyan and Al-Khwarizmi including:...

 (Segovia
Segovia
Segovia is a city in Spain, the capital of Segovia Province in the autonomous community of Castile and León. It is situated north of Madrid, 30 minutes by high speed train. The municipality counts some 55,500 inhabitants.-Etymology:...

, 1145) hence "algebra", and also by Gerard of Cremona
Gerard of Cremona
Gerard of Cremona was an Italian translator of Arabic scientific works found in the abandoned Arab libraries of Toledo, Spain....

. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept in Cambridge.

It provided an exhaustive account of solving polynomial equations up to the second degree, and discussed the fundamental methods 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.

Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and c are positive integers)
  • squares equal roots (ax2 = bx)
  • squares equal number (ax2 = c)
  • roots equal number (bx = c)
  • squares and roots equal number (ax2 + bx = c)
  • squares and number equal roots (ax2 + c = bx)
  • roots and number equal squares (bx + c = ax2)


by dividing out the coefficient of the square and using the two operations ( “restoring” or “completion”) and ("balancing"). is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example, x2 = 40x − 4x2 is reduced to 5x2 = 40x. is the process of bringing quantities of the same type to the same side of the equation. For example, x2 + 14 = x + 5 is reduced to x2 + 9 = x.

The above discussion uses modern mathematical notation for the types of problems which the book discusses. However, in al-Khwārizmī's day, most of this notation had not yet been invented
History of mathematical notation
Mathematical notation comprises the symbols used to write mathematical equations and formulas. It includes Hindu-Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.The development of...

, so he had to use ordinary text to present problems and their solutions. For
example, for one problem he writes, (from an 1831 translation)

In modern notation this process, with 'x' the "thing" (shay') or "root", is given by the steps,
Let the roots of the equation be 'p' and 'q'. Then , and
So a root is given by

Several authors have also published texts under the name of , including |Abū Ḥanīfa al-Dīnawarī
Al-Dinawari
Ābu Ḥanīfah Āḥmad ibn Dawūd Dīnawarī was a Persian polymath excelling as much in astronomy, agriculture, botany and metallurgy and as he did in geography, mathematics and history. He was born in Dinawar, . He studied astronomy, mathematics and mechanics in Isfahan and philology and poetry in...

, Abū Kāmil Shujā ibn Aslam, Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk
'Abd al-Hamid ibn Turk
' , known also as ' was a ninth century Turkic Muslim mathematician. Not much is known about his biography. The two records of him, one by Ibn Nadim and the other by al-Qifti are not identical. However al-Qifi mentions his name as ʿAbd al-Hamīd ibn Wase ibn Turk Jili...

, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn al-Ṭūsī.

J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive
MacTutor History of Mathematics archive
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland...

:
R. Rashed and Angela Armstrong write:

Arithmetic

Al-Khwārizmī's second major work was on the subject of arithmetic, which survived in a Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 translation but was lost in the original Arabic
Arabic language
Arabic is a name applied to the descendants of the Classical Arabic language of the 6th century AD, used most prominently in the Quran, the Islamic Holy Book...

. The translation was most likely done in the twelfth century by Adelard of Bath
Adelard of Bath
Adelard of Bath was a 12th century English scholar. He is known both for his original works and for translating many important Greek and Arabic scientific works of astrology, astronomy, philosophy and mathematics into Latin from Arabic versions, which were then introduced to Western Europe...

, who had also translated the astronomical tables in 1126.

The Latin manuscripts are untitled, but are commonly referred to by the first two words with which they start: Dixit algorizmi ("So said al-Khwārizmī"), or Algoritmi de numero Indorum ("al-Khwārizmī on the Hindu Art of Reckoning"), a name given to the work by Baldassarre Boncompagni
Baldassarre Boncompagni
Prince Baldassarre Boncompagni-Ludovisi , was an Italian historian of mathematics and aristocrat.-Biography:...

 in 1857. The original Arabic title was possibly ("The Book of Addition and Subtraction According to the Hindu Calculation")

Al-Khwarizmi's work on arithmetic was responsible for introducing the Arabic numerals
Arabic numerals
Arabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral...

, based on the Hindu-Arabic numeral system
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...

 developed in Indian mathematics
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...

, to the Western world
Western world
The Western world, also known as the West and the Occident , is a term referring to the countries of Western Europe , the countries of the Americas, as well all countries of Northern and Central Europe, Australia and New Zealand...

. The term "algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

" is derived from the algorism
Algorism
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist...

, the technique of performing arithmetic with Hindu-Arabic numerals developed by al-Khwarizmi. Both "algorithm" and "algorism" are derived from the Latinized forms of al-Khwarizmi's name, Algoritmi and Algorismi, respectively.

Astronomy

(Arabic: زيج "astronomical tables of Sind
Sindh
Sindh historically referred to as Ba'ab-ul-Islam , is one of the four provinces of Pakistan and historically is home to the Sindhi people. It is also locally known as the "Mehran". Though Muslims form the largest religious group in Sindh, a good number of Christians, Zoroastrians and Hindus can...

 and Hind
Indian subcontinent
The Indian subcontinent, also Indian Subcontinent, Indo-Pak Subcontinent or South Asian Subcontinent is a region of the Asian continent on the Indian tectonic plate from the Hindu Kush or Hindu Koh, Himalayas and including the Kuen Lun and Karakoram ranges, forming a land mass which extends...

") is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

 values. This is the first of many Arabic Zij
Zij
Zīj is the generic name applied to Islamic astronomical books that tabulate parameters used for astronomical calculations of the positions of the Sun, Moon, stars, and planets. The name is derived from the Middle Persian term zih or zīg, meaning cord...

es
based on the Indian astronomical methods known as the sindhind. The work contains tables for the movements of the sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, the moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

 and the five planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

s known at the time. This work marked the turning point in Islamic astronomy
Islamic astronomy
Islamic astronomy or Arabic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age , and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa, and...

. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge.

The original Arabic version (written c. 820) is lost, but a version by the Spanish astronomer Maslamah Ibn Ahmad al-Majriti
Maslamah Ibn Ahmad al-Majriti
Maslama al-Majriti or Abu al-Qasim al-Qurtubi al-Majriti was a Muslim astronomer, chemist, mathematician, economist and Scholar in Islamic Spain...

 (c. 1000) has survived in a Latin translation, presumably by Adelard of Bath
Adelard of Bath
Adelard of Bath was a 12th century English scholar. He is known both for his original works and for translating many important Greek and Arabic scientific works of astrology, astronomy, philosophy and mathematics into Latin from Arabic versions, which were then introduced to Western Europe...

 (January 26, 1126). The four surviving manuscripts of the Latin translation are kept at the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Bibliotheca Nacional (Madrid) and the Bodleian Library (Oxford).

Trigonometry

Al-Khwārizmī's Zīj al-Sindhind also contained tables for the trigonometric functions of sines and cosine. A related treatise on spherical trigonometry
Spherical trigonometry
Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...

 is also attributed to him.

Geography



Al-Khwārizmī's third major work is his (Arabic: كتاب صورة الأرض "Book on the appearance of the Earth" or "The image of the Earth" translated as Geography), which was finished in 833. It is a revised and completed version of Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

's Geography
Geographia (Ptolemy)
The Geography is Ptolemy's main work besides the Almagest...

, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.

There is only one surviving copy of , which is kept at the Strasbourg University Library. A Latin translation is kept at the Biblioteca Nacional de España
Biblioteca Nacional de España
The Biblioteca Nacional de España is a major public library, the largest in Spain.It is located in Madrid, on the Paseo de Recoletos.-History:...

 in Madrid
Madrid
Madrid is the capital and largest city of Spain. The population of the city is roughly 3.3 million and the entire population of the Madrid metropolitan area is calculated to be 6.271 million. It is the third largest city in the European Union, after London and Berlin, and its metropolitan...

. The complete title translates as Book of the appearance of the Earth, with its cities, mountains, seas, all the islands and rivers, written by Abu Ja'far Muhammad ibn Musa al-Khwārizmī, according to the geographical treatise written by Ptolemy the Claudian.

The book opens with the list of latitudes and longitudes, in order of "weather zones", that is to say in blocks of latitudes and, in each weather
Weather
Weather is the state of the atmosphere, to the degree that it is hot or cold, wet or dry, calm or stormy, clear or cloudy. Most weather phenomena occur in the troposphere, just below the stratosphere. Weather refers, generally, to day-to-day temperature and precipitation activity, whereas climate...

 zone, by order of longitude. As Paul Gallez
Paul Gallez
Paul Gallez was an Argentinian cartographer and historian, born in Brussels, and based on the city of Bahía Blanca, province of Buenos Aires, Argentina....

points out, this excellent system allows the deduction of many latitudes and longitudes where the only extant document is in such a bad condition as to make it practically illegible.

Neither the Arabic copy nor the Latin translation include the map of the world itself; however, Hubert Daunicht was able to reconstruct the missing map from the list of coordinates. Daunicht read the latitudes and longitudes of the coastal points in the manuscript, or deduces them from the context where they were not legible. He transferred the points onto graph paper
Graph paper
Graph paper, graphing paper, grid paper or millimeter paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting mathematical functions or experimental data and drawing diagrams. It is commonly found in mathematics and...

 and connected them with straight lines, obtaining an approximation of the coastline as it was on the original map. He then does the same for the rivers and towns.

Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea
Mediterranean Sea
The Mediterranean Sea is a sea connected to the Atlantic Ocean surrounded by the Mediterranean region and almost completely enclosed by land: on the north by Anatolia and Europe, on the south by North Africa, and on the east by the Levant...

 from the Canary Islands
Canary Islands
The Canary Islands , also known as the Canaries , is a Spanish archipelago located just off the northwest coast of mainland Africa, 100 km west of the border between Morocco and the Western Sahara. The Canaries are a Spanish autonomous community and an outermost region of the European Union...

 to the eastern shores of the Mediterranean; Ptolemy overestimated it at 63 degrees of longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

, while al-Khwarizmi almost correctly estimated it at nearly 50 degrees of longitude. He "also depicted the Atlantic
Atlantic Ocean
The Atlantic Ocean is the second-largest of the world's oceanic divisions. With a total area of about , it covers approximately 20% of the Earth's surface and about 26% of its water surface area...

 and Indian Ocean
Indian Ocean
The Indian Ocean is the third largest of the world's oceanic divisions, covering approximately 20% of the water on the Earth's surface. It is bounded on the north by the Indian Subcontinent and Arabian Peninsula ; on the west by eastern Africa; on the east by Indochina, the Sunda Islands, and...

s as open bodies of water
Ocean
An ocean is a major body of saline water, and a principal component of the hydrosphere. Approximately 71% of the Earth's surface is covered by ocean, a continuous body of water that is customarily divided into several principal oceans and smaller seas.More than half of this area is over 3,000...

, not land-locked sea
Sea
A sea generally refers to a large body of salt water, but the term is used in other contexts as well. Most commonly, it means a large expanse of saline water connected with an ocean, and is commonly used as a synonym for ocean...

s as Ptolemy had done." Al-Khwarizmi thus set the Prime Meridian
Prime Meridian
The Prime Meridian is the meridian at which the longitude is defined to be 0°.The Prime Meridian and its opposite the 180th meridian , which the International Date Line generally follows, form a great circle that divides the Earth into the Eastern and Western Hemispheres.An international...

 of the Old World
Old World
The Old World consists of those parts of the world known to classical antiquity and the European Middle Ages. It is used in the context of, and contrast with, the "New World" ....

 at the eastern shore of the Mediterranean, 10–13 degrees to the east of Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...

 (the prime meridian previously set by Ptolemy) and 70 degrees to the west of Baghdad
Baghdad
Baghdad is the capital of Iraq, as well as the coterminous Baghdad Governorate. The population of Baghdad in 2011 is approximately 7,216,040...

. Most medieval Muslim geographers continued to use al-Khwarizmi's prime meridian.

Jewish calendar

Al-Khwārizmī wrote several other works including a treatise on the Hebrew calendar
Hebrew calendar
The Hebrew calendar , or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits , and daily Psalm reading, among many ceremonial uses...

 ( "Extraction of the Jewish Era"). It describes the 19-year intercalation cycle
Metonic cycle
In astronomy and calendar studies, the Metonic cycle or Enneadecaeteris is a period of very close to 19 years which is remarkable for being very nearly a common multiple of the solar year and the synodic month...

, the rules for determining on what day of the week the first day of the month Tishrī
Tishrei
Tishrei or Tishri , Tiberian: ; from Akkadian "Beginning", from "To begin") is the first month of the civil year and the seventh month of the ecclesiastical year in the Hebrew calendar. The name of the month is Babylonian. It is an autumn month of 30 days...

 shall fall; calculates the interval between the Jewish era
Anno Mundi
' , abbreviated as AM or A.M., refers to a Calendar era based on the Biblical creation of the world. Numerous efforts have been made to determine the Biblical date of Creation, yielding varying results. Besides differences in interpretation, which version of the Bible is being referenced also...

 (creation of Adam) and the Seleucid era
Seleucid era
The Seleucid era was a system of numbering years in use by the Seleucid Empire and other countries among the ancient Hellenistic civilizations. The era dates from the return of Seleucus I Nicator to Babylon in 311 BC after his exile in Ptolemaic Egypt, considered by Seleucus and his court to mark...

; and gives rules for determining the mean longitude of the sun and the moon using the Jewish calendar. Similar material is found in the works of al-Bīrūnī
Al-Biruni
Abū al-Rayḥān Muḥammad ibn Aḥmad al-BīrūnīArabic spelling. . The intermediate form Abū Rayḥān al-Bīrūnī is often used in academic literature...

 and Maimonides
Maimonides
Moses ben-Maimon, called Maimonides and also known as Mūsā ibn Maymūn in Arabic, or Rambam , was a preeminent medieval Jewish philosopher and one of the greatest Torah scholars and physicians of the Middle Ages...

.

Other works

Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo and Paris contain further material that surely or with some probability comes from al-Khwārizmī. The Istanbul manuscript contains a paper on sundials, which is mentioned in the Fihrist. Other papers, such as one on the determination of the direction of Mecca
Mecca
Mecca is a city in the Hijaz and the capital of Makkah province in Saudi Arabia. The city is located inland from Jeddah in a narrow valley at a height of above sea level...

, are on the spherical astronomy
Spherical astronomy
Spherical astronomy or positional astronomy is the branch of astronomy that is used to determine the location of objects on the celestial sphere, as seen at a particular date, time, and location on the Earth. It relies on the mathematical methods of spherical geometry and the measurements of...

.

Two texts deserve special interest on the morning width
Morning width
In astronomy, the morning width or rise width is the horizontal angular distance between the rise azimuth of a celestial body and the East direction....

 (Maʿrifat saʿat al-mashriq fī kull balad) and the determination of the azimuth from a height (Maʿrifat al-samt min qibal al-irtifāʿ).

He also wrote two books on using and constructing astrolabe
Astrolabe
An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...

s. Ibn al-Nadim
Ibn al-Nadim
Abu'l-Faraj Muhammad bin Is'hāq al-Nadim , whose father was known as al-Warrāq was a Shia Muslim scholar and bibliographer. Some scholars regard him as a Persian, but this is not certain. He is famous as the author of the Kitāb al-Fihrist...

 in his (an index of Arabic books) also mentions (the book on sundial
Sundial
A sundial is a device that measures time by the position of the Sun. In common designs such as the horizontal sundial, the sun casts a shadow from its style onto a surface marked with lines indicating the hours of the day. The style is the time-telling edge of the gnomon, often a thin rod or a...

s) and (the book of history
History
History is the discovery, collection, organization, and presentation of information about past events. History can also mean the period of time after writing was invented. Scholars who write about history are called historians...

) but the two have been lost.

See also

  • Al-Khwarizmi (crater)
    Al-Khwarizmi (crater)
    Al-Khwarizmi is a lunar impact crater located on the far side of the Moon. It lies to the southeast of the crater Moiseev, and northeast of Saenger....

     — A crater on the far side of the moon named after al-Khwārizmī.
  • Khwarizmi International Award
    Khwarizmi International Award
    The Khwarizmi International Award is given annually by the Iranian Research Organization for Science and Technology to individuals who have made outstanding achievements in research, innovation and invention, in fields related to science and technology....

     — An Iranian award named after al-Khwārizmī.
  • Mathematics in medieval Islam
  • Astronomy in medieval Islam
  • Hindu and Buddhist contribution to science in medieval Islam
    Hindu and Buddhist contribution to science in medieval Islam
    Hindu and Buddhist contributions to science in medieval Islam have been numerous, affecting such varied areas as medicine, astronomy and mathematics...



Further reading

Biographical
  • Brentjes, Sonja (2007). "Khwārizmī: Muḥammad ibn Mūsā al‐Khwārizmī" in Thomas Hockey et al. (eds.). The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007, pp. 631–633. (PDF version)
  • Fuat Sezgin
    Fuat Sezgin
    Fuat Sezgin is an orientalist who specializes in the history of Arabic-Islamic science. He is professor emeritus of the History of Natural Science at Johann Wolfgang Goethe University in Frankfurt, Germany and the founder and honorary director of the Institute of the History of the Arab Islamic...

    . Geschichte des arabischen Schrifttums. 1974, E. J. Brill, Leiden, the Netherlands.
  • Sezgin, F., ed., Islamic Mathematics and Astronomy, Frankfurt: Institut für Geschichte der arabisch-islamischen Wissenschaften, 1997–9.

Algebra
  • Barnabas Hughes. Robert of Chester's Latin translation of al-Khwarizmi's al-Jabr: A new critical edition. In Latin. F. Steiner Verlag Wiesbaden (1989). ISBN 3-515-04589-9.

Arithmetic
Astronomy
  • Suter, Heinrich
    Heinrich Suter
    Heinrich Suter was a historian of science specializing in Islamic mathematics and astronomy.- Books :...

    . [Ed.]: Die astronomischen Tafeln des Muhammed ibn Mûsâ al-Khwârizmî in der Bearbeitung des Maslama ibn Ahmed al-Madjrîtî und der latein. Übersetzung des Athelhard von Bath auf Grund der Vorarbeiten von A. Bjørnbo und R. Besthorn in Kopenhagen. Hrsg. und komm. Kopenhagen 1914. 288 pp. Repr. 1997 (Islamic Mathematics and Astronomy. 7). ISBN 3-8298-4008-X.
  • Van Dalen, B. Al-Khwarizmi's Astronomical Tables Revisited: Analysis of the Equation of Time.

Jewish calendar
Geography
Spherical trigonometry
  • B. A. Rozenfeld. "Al-Khwarizmi's spherical trigonometry" (Russian), Istor.-Mat. Issled. 32-33 (1990), 325-339.


General references

For a more extensive bibliography see: History of mathematics
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....

, Mathematics in medieval Islam, and Astronomy in medieval Islam.

  • Roshdi Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.
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