Muḥammad ibn Mūsā al-Khwārizmī[note 1] (Persian: محمد بن
موسى خوارزمی; c. 780 – c. 850), formerly
Latinized as Algoritmi,[note 2] was a Persian scholar who
produced works in mathematics, astronomy, and geography under the
patronage of the Caliph
1 Life 2 Contributions
2.1 Algebra 2.2 Arithmetic 2.3 Astronomy 2.4 Trigonometry 2.5 Geography 2.6 Jewish calendar 2.7 Other works
3 See also 4 Notes 5 References 6 Further reading
6.1 Specific references 6.2 General references
Few details of al-Khwārizmī's life are known with certainty. He was
born into a Persian family and
Ibn al-Nadim gives his birthplace as
There is no need to be an expert on the period or a philologist to see that al-Tabari's second citation should read "Muhammad ibn Mūsa al-Khwārizmī and al-Majūsi al-Qutrubbulli," and that there are two people (al-Khwārizmī and al-Majūsi al-Qutrubbulli) between whom the letter wa [Arabic 'و' for the conjunction 'and'] has been omitted in an early copy. This would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, G. J. Toomer ... with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader.
Regarding al-Khwārizmī's religion, Toomer writes:
Another epithet given to him by al-Ṭabarī, "al-Majūsī," would seem to indicate that he was an adherent of the old Zoroastrian religion. This would still have been possible at that time for a man of Iranian origin, but the pious preface to al-Khwārizmī's Algebra shows that he was an orthodox Muslim, so al-Ṭabarī's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.
However, Rashed put a rather different interpretation on the same words by Al-Tabari:
... Al-Tabari's words should read: "
Muhammad ibn Musa al-Khwarizmi
Ibn al-Nadīm'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
A page from al-Khwārizmī's Algebra
Al-Khwārizmī's contributions to mathematics, geography, astronomy,
and cartography established the basis for innovation in algebra and
trigonometry. His systematic approach to solving
linear and quadratic equations led to algebra, a word derived from the
title of his book on the subject, "The Compendious Book on Calculation
by Completion and Balancing".
On the Calculation with Hindu Numerals written about 820, was
principally responsible for spreading the Hindu–Arabic numeral
system throughout the
The Compendious Book on Calculation by Completion and Balancing
(Arabic: الكتاب المختصر في حساب الجبر
والمقابلة al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr
wal-muqābala) is a mathematical book written approximately 820 CE.
The book was written with the encouragement of Caliph al-Ma'mun 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" is derived from the name of one of
the basic operations with equations (al-jabr, meaning "restoration",
referring to adding a number to both sides of the equation to
consolidate or cancel terms) described in this book. The book was
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 al-jabr (Arabic: الجبر "restoring" or "completion") and al-muqābala ("balancing"). Al-jabr 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. Al-muqābala 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, so he had to use ordinary text to present problems and their solutions. For example, for one problem he writes, (from an 1831 translation)
If some one say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times." Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.
In modern notation this process, with 'x' the "thing" (شيء shayʾ) or "root", is given by the steps,
( 10 − x
= 81 x
displaystyle (10-x)^ 2 =81x
− 20 x + 100 = 81 x
displaystyle x^ 2 -20x+100=81x
+ 100 = 101 x
displaystyle x^ 2 +100=101x
Let the roots of the equation be 'p' and 'q'. Then
p + q
displaystyle tfrac p+q 2 =50 tfrac 1 2
p q = 100
p − q
p + q
− p q
displaystyle frac p-q 2 = sqrt left( frac p+q 2 right)^ 2 -pq = sqrt 2550 tfrac 1 4 -100 =49 tfrac 1 2
So a root is given by
x = 50
displaystyle x=50 tfrac 1 2 -49 tfrac 1 2 =1
Several authors have also published texts under the name of Kitāb al-jabr wal-muqābala, including Abū Ḥanīfa Dīnawarī, Abū Kāmil Shujāʿ ibn Aslam, Abū Muḥammad al-‘Adlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn ‘Alī, Sahl ibn Bišr, and Sharaf al-Dīn al-Ṭūsī. 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.
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.
According to Swiss-American historian of mathematics, Florian
Cajori,Al-Khwarizmi's algebra was different from the work of Indian
mathematicians,for Indians had no rules like the ''restoration'' and
''reduction''.Regarding dissimilarity and significance of
Al-Khwarizmi's algebraic work from that of Indian Mathematician
Carl Benjamin Boyer
It is quite unlikely that al-Khwarizmi knew of the work of Diophantus,
but he must have been familiar with at least the astronomical and
computational portions of Brahmagupta; yet neither al-Khwarizmi nor
other Arabic scholars made use of syncopation or of negative
numbers.Nevertheless,the Al-jabr comes closer to elementary algebra of
today than the works of either
Page from a
Al-Khwārizmī's second major work was on the subject of arithmetic,
which survived in a
Page from Corpus Christi College MS 283. A
Al-Khwārizmī's Zīj al-Sindhind (Arabic: زيج السند
هند, "astronomical tables of Siddhanta") 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 values.
This is the first of many Arabic Zijes based on the Indian
astronomical methods known as the sindhind. The work contains
tables for the movements of the sun, the moon and the five planets
known at the time. This work marked the turning point in Islamic
Daunicht's reconstruction of the section of al-Khwārizmī's world map concerning the Indian Ocean.
A 15th-century version of
A stamp issued September 6, 1983 in the Soviet Union, commemorating al-Khwārizmī's (approximate) 1200th birthday.
Statue of Al-Khwārizmī in his birth town Khiva, Uzbekistan.
Al-Khwārizmī's third major work is his Kitāb Ṣūrat al-Arḍ
(Arabic: كتاب صورة الأرض, "Book of the Description of
the Earth"), also known as his Geography, which was finished in
833. It is a major reworking of Ptolemy's 2nd-century Geography,
consisting of a list of 2402 coordinates of cities and other
geographical features following a general introduction.
There is only one surviving copy of Kitāb Ṣūrat al-Arḍ, which is
kept at the Strasbourg University Library. A
Wikiquote has quotations related to: al-Khwārizmī
Wikimedia Commons has media related to Muhammad ibn Musa al-Khwarizmi.
Al-Khwarizmi (crater) — A crater on the far side of the moon named for al-Khwārizmī. Astronomy in the medieval Islamic world Indian influence on Islamic science List of pioneers in computer science Khwarizmi International Award — An Iranian award named after al-Khwārizmī. Mathematics in medieval Islam Al-Khwarizmi Institute of Computer Science (KICS)- A Pakistani research institute named after al-Khwārizmī.
^ There is some confusion in the literature on whether
al-Khwārizmī's full name is ابو عبد الله محمد بن
موسى الخوارزمي Abū ʿAbdallāh Muḥammad ibn Mūsā
al-Khwārizmī or ابو جعفر محمد بن موسی
الخوارزمی Abū Ja‘far Muḥammad ibn Mūsā
al-Khwārizmī. Ibn Khaldun notes in his encyclopedic work: "The first
who wrote upon this branch [algebra] was Abu ‘Abdallah
al-Khowarizmi, after whom came
Abu Kamil Shoja‘ ibn Aslam."
(MacGuckin de Slane). (Rosen 1831, pp. xi–xiii) mentions that "[Abu
Abdallah Mohammed ben Musa] lived and wrote under the caliphate 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
^ Berggren 1986; Struik 1987, p. 93
^ O'Connor, John J.; Robertson, Edmund F., "Abū Kāmil Shujā‘ ibn
Aslam", MacTutor History of Mathematics archive, University of St
^ Saliba, George (September 1998). "Science and medicine". Iranian
Studies. 31 (3-4): 681–690. doi:10.1080/00210869808701940. Take, for
example, someone like Muhammad b. Musa al-Khwarizmi (fl. 850) who may
present a problem for the EIr, for although he was obviously of
Persian descent, he lived and worked in
"The Arabs in general loved a good clear argument from premise to
conclusion, as well as systematic organization — respects in
^ (Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain
just what the terms al-jabr and muqabalah mean, but the usual
interpretation is similar to that implied in the translation above.
The word al-jabr presumably meant something like "restoration" or
"completion" and seems to refer to the transposition of subtracted
terms to the other side of an equation; the word muqabalah is said to
refer to "reduction" or "balancing" — that is, the cancellation
of like terms on opposite sides of the equation."
^ a b O'Connor, John J.; Robertson, Edmund F., "Muhammad ibn Musa
al-Khwarizmi", MacTutor History of Mathematics archive, University of
St Andrews .
^ Rashed, R.; Armstrong, Angela (1994). The Development of Arabic
Mathematics. Springer. pp. 11–2. ISBN 0-7923-2565-6.
Florian Cajori (1919). A History of Mathematics. p. 103. That
it came from Indian source is impossible,for Hindus had no rules like
"restoration" and "reduction" .They were never in the habit of making
all terms in an equation positive, as is done in the process of
Carl Benjamin Boyer
Further reading Specific references
Toomer, Gerald (1990). "Al-Khwārizmī, Abu Ja'far Muḥammad ibn Mūsā". In Gillispie, Charles Coulston. Dictionary of Scientific Biography. 7. New York: Charles Scribner's Sons. ISBN 0-684-16962-2. 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) Dunlop, Douglas Morton (1943). "Muḥammad b. Mūsā al-Khwārizmī". The Journal of the Royal Asiatic Society of Great Britain and Ireland. Cambridge University (2): 248–250. JSTOR 25221920. Hogendijk, Jan P., Muhammad ibn Musa (Al-)Khwarizmi (ca. 780-850 CE) – bibliography of his works, manuscripts, editions and translations. O'Connor, John J.; Robertson, Edmund F., "Abu Ja'far Muhammad ibn Musa Al-Khwarizmi", MacTutor History of Mathematics archive, University of St Andrews . Fuat Sezgin. 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.
Gandz, Solomon (November 1926). "The Origin of the Term "Algebra"".
The American Mathematical Monthly. The American Mathematical Monthly,
Vol. 33, No. 9. 33 (9): 437–440. doi:10.2307/2299605.
ISSN 0002-9890. JSTOR 2299605.
Gandz, Solomon (1936). "The Sources of al-Khowārizmī's Algebra".
Osiris. 1 (1): 263–277. doi:10.1086/368426. ISSN 0369-7827.
Gandz, Solomon (1938). "The
Folkerts, Menso (1997). Die älteste lateinische Schrift über das indische Rechnen nach al-Ḫwārizmī (in German and Latin). München: Bayerische Akademie der Wissenschaften. ISBN 3-7696-0108-4. Vogel, Kurt (1968). Mohammed ibn Musa Alchwarizmi's Algorismus; das früheste Lehrbuch zum Rechnen mit indischen Ziffern. Nach der einzigen (lateinischen) Handschrift (Cambridge Un. Lib. Ms. Ii. 6.5) in Faksimile mit Transkription und Kommentar herausgegeben von Kurt Vogel. Aalen, O. Zeller.
Goldstein, B. R. (1968). Commentary on the Astronomical Tables of
Al-Khwarizmi: By Ibn Al-Muthanna. Yale University Press.
Hogendijk, Jan P. (1991). "Al-Khwārizmī's Table of the "
B. A. Rozenfeld. "Al-Khwarizmi's spherical trigonometry" (Russian), Istor.-Mat. Issled. 32–33 (1990), 325–339.
Kennedy, E. S. (1964). "Al-Khwārizmī on the Jewish Calendar". Scripta Mathematica. 27: 55–59.
Daunicht, Hubert (1968–1970). Der Osten nach der Erdkarte al-Ḫuwārizmīs : Beiträge zur historischen Geographie und Geschichte Asiens (in German). Bonner orientalistische Studien. N.S.; Bd. 19. LCCN 71468286. Mžik, Hans von (1915). "Ptolemaeus und die Karten der arabischen Geographen". Mitteil. D. K. K. Geogr. Ges. In Wien. 58: 152. Mžik, Hans von (1916). "Afrika nach der arabischen Bearbeitung der γεωγραφικὴ ὑφήγησις des Cl. Ptolomeaus von Muh. ibn Mūsa al-Hwarizmi". Denkschriften d. Akad. D. Wissen. In Wien, Phil.-hist. Kl. 59. Mžik, Hans von (1926). Das Kitāb Ṣūrat al-Arḍ des Abū Ǧa‘far Muḥammad ibn Mūsā al-Ḫuwārizmī. Leipzig. Nallino, C. A. (1896), "Al-Ḫuwārizmī e il suo rifacimento della Geografia di Tolemo", Atti della R. Accad. dei Lincei, Arno 291, Serie V, Memorie, Classe di Sc. Mor., Vol. II, Rome Ruska, Julius (1918). "Neue Bausteine zur Geschichte der arabischen Geographie". Geographische Zeitschrift. 24: 77–81. Spitta, W. (1879). "Ḫuwārizmī's Auszug aus der Geographie des Ptolomaeus". Zeitschrift Deutschen Morgenl. Gesell. 33.
General references For a more extensive bibliography, see History of mathematics, Mathematics in medieval Islam, and Astronomy in medieval Islam.
Berggren, J. Lennart (1986). Episodes in the Mathematics of Medieval
Islam. New York: Springer Science+Business Media.
Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics
(Second ed.). John Wiley & Sons, Inc.
Daffa, Ali Abdullah al- (1977). The
v t e
Mathematics in medieval Islam
'Abd al-Hamīd ibn Turk Sind ibn Ali al-Jawharī Al-Ḥajjāj ibn Yūsuf Al-Kindi Al-Mahani al-Dinawari Banū Mūsā Hunayn ibn Ishaq al-Khwārizmī Yusuf Al-Khuri ibn Qurra Na'im ibn Musa Sahl ibn Bishr al-Marwazi Abu Said Gorgani
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al-Umawī Ibn al-Banna' Ibn Shuayb Ibn al-Shatir Kamāl al-Dīn al-Fārisī Al-Khalili Qutb al-Din al-Shirazi Ahmad al-Qalqashandi Ibn al-Durayhim
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Al-Azhar University Al-Mustansiriya University House of Knowledge House of Wisdom Constantinople observatory of Taqi al-Din Madrasa Maktab Maragheh observatory University of Al Quaraouiyine
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Astronomy in the medieval Islamic world
by century (CE AD)
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Ibn al-Banna' al-Marrakushi Ibn al‐Ha'im al‐Ishbili Jamal ad-Din al-Hanafi Muhyi al-Dīn al-Maghribī Nasir al-Din al-Tusi Qutb al-Din al-Shirazi Shams al-Dīn al-Samarqandī Zakariya al-Qazwini Ibn Abi al-Shukr al-ʿUrḍī al-Abhari Muhammad ibn Abi Bakr al‐Farisi Abu Ali al-Hasan al-Marrakushi Al-Ashraf Umar II
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Ali Kuşçu ʿAbd al‐Wājid Jamshīd al-Kāshī Kadızade Rumi Ulugh Beg Sibt al-Maridini Ibn al-Majdi al-Wafa' al-Kubunani
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Almucantar Apogee Astrology in medieval Islam Astrophysics Axial tilt Azimuth Celestial mechanics Celestial spheres Circular orbit Deferent and epicycle Earth's rotation Eccentricity Ecliptic Elliptic orbit Equant Galaxy Geocentrism Gravitational potential energy Gravity Heliocentrism Inertia Islamic cosmology Moonlight Multiverse Obliquity Parallax Precession Qibla Salah times Specific gravity Spherical Earth Sublunary sphere Sunlight Supernova Temporal finitism Trepidation Triangulation Tusi couple Universe
Al-Azhar University House of Knowledge House of Wisdom University of Al Quaraouiyine Observatories
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v t e
Geography and cartography in medieval Islam
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Sulaiman Al Mahri Piri Reis Mir Ahmed Nasrallah Thattvi Amīn Rāzī
Book of Roads and Kingdoms (al-Bakrī)
Book of Roads and Kingdoms (ibn Khordadbeh)
Kitab al-Rawd al-Mitar
The Meadows of Gold
WorldCat Identities VIAF: 365144782982270357614 LCCN: n84020660 ISNI: 0000 0001 2030 4018 GND: 118676180 SELIBR: 33137 SUDOC: 030896711 BNF: cb165923408 (data) NLA: 35538363 NKC: ola2002161