Muḥammad
ibn Mūsā al-Khwārizmī[ (Persian: محمد بن موسى خوارزمی; c. 780 – c. 850), formerly
Latinized as Algoritmi, was a Persian Muslim scholar in the House of Wisdom in
Baghdad who produced works in mathematics, astronomy, and geography during the
Abbasid Caliphate.

In
the 12th century, Latin translations of his work on the Indian numerals
introduced the decimal positional number system to the Western world.
Al-Khwārizmī's The Compendious Book on Calculation by Completion and Balancing
presented the first systematic solution of linear and quadratic equations in
Arabic. Because he is the first to teach algebra as an independent discipline
and introduced the methods of "reduction" and "balancing"
(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), he has been
described as the father] or founder of algebra.The Compendious Book on
Calculation by Completion and Balancing,translated into Latin by Robert of
Chester in 1145, was used until the sixteenth century as the principal
mathematical text-book of European universities.

He
revised Ptolemy's Geography and wrote on astronomy and astrology.

Some
words reflect the importance of al-Khwārizmī's contributions to mathematics.
"Algebra" is derived from al-jabr, one of the two operations he used
to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the
Latin form of his name. His name is also the origin of (Spanish) guarismo and of
(Portuguese) algarismo, both meaning digit.

**Life**
Few
details of al-Khwārizmī's life are known with certainty. He was born in a
Persian[4] family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater
Khorasan (modern Khiva, Xorazm Region, Uzbekistan).

Muhammad
ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy
al-Majūsiyy al-Quṭrubbaliyy (محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ).
The epithet al-Qutrubbulli could indicate he might instead have come from
Qutrubbul (Qatrabbul), a viticulture district near Baghdad. However, Rashed suggests:

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 and al-Majusi al-Qutrubbulli ...",
(and that there are two people al-Khwarizmi and al-Majusi al-Qutrubbulli): the
letter "wa" was omitted in the early copy. This would not be worth
mentioning if a series of conclusions about al-Khwarizmi's personality,
occasionally even the origins of his knowledge, had not been drawn. In his
article ([1]) G J Toomer, with naive confidence, constructed an entire fantasy
on the error which cannot be denied the merit of making amusing reading.

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 Muslim conquest of Persia, Baghdad
became the centre of scientific studies and trade, and many merchants and
scientists from as far as China and India traveled to this city, as did
al-Khwārizmī[citation needed]. He worked in Baghdad as a scholar at the House
of Wisdom established by Caliph al-Ma’mūn, where he studied the sciences and
mathematics, which included the translation of Greek and Sanskrit scientific
manuscripts.

Douglas
Morton 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.

**Contributions**

Al-Khwārizmī's
contributions to mathematics, geography, astronomy, and cartography established
the basis for innovation in algebra and trigonometry[citation needed]. 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 Middle
East and Europe. It was translated into Latin as Algoritmi de numero Indorum.
Al-Khwārizmī, rendered as (Latin) Algoritmi, led to the term
"algorithm".

Some
of his work was based on Persian and Babylonian astronomy, Indian numbers, and
Greek mathematics.

Al-Khwārizmī
systematized and corrected Ptolemy's data for Africa and the Middle 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 but with improved values for the Mediterranean Sea,
Asia, and Africa.

He
also wrote on mechanical devices like the astrolabe and sundial.

He
assisted a project to determine the circumference of the Earth and in making a
world map for al-Ma'mun, the caliph, overseeing 70 geographers.

When,
in the 12th century, his works spread to Europe through Latin translations, it
had a profound impact on the advance of mathematics in Europe.

**1.Algebra**

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.[26] 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
translated in Latin as Liber algebrae et almucabala by Robert of Chester (Segovia,
1145) hence "algebra", and also by Gerard of Cremona. 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 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 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.

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. 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.

**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
Brahmagupta, Carl Benjamin Boyer write:

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 Diophantus or Brahmagupta, because the book is not
concerned with difficult problems in indeterminant analysis but with a straight
forward and elementary exposition of the solution of equations, especially that
of second degree.The Arabs in general loved a good clear argument from premise
to conclusion,as well as systematic organization – respects in which neither
Diophantus nor the Hindus excelled.

**2.Arithmetic**

Al-Khwārizmī's
second major work was on the subject of arithmetic, which survived in a Latin
translation but was lost in the original Arabic. The translation was most
likely done in the 12th century by Adelard of Bath, 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"), or
Algoritmi de numero Indorum ("al-Khwārizmī on the Hindu Art of
Reckoning"), a name given to the work by Baldassarre Boncompagni in 1857.
The original Arabic title was possibly Kitāb al-Jam‘ wat-Tafrīq bi-Ḥisāb
al-Hind ("The Book of Addition and Subtraction According to the Hindu
Calculation").

Al-Khwārizmī's
work on arithmetic was responsible for introducing the Arabic numerals, based
on the Hindu–Arabic numeral system developed in Indian mathematics, to the
Western world. The term "algorithm" is derived from the algorism, the
technique of performing arithmetic with Hindu-Arabic numerals developed by
al-Khwārizmī. Both "algorithm" and "algorism" are derived
from the Latinized forms of al-Khwārizmī's name, Algoritmi and Algorismi,
respectively.

**3.Astronomy**

Al-Khwārizmī's
Zīj al-Sindhind[22] (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 astronomy. 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 (c. 1000) has survived in a Latin
translation, presumably by Adelard of Bath (January 26, 1126). The four
surviving manuscripts of the Latin translation are kept at the Bibliothèque
publique (Chartres), the Bibliothèque Mazarine (Paris), the Biblioteca Nacional
(Madrid) and the Bodleian Library (Oxford).

**4.Trigonometry **

Al-Khwārizmī's
Zīj al-Sindhind also contained tables for the trigonometric functions of sines
and cosine. A related treatise on spherical trigonometry is also attributed to
him.

**5.Geography**

**Daunicht's reconstruction of the section of al-Khwārizmī's world map concerning the Indian Ocean.**

**A 15th-century version of Ptolemi's Geography for comparison. **

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 Latin translation is kept at the Biblioteca
Nacional de España in Madrid.[citation needed] 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 zone, by order of longitude. As
Paul Gallez[dubious – discuss] 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 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
from the Canary Islands to the eastern shores of the Mediterranean; Ptolemy
overestimated it at 63 degrees of longitude, while al-Khwārizmī almost
correctly estimated it at nearly 50 degrees of longitude. He "also
depicted the Atlantic and Indian Oceans as open bodies of water, not
land-locked seas as Ptolemy had done. Al-Khwārizmī's Prime Meridian at the
Fortunate Isles was thus around 10° east of the line used by Marinus and
Ptolemy. Most medieval Muslim gazetteers continued to use al-Khwārizmī's prime
meridian.

**6.Jewish calendar**

Al-Khwārizmī
wrote several other works including a treatise on the Hebrew calendar, titled
Risāla fi istikhrāj ta’rīkh al-yahūd (Arabic: رسالة في إستخراج تأريخ اليهود,
"Extraction of the Jewish Era"). It describes the Metonic cycle, a
19-year intercalation cycle; the rules for determining on what day of the week
the first day of the month Tishrei shall fall; calculates the interval between
the Anno Mundi or Jewish year and the Seleucid era; and gives rules for
determining the mean longitude of the sun and the moon using the Hebrew
calendar. Similar material is found in the works of Abū Rayḥān al-Bīrūnī and
Maimonides.

**7.Other works**

Ibn
al-Nadim's Kitāb al-Fihrist, an index of Arabic books, mentions al-Khwārizmī's
Kitāb al-Taʾrīkh (Arabic: كتاب التأريخ), a book of annals. No direct
manuscript survives; however, a copy had reached Nusaybin by the 11th century,
where its metropolitan bishop, Mar Elyas bar Shinaya, found it. Elias's
chronicle quotes it from "the death of the Prophet" through to 169
AH, at which point Elias's text itself hits a lacuna.

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; the Fihrist credits
al-Khwārizmī with Kitāb ar-Rukhāma(t) (Arabic: كتاب الرخامة). Other papers,
such as one on the determination of the direction of Mecca, are on the
spherical astronomy.

Two
texts deserve special interest on the morning width (Ma‘rifat sa‘at al-mashriq
fī kull balad) and the determination of the azimuth from a height (Ma‘rifat
al-samt min qibal al-irtifā‘).

He
also wrote two books on using and constructing astrolabes.

https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi