Abu Abdullah Muhammad ibn Musa al-Khwarizmi

Abu Abdullah Muhammad ibn Musa al-Khwarizmi
(c. 780, – c. 850)

-- Manoj Subedi (2010)

Abu Abdullah Muhammad ibn Musa al-Khwarizmi was a Persian mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad.

His Kitab al-Jabr wa-l-Muqabala presented the first systematic solution of linear and quadratic equations. He is considered the founder of algebra, In the twelfth century, Latin translations of his work on the Indian numerals, introduced the decimal positional number system to the Western world. He revised Ptolemy's Geography and wrote on astronomy and astrology.
His contributions had a great impact on language on 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 the origin of Spanish meaning digit.

Life

Few details of Al-Khwarizmi’s life are known with certainty, even his birthplace is unsure. His name may indicate that he came from Khwarezm which occupied the eastern part of the Greater Iran, now in Uzbekistan.

Harun al-Rashid became the fifth Caliph of the Abbasid dynasty on 14 September 786, about the time that al-Khwarizmi was born. Harun ruled, from his court in the capital city of Baghdad, over the Islam empire which stretched from the Mediterranean to India. He brought culture to his court and tried to establish the intellectual disciplines which at that time were not flourishing in the Arabic world. He had two sons, the eldest was Al-Amin while the younger was Al-Mamun. Harun died in 809 and there was an armed conflict between the brothers.

Al-Mamun won the armed struggle and Al-Amin was defeated and killed in 813. Following this, Al-Mamun became Caliph and ruled the empire from Baghdad. He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. He also built up a library of manuscripts, the first major library to be set up since that at Alexandria, collecting important works from Byzantium. In addition to the House of Wisdom, Al-Mamun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples.

Al-Khwarizmi and his colleagues the Banu Musa were scholars at the House of Wisdom in Baghdad. Their tasks there involved the translation of Greek scientific manuscripts and they also studied, and wrote on, algebra, geometry and astronomy. Certainly al-Khwarizmi worked under the patronage of Al-Mamun and he dedicated two of his texts to the Caliph. These were his treatise on algebra and his treatise on astronomy. The algebra treatise Hisab al-jabr w'al-muqabala was the most famous and important of all of al-Khwarizmi's works. It is the title of this text that gives us the word "algebra" and, in a sense that we shall investigate more fully below, it is the first book to be written on algebra. Regarding Al-Khwarizmi’s religion, he was an orthodox Muslim.

Al-Khwarizmi accomplished most of his work in the period between 813 and 833. After the Islamic 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-Khwarizmi. He worked in Baghdad as a scholar at the House of Wisdom established by Caliph Al-Mamun, where he studied the sciences and mathematics, which included the translation of Greek and Sanskrit scientific manuscripts.

Contributions

Algebra
Al-Kitab al-mukhtaSar fi ḥisab al-jabr wa-l-muqabala
“The Compendious Book on Calculation by Completion and Balancing”)
Arithmetic
Al-Khwarizmi on the Hindu Art of Reckoning.
Trigonometry
Zij al-Sindhind,( tables for the trigonometric functions sine, cosine and for the first time for tangent)
Astronomy (construction of sundials and quadrants)
Geography
Kitab ṣurat al-Arḍ ("Book on the appearance of the Earth" or "The image of the Earth" translated as Geography)
Jewish calendar
Al-Khwarizmi’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" (Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala.
On the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Indian system of numeration throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum. Al-Khwarizmi 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-Khwarizmi 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. He introduced Arabic numerals into the Latin West, based on a place-value decimal system developed from Indian sources.

Algebra

Al-Khwarizmi is sometimes called the “Father of Algebra”.Al-Khwarizmi’s most important work Hisab al-jabr w’al-muqabala written in 830 gives us the word algebra. This treatise classifies the solution of quadratic equations and gives geometric methods for completing the square. No symbols are used and no negative or zero coefficients were allowed.

Al-Kitab al-jabr wa-l-muqabala “The Compendious Book on Calculation by Completion and Balancing”) is a mathematical book written approximately 830 CE. The term algebra is derived from the name of one of the basic operations with equations (al-jabr, meaning completion, or, subtracting a number from both sides of the equation) described in this book.

The al-jabr is considered the foundational text of modern algebra. It provided an exhaustive account of solving polynomial equations up to the second degree, and introduced 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.

His book Al-Kitab al-mukhtaSar fi ḥisab al-jabr wa-l-muqabala on algebra, was translated into Latin and used for generations in Europe.

Some featurers of his book Al-jabr wa-l-muqabala

• It is strictly rhetorical – even the numerals are expressed as words, and it is more elementary than Arithmetica by Diophantus.
• It is a practical work, by design, being concerned with straightforward solutions of deterministic problems, linear and especially quadratic.
• Chapters I–VI covers cases of all quadratics with a positive solution in a systematic and exhaustive way.
• It would have been easy for any student to master the solutions. Mostly he shows his methods using examples – as others have done.
• Al-Khwarizmi then establishes geometric proofs for the same solutions of these quadratics. However, the proofs are more in the Babylonnian style.
• He dealt with three types of quantities: the square of a number, the root of the square (i.e. the unknown), and absolute numbers.
• He then solves the equation using essentially a rhetorical form for the quadratic equation.
He notes six different types of quadratics:
ax2 = bx
ax2 = c
bx = c
ax2 + bx = c
ax2 + c = bx
ax2 = bx + c
The treatise Al-Kitab al-mukhtaSar fi ḥisab al-jabr wa-l-muqabala was the most famous and important of all of Al-Khwarizmi's works. In this book, which has givenus the word ‘algebra’, Al-khwarizmi gives a complete solution to all possible types of quadratic equation.
Here is Al-Khwarizmi's solution of the equation.
x2 + 21 = 10x.
What is most remarkable is that in this case he knows that the quadratic has two solutions
Halve the number of the roots. It is 5. Multiply this by itself and the product is 25. Subtract from this the 21 added to the square term and the remainder is 4. Extract its square root, 2, and subtract this from half the number of roots, 5. There remains 3. This is the root you wanted, whose square is 9. Alternatively, you may add the square root to half the number of roots and the sum is 7. This is then the root you wanted and the square is 49. The above discussion uses modern mathematical notation for the types of problems which the book discusses. However, in Al-Khwarizmi’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,
You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.
In modern notation this process, with 'x' the "thing" or "root” is given by the steps,
(10 − x)2 = 81x
or, x2 + 100 = 101x
Arithmetic
Al-Khwarizmi's second major work was on the subject of arithmetic, which survived in a Latin translation but was lost in the original Arabic.
Al-Khwarizmi also wrote on Hindu-Arabic numerals. The Arabic text is lost, but a Latin translation, Algoritmi de numero Indorum, which in English is Al-Khwarizmi on the Hindu Art of Reckoning, gave rise to the word algorithm deriving from his name in the title.
The Latin manuscripts are untitled, but are commonly referred to by the first two words with which they start: Dixit algorizmi ("So said Al-Khwarizmi”), or Algoritmi de numero Indorum ("Al-Khwarizmi's on the Hindu Art of Reckoning").
Al-Khwarizmi'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-Khwarizmi. Both "algorithm" and "algorism" are derived from the Latinized forms of al-Khwarizmi's name, Algoritmi and Algorismi, respectively.
Trigonometry
In trigonometry, Al-Khwarizmi produced tables for the trigonometric functions of sine and cosine in the Zij al-Sindhind, alongside the first tables for tangents. He was also an early pioneer in spherical trigonometry, and wrote a treatise on the subject.
Astronomy
Al-Khwarizmi made several important improvements to the theory and construction of sundials, which he inherited from his Indian and Greek predecessors. He made tables for these instruments which considerably shortened the time needed to make specific calculations. His sundial was universal and could be observed from anywhere on the Earth. From then on, sundials were frequently placed on mosques to determine the time of prayer. The shadow square, to find the height of a body was also invented by Al-Khwarizmi in ninth-century Baghdad.
The first quadrant was invented by al-Khwarizmi in ninth century Baghdad. The sine quadrant, invented by Al-Khwarizmi was used for astronomical calculations. The first horary quadrant for specific latitudes, was also invented by Al-Khwarizmi in Baghdad, then center of the development of quadrants. It was used to determine time.
Geography
Al-Khwarizmi’s third major work is his Kitab ṣurat al-Arḍ ("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's Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.
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
Jewish calendar
Al-Khwarizmi wrote several other works including a treatise on the Hebrew calendar "Extraction of the Jewish Era"). It describes the 19-year cycle, the rules for determining on what day of the week the first day of the month.
Honor and Tribute
He is sometime also known as father of Algebra.
The words Algebra and Algorithm are termed after his name.
Khwarizmi is a lunar impact crater located on the far side of the Moon named by NASA on his tribute.
The Khwarizmi International Award is given annually by the Iranian Research Organization for Science and Technology (IROST) to individuals who have made outstanding achievements in research, innovation and invention, in fields related to science and technology.
References
History of mathematics.
http://en.wikipedia.org/wiki/Khwarezm"
http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Khwarizmi.html
http://en.wikipedia.org/wiki/Muhammad_ibn_Mūsā_al-Khwārizmī