Rabu, 24 Juni 2015

980 – 1030 Al-Karajī Abū Bakr ibn Muḥammad ibn al-Ḥusayn al-Karajī


Al-Karajī Abū Bakr ibn Muammad ibn al-usayn al-Karajī   980 – 1030 Persian mathematician and engineer
 
al-KarajīPersian mathematician and engineer
Also known as al-Karkhī
Born c. 980 Karaj, Persia Died c. 1030

Al-Karajī, also known as al-Karkhī, in full, Abū Bakr ibn Muammad ibn al-usayn al-Karajī   (born c. 980, most likely Karaj, Persia, rather than Karkh, near Baghdad, Iraq—died c. 1030), mathematician and engineer who held an official position in Baghdad (c. 1010–1015), perhaps culminating in the position of vizier, during which time he wrote his three main works, al-Fakhrī fīʾl-jabr wa’l-muqābala(“Glorious on algebra”), al-Badī‘ fī’l-hisāb (“Wonderful on calculation”), and al-Kāfī fī’l-hisāb (“Sufficient on calculation”). A now lost work of his contained the first description of what later became known as Pascal’s triangle (see binomial theorem).
Al-Karajī combined tradition and novelty in his mathematical exposition. Like his Arabic predecessors he did not use symbolism—even writing numbers as words rather than using Indian numerals (except for large numbers and in numerical tables). However, with his writings Arabic algebra began to free itself from the early tradition of illustrating formulas and the resolutions of equations with geometric diagrams.
As part of his official duties al-Karajī composed his Sufficient, an arithmetic textbook for civil servants on calculating with integers and fractions (in both base 10 and base 60), extracting square roots, and determining areas and volumes. He also composed a small and very elementary compendium of basic algebra.
The Glorious and the Wonderful are more advanced algebraic texts and contain a large collection of problems. In particular, the Wonderful contains a useful introduction to the basic algebraic methods of Diophantus of Alexandria (fl. c. 250).
Although much of his work was taken from others’ writings, there is no doubt that al-Karajī was an able mathematician, and traces of his influence were frequent in the following centuries. However, the quality of his work was uneven; he seems to have worked too hastily at times, as he confessed in the closing words of theSufficient.
After leaving Baghdad for Persia, al-Karajī wrote an engineering work 

The first comment that we must make regards al-Karaji's name. It appears both as al-Karaji and as al-Karkhi but this is not a simple matter of two different transliterations of the same Arabic name. The significance is that Karaj is a city in Iran and if the mathematician's name is al-Karaji then certainly his family were from that city. On the other hand Karkh is one of the original suburbs of Baghdad which grew up outside the southern gate of the original city. The name al-Karkhi would indicate that the mathematician came from the suburb of Baghdad.
Historians seem divided as to which of these interpretations is correct. The version al-Karkhi was proposed by Woepcke (see [7] or [8]) but al-Karaji, the version which is most often used in texts today, was suggested as most likely by della Vida in 1933. Rashed comments (see [1] or [5]):-
In the present state of our knowledge delle Vida's argument is plausible but not decisive. On the basis of the manuscripts consulted it is far from easy to decide in favour of either name.
Certainly we know that al-Karaji lived in Baghdad for most of his life and that his chief mathematical works were written during the time when he lived in that city. His important treatise on algebra Al-Fakhri was dedicated to the ruler of Baghdad and was written in the city. However, at some later point in his career, al-Karaji left Baghdad to live in what are described as the "mountain countries". He seems to have given up mathematics at this time and concentrated on engineering topics such as the drilling of wells.
The importance of al-Karaji in the development of mathematics is viewed rather differently by different authors. The reason for this, rather in the same spirit as the different views on al-Khwarizmi, depends on the significance one attaches to the style of his mathematics. Some consider that his work is merely reworking ideas from earlier mathematicians while others see him as the first person to completely free algebra from geometrical operations and replace them with the arithmetical type of operations which are at the core of algebra today.
Crossley [3] sounds relatively unimpressed by al-Karaji's contributions (although he describes the content accurately):-
[Al-Karaji] gives rules for the arithmetic operations including (essentially) the multiplication of polynomials. ... al-Karaji usually gives a numerical example for his rules but does not give any sort of proof beyond giving geometrical pictures. Often he explicitely says that he is giving a solution in the style of Diophantus. He does not treat equations above the second degree except for ones which can easily be reduced to at most second degree equations followed by the extraction of roots. The solutions of quadratics are based explicitly on the Euclidean theorems ...


Woepcke in [7] (see also the reprint [8]) was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He describes it as the first appearance of a:-
... theory of algebraic calculus ... .
Rashed (see [5] which contains Rashed's article from [1] and other writings by Rashed on al-Karaji) agrees with Woepcke's interpretation and perhaps goes even further in stressing al-Karaji's importance. He writes:-
... the more-or-less explicit aim of [al-Karaji's] exposition was to find the means of realising the autonomy and specificity of algebra, so as to be in a position to reject, in particular, the geometric representation of algebraic operations.
To give another quote from Rashed's description of al-Karaji's contribution:-
Al-Karaji's work holds an especially important place in the history of mathematics. ... the discovery and reading of the arithmetical work of Diophantus, in the light of the algebraic conceptions and methods of al-Khwarizmi and other Arab algebraists, made possible a new departure in algebra by Al-Karaji ...
So what was this new departure in algebra? Perhaps it is best described by al-Samawal, one of al-Karaji's successors, who described it as [5]:-
... operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.
What al-Karaji achieved in Al-Fakhri was first to define the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and to give rules for products of any two of these. So what he achieved here was defining the product of these terms without any reference to geometry. In fact he almost gave the formula
xnxm = xm+n for all integers n and m
but he failed to make the definition x0 = 1 so he fell just a little short.
Having given rules for multiplication and division of monomials al-Karaji then looked at "composite quantities" or sums of monomials. For these he gave rules for addition, subtraction and multiplication but not for division in the general case, only giving rules for the division of a composite quantity by a monomial. He was able to give a rule for finding the square root of a composite quantity which is not completely general since it required the coefficients to be positive, but it is still a remarkable achievement.
Al-Karaji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle. Basically what al-Karaji does is to demonstrate an argument for n = 1, then prove the case n = 2 based on his result for n = 1, then prove the case n = 3 based on his result for n = 2, and carry on to around n = 5 before remarking that one can continue the process indefinitely. Although this is not induction proper, it is a major step towards understanding inductive proofs.
One of the results on which al-Karaji uses this form of induction comes from his work on the binomial theorem, the binomial coefficients and the Pascal triangle. In Al-Fakhri al-Karaji computed (a+b)3 and in Al-Badi he computed (a-b)3 and (a+b)4. The general construction of the Pascal triangle was given by al-Karaji in work described in the later writings of al-Samawal. In the translation by Rashed and Ahmad (see for example [5]) al-Samawal writes:-
Let us now recall a principle for knowing the necessary number of multiplications of these degrees by each other, for any number divided into two parts. Al-Karaji said that in order to succeed we must place 'one' on a table and 'one' below the first 'one', move the first 'one' into a second column, add the first 'one' to the 'one' below it. Thus we obtain 'two', which we put below the transferred 'one' and we place the second 'one' below the 'two'. We have therefore 'one', 'two', and 'one'.
To see how the second column of 1,2,1 corresponds to squaring a+b al-Samawal continues to describe Al-Karaji's work writing:-
This shows that for every number composed of two numbers, if we multiple each of them by itself once - since the two extremes are 'one' and 'one' - and if we multiply each one by the other twice - since the intermediate term is 'two' - we obtain the square of this number.
This is a beautiful description of the binomial theorem using the Pascal triangle. The description continues up to the binomial coefficients which give (a+b)5 but we shall only quote how al-Karaji constructs the third column from the second:-
If we transfer the 'one' in the second column into a third column, then add 'one' from the second column to 'two' below it, we obtain 'three' to be written under the 'one' in the third column. If we then add 'two' from the second column to ''one' below it we have 'three' which is written under the 'three', then we write 'one' under this 'three'; we thus obtain a third column whose numbers are 'one', 'three', 'three', and 'one'.

The table al-Karaji constructed looks like the Pascal triangle on its side.
Other results obtained by al-Karaji include summing the first n natural numbers, the squares of the first n natural numbers and the cubes of these numbers. He proved that the sum of the first n natural numbers was ½ n(n + 1). He also gave (in Rashed and Ahmad's translation, see for example [5]):-
The sum of the squares of the numbers that follow one another in natural order from one is equal to the sum of these numbers and the product of each of them by its predecessor.
In modern notation this result is

∑ i 2 = ∑ i + ∑ i (i - 1).

Al-Karaji also considered sums of the cubes of the first n natural numbers writing (in Rashed and Ahmad's translation, see for example [5]):-
If we want to add the cubes of the numbers that follow one another in their natural order we multiply their sum by itself.
In modern notation

∑ i 3 = (∑ i)2.

Al-Karaji showed that (1 + 2 + 3 + ... + 10)2 was equal to 13 + 23 + 33 + ... + 103. He did this by first showing that (1 + 2 + 3 + ... + 10)2 = (1 + 2 + 3 + ... + 9)2 + 103. He could now use the same rule on (1 + 2 + 3 + ... + 9)2, then on (1 + 2 + 3 + ... + 8)2 etc. to get

( 1 + 2 + ... + 10)2
= (1 + 2 + 3 + ... + 8)2 + 93 + 103
= (1 + 2 + 3 + ... + 7)2 + 83 + 93 + 103
= . . .
= 13 + 23 + 33 + ... + 103.

Finally we should mention the influence of Diophantus on al-Karaji. The first five books of Diophantus's Arithmetica had been translated into Arabic by ibn Liqa around 870 and these were studied by al-Karaji. Woepcke in his introduction to Al-Fakhri ([7] or [8]) writes that he found:-
... more than a third of the problems of the first book of Diophantus, the problems of the second book starting with the eighth, and virtually all the problems of the third book were included by al-Karaji in his collection.
Al-Karaji also invented many new problem of his own but even those of Diophantus were certainly not just taken without further development. He always tried to generalise Diophantus's results and to find methods which were more generally applicable.

It was not only to algebra that al-Karaji contributed. The paper [9] discusses some of his geometrical work. This occurs in a chapter entitled On measurement and balances for measuring of buildings and structures. al-Karaji defines points, lines, surfaces, solids and angles. He also gives rules for measuring both plane and solid figures, often using arches as examples. He also gives methods of weighing different substances.


 Muslim contributions in the field of mathematics have been both varied and far reaching. This article by Mahbub Ghani (from the Department of Electronic Engineering at King's College, London University) considers some Muslim contributions in algebra, focussing in particular on the contributions of scholars such as Al-Khwarizmi, Ibn Qurra, Al-Karaji and Al-Samaw'al.
 Some geometrical shapes from Suhayl al-Quhî's book "Fî istihraci mesaha al-muhassama al-maqafî or Risala-i abu Sahl". Suleymaniya Library, Ayasofya 4832.
There are many occasions where we are confronted by problems in which we are concerned with determining the value of an unknown quantity. Often, these problems are problems of geometry: for instance, given a line segment AB, we wish to divide it into segments AG and GB, such that the rectangle whose sides are AB and GB is equal to the square whose side is AG. The unknown item here is the segment AG. Such puzzles were very popular amongst the Greeks, many of them stated and solved in Euclid's celebrated work, the Elements. Problems involving unknowns are not limited to geometry, however; the famous mathematician of antiquity, Diophantus solved the following purely numerical problem in his Arithmetica: To find three numbers so that the product of any two added to the third gives a square. Nowadays, from an early age, students are taught to treat problems like the above in a unified way using the tools and techniques of algebra. This unified approach was developed and placed on firm foundations by Muslim scholars, one of the earliest and well-known being Mohammed bin Musa Al-Khwarizmi. It is due to Al-Khwarizmi that we have acquired the name algebra, transformed from the Arabic word al-Jabr appearing in the title of his most famous treatise, Kitab al-Jabr Wa l-Muqabala, literally meaning, "The book of restoring and balancing".
Confident in their own values and traditions, Muslim mathematicians benefited from their encounters with great civilisations, often integrating their ideas and techniques within a broader, more general framework. This was certainly the case with algebra. On the one hand, Muslim scholars were thoroughly versed with the work of the Greeks in geometry, having translated and produced critical commentary on crucial works such as Euclid's Elements and Archimedes' Sphere and Cylinder. The numerical and arithmetic work of the Babylonians also came under the scrutiny of the curious Muslim intellect. Of special interest to Muslim scholars were the investigations carried out by Hindu mathematicians as early as the late fifth century CE. For instance, Brahmagupta in the first half of the seventh century CE is interested, like the Babylonians, in what we today know as quadratic equations, and gave numerical procedures for obtaining their solutions. Recognising the effectiveness of numerical methods of the Hindus and Babylonians and the certainty provided by the axiomatic approach based on proof from the Greeks, the Muslims drew together these two strands to produce the new science of Algebra.

Gambar 3. miniatur Ala al-Din al-Aswad dari Tarjama-i Shakaik al-Numaniya, TSMK, H 1263.
Seorang sarjana kontemporer, Ruth McNeill, reminisces tentang bagaimana aturan tersebut membawanya untuk meninggalkan matematika:
. "Apa yang saya adalah gagasan bahwa negatif kali jumlah angka negatif keluar untuk angka positif ini tampak (dan masih tampak) inheren tidak mungkin - berlawanan dengan intuisi, sebagai matematikawan mengatakan saya bergumul dengan ide untuk apa yang saya bayangkan untuk menjadi. beberapa minggu, mencoba untuk mendapatkan penjelasan yang masuk akal dari guru saya, teman sekelas saya, orang tua saya, siapa pun. "
Ini, kemudian, membuat pernyataan al-Samaw'al tentang hubungan hilang semua lebih luar biasa. Pernyataan itu muncul dalam karya eksotis berjudul, Al-Bahir fi'l - Hasib (The Shining Buku Perhitungan), yang ia tulis saat ia hanya sembilan belas: "... jika kita mengurangi angka kekurangan dari sejumlah kekurangan yang lebih besar dari itu, masih ada perbedaan [misalnya - 5 - (- 2) = - (5-2)], kekurangan, tetapi dalam kasus lain masih ada perbedaan mereka, kelebihan [misalnya - 2 - (- 5) = + (5 -. 2)]. "
Kehidupan pribadi Al-Samaw'al membuat untuk bacaan menarik. Dia sebenarnya lahir dalam keluarga Yahudi dan dipaksa untuk menyelesaikan studi dari volume sisa Elemen Euclid sendiri. Ini karena tidak menemukan seorang guru yang cukup kompeten Matematika di Baghdad pada saat itu. Dia melanjutkan untuk belajar, lagi sendiri, karya Al-Karaji, yang kemudian dijabarkan dan diperpanjang. Masuk Islam, menurut otobiografinya, terinspirasi oleh mimpi yang ada dalam 1163. Ia menghabiskan hidupnya bepergian sebagai seorang dokter, mengobati Princes pada kesempatan, dan meninggal di Maragha, Iran utara, sekitar 1180. Secara total, Al prestasi ensiklopedis -Samaw'al yang mencakup matematika, astronomi, kedokteran, dan teologi, mengisi delapan puluh lima karya, hanya beberapa yang selamat. Seiring dengan aturan yang berkaitan dengan memanipulasi angka negatif yang dijelaskan di atas, hukum eksponen dan pembagian polinomial semua dianggap di salah satu bertahan studi matematika Al-Samawa'al itu, The Shining. Apa yang akan kita mengungkapkan saat ini dalam notasi modern x-3 x 4 = x--7, catatan Al-Samaw'al dalam bahasa waktunya seperti dalam kutipan ini:
"Opposite [atas] urutan bagian dari kubus adalah 3 dan berlawanan bagian dari mal mal adalah 4. Kami menambahkan mereka untuk mendapatkan 7 dan sebaliknya itu adalah urutan bagian dari mal mal kubus."
Kunjungan tersebut dalam dunia eksponen dibantu Al-Samaw'al saat ia membulatkan tajam untuk masalah membagi satu polinomial dengan yang lain. Rincian prosedur tidak perlu menjadi perhatian kita di sini; itu sudah cukup untuk mereproduksi ringkasan Berggren ini:
"... Penemuan prosedur ini pembagian panjang, yang dalam semua perhitungan yang tepat kita sekarang-hari pertama, merupakan kontribusi yang baik untuk sejarah matematika, dan tampaknya menjadi prestasi bersama al-Karaji dan al-Samaw 'al. "
Umar al-Khayyami, lahir di Nishapur sekitar tahun 1048 terkenal dan dikagumi di kalangan populer lebih untuk puisinya, terutama Rub'ayat, daripada prestasi yang luas dan luar biasa dalam matematika. Sebelum kita meneliti kontribusi untuk aljabar, perlu dicatat bahwa wawasan ke dalam rasio besaran (misalnya, menghasilkan atau) "sebesar pengenalan bilangan real positif," seperti dicatat oleh Berggren, yang dikomunikasikan kepada Eropa matematikawan melalui Nasir al-Din al-Tusi. Umar al-Khayyami terutama tertarik untuk mengklasifikasikan dan memecahkan persamaan kubik. Dia mencatat dalam pengantar untuk Aljabar bahwa ia bermaksud untuk mengejar pengobatan aljabar masalah sampai sekarang tidak diberikan perhatian yang sama, sampai penulis modern seperti Abu 'Abdullah al-Mahani. Jenis masalah yang menarik bagi umat Islam yang dicontohkan oleh masalah Archimedes 'memotong bola dengan pesawat sehingga volume dari dua bagian bola yang terkait satu sama lain dengan rasio tertentu. Masalahnya menyebabkan persamaan bentuk x3 + m = NX2, kelas tertentu kubik.

Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī (or al-Karkhī) (c. 953 – c. 1029) was a 10th-century mathematician and engineer who flourished at Baghdad. His three principal surviving works are mathematical: Al-Badi' fi'l-hisab (Wonderful on calculation), Al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra), and Al-Kafi fi'l-hisab (Sufficient on calculation).
Believed to be the "first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He was first to define the monomials x, x2, x3, ... and 1 / x, 1 / x2, 1 / x3, ... and to give rules for products of any two of these. He started a school of algebra which flourished for several hundreds of years" . Discovered the binomial theorem for integer exponents. [1] states that this "was a major factor in the development of numerical analysis based on the decimal system." 

Name

There is ambiguity in what his last name was. Some medieval Arabic documents have al-Karajī and others have al-Karkhī.[1] Arabic documents from the Baghdad of that era are sometimes written without diacritical points, whereby the written name is inherently ambiguous and can be read in Arabic as Karajī (reading ج) or Karkhī (reading خ) or Karahī or Karhī (reading ح) -- see Arabic rasm notation, i.e. the absence of i'jam diacritic distinctions of consonants. His name could have been al-Karkhī, indicating that he was born inKarkh, a suburb of Baghdad, or al-Karajī indicating his family came from the city of Karaj in Iran. He certainly lived and worked for most of his life in Baghdad, however, which was the scientific and trade capital of the Islamic world.

Work

Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus)[2] but most regard him as more original, in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book al-fakhri fi al-jabr wa al-muqabala, which survives from the medieval era in at least four copies.[1]
He systematically studied the algebra of exponents, and was the first to realise that the sequence x, x^2, x^3,... could be extended indefinitely; and the reciprocals 1/x, 1/x^2, 1/x^3,... . However, since for example the product of a square and a cube would be expressed, in words rather than in numbers, as a square-cube, the numerical property of adding exponents was not clear.[3]
His work on algebra and polynomials gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials.
He wrote on the binomial theorem and Pascal's triangle.
In a now lost work known only from subsequent quotation by al-Samaw'al Al-Karaji introduced the idea of argument by mathematical induction. As Katz says
Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary n. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. [...] Al-Karaji's argument includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 13) and the deriving of the truth for n = k from that of n = k - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in al-Fakhri is the earliest extant proof ofthe sum formula for integral cubes.[4]

 

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