Al-Karajī Abū Bakr ibn Muḥammad ibn al-Ḥusayn al-Karajī 980 –
1030 Persian mathematician and
engineer
al-KarajīPersian mathematician and engineer
Also known
as al-Karkhī
Al-Karajī, also known as al-Karkhī, in full, Abū
Bakr ibn Muḥammad ibn al-Ḥusayn
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.
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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