Matematik pada zaman pertengahan Islam

Dalam sejarah matematik, **"Matematik pada zaman pertengahan Islam"** merujuk kepada matematik yang dikembangkan oleh ahli matematik dari budaya Islam dari bermulanya Islam sehingga abad ke-17, kebanyakannya termasuk ahli matematik Arab dan Parsi, dan juga umat Muslim yang lain dan bukan-Muslim yang sebahagian dari kebudayaan Islam. Ahli matematik Islam juga dikenali sebagai **ahli matematik Arab** oleh kerana berasaskan teks matematik Islam ditulis dalam Bahasa Arab. Ahli matematik Islam adalah aspek utama pada sejarah lebih besar sains Islam, dan juga sebahagian penting bagi sejarah matematik.

Sains and matematik Islam berkembang di bawah pemerintahan Khalifah Islam (juga dikenali sebagai Empayar Arab atau Empayar Islam ditubuhkan di merata-rata Timur Tengah, Asia Tengah, Afrika Utara, Sicily, Semenanjung Iberia, dan sesetengah bahagian Perancis dan Pakistan (dikenali sebagai India pada waktu itu) pada abad ke-8. Walaupun kebanyakan teks Islam pada matematik dituliskan dalam Bahasa Arab, ia tidak semua ditulis oleh orang Arab, semenjak - seperti taraf Bahasa Yunani di dunia Hellenistik — Bahasa Arab digunakan sebagai Bahasa sarjana bukan Arab di merata-rata dunia Islam pada zaman itu. Banyak ahli matematik Islam adalah orang Parsi.

Kajian terkini melukiskan sebuah gambaran baru tentang simtem hutang yang kita harus berterima kasih kepada matematik Islam. Sudah tentu banyak idea yang dahulu dianggapkan sebagai konsep baru yang bijak oleh kerana ahli matematik Eropah pada abad ke-16, ke-17, dan ke-18 sekarang telah diketahui pada perkembangannya oleh ahli matematik Arab/Islam sekitar empat abad yang lebih awal. Dengan hormatnya, pelajaran ahli matematik hari ini lebih serupa dengan gaya matematik Islam daripada matematik Hellenistik.

## Sejarah
## Pengaruh

Matematik Yunani dan matematik India memainkan peranan penting dalam kemajuan matematik Islam terawal, terutamanya karya-karya seperti karya geometri klasik Euclid, karya trigonometri Aryabhata dan karya arimetik Brahmagupta, dan sarjana menganggap karya-karya itu menyumbang pada era inovasi saintifik Islam yang berlanjutan hingga kurun ke-14. Banyak teks-teks Yunani kuno hanya bertahan dalam bentuk terjemahan ke bahasa Arab yang dilakukan ilmuan Islam. Kemungkinan sumbangan paling penting dalam bidang matematik dari India ialah sistem angka India-Arab yang berasaskan tempat perpuluhan, yang juga dikenali sebagai angka Hindu. Sejarawan Parsi al-Biruni (sekitar 1050M) dalam buku beliau *Tahqiq ma li al-Hind* menyatakan bahawa Khalifah al-Ma'mun menerima kedatangan duta dari India dan membawa bersama sebuah buku yang diterjemahkan ke bahasa Arab sebagai *Sindhind*. Berkemungkinan *Sindhind* tidak lain tidak bukan *[[Brahmasphutasiddhanta|Brahmasphuta-siddhanta]]* tulisan Brahmagupta.

## Ahli matematik Islam termasyhur
### Muhammad ibn Musa al-Khwarizmi

Seorang tokoh penting dalam matematik Islam ialah  (780M-850M), juga dikenali sebagai al-Khwarizmi, ahli matematik dan ahli falak berketurunan Parsi yang bekerja untuk Khalifah di Baghdad. Beliau menulis beberapa buah buku penting dan hari ini dikenali kerana memperkenalkan sistem angka desimal (perpuluhan) yang kita gunakan sekarang. Sistem tersebut dibangunkan di India pada kurun ke-6M, namun sistem ini hanya diketahui orang Eropah pada kurun ke-13, melalui terjemahan Latin karya al-Khwarizmi.
Karya-karya matematik Eropah pada Zaman Pertengahan menggunakan frasa "dixit Algorismi" ("begitulah kata al-Khwarizmi") ketika menggunakan sistem angka desimal; dari sini perkataan "algorithm" terbit. Juga perkataan "algebra" terbit daripada salah satu karya beliau, *[[Al-Jabr wa-al-Muqabalah]]*, yang membicarakan persamaan matematik, polinomial, pecahan dsb. - khususnya karya tersebut menerangkan cara mengetahui kuantiti majhul dalam persamaan dengan melakukan imbangan yang memelihara persamaan.
Walaupun terdapat dakwaan bahawa beliau beragama Majusi, sukar untuk menafikan kedudukan beliau sebagai seorang Muslim berdasarkan nama beliau sempena nama Nabi Muhammad.
Apa pun, karya beliau akan selalu kekal dalam arus perdana sejarah intelek Islam. Al-Khwarizmi sering dianggap sebagai bapa algebra dan algorithm disebabkan karya-karya penting beliau dalam bidang-bidang tersebut.

Barangkali salah satu kemajuan signifikan yang dikeluarkan oleh matematik Islam bermula dengan karya al-Khwarizmi, terutamanya permulaan algebra. Penting untuk difahami betapa signifikannya idea baru ini. Ia beralih secara revolusi dari konsep orang Yunani mengenai matematik yang pada dasarnya adalah geometri.

Algebra ialah teori penyatuan yang menganggap nombor nisbah, nombor bukan nisbah, magnitud geometri, dsb. sebagai "objek algebra".
Ia memberi matematik satu landasan pembangunan baru sepenuhnya yang lebih luas dalam konsep yang wujud sebelum ini, dan menyediakan kenderaan bagi pembangunan akan datang.
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.

Al-Khwarizmi's successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.

Though Al-Khwarizmi's approach to mathematics was mostly algebraic, he did contribute to the study of practical geometry.

### Al-Mahani
Persian mathematician al-Mahani (b. 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.

### 'Abd al-Hamid ibn Turk
'Abd al-Hamid ibn Turk contributed to the study of quadratic equations.

### Thabit ibn Qurra
Arab mathematician and geometer Thabit ibn Qurra (b. 836) made many contributions to mathematics, particularly geometry. In his work on number theory, he discovered an important theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other. Amicable numbers later played a large role in Islamic mathematics.

Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. Thabit ibn Qurra studied curves required in the construction of sundials. Thabit ibn Qurra also undertook both theoretical and observational work in astronomy.

### Abu Kamil
Egyptian mathematician Abu Kamil ibn Aslam (850) forms an important link in the development of algebra between al-Khwarizmi and al-Karaji. He had begun to understand what we would write in symbols as \ x^n.x^m = x^{m+n}. He also studied algebra using irrational numbers.

### Al-Batanni
Arab mathematician and astronomer Abu Abdallah Muhammad ibn Jabir al-Battani (868-929) made accurate astronomical observations which allowed him to improve on Ptolemy's data for the Sun and the Moon. He also produced a number of trigonometrical relationships:

::\tan a = \frac{\sin a}{\cos a}

::\sec a = \sqrt{1 + \tan^2 a }

He also solved the equation sin *x* = *a* cos *x* discovering the formula:

::\sin x = \frac{a}{\sqrt{1 + a^2}}

and used al-Marwazi's idea of tangents ("shadows") to develop equations for calculating tangents and cotangents, compiling tables of them.

### Sinan ibn Thabit
Arab scientist Sinan ibn Thabit ibn Qurra (c. 880-943) was the son of Thabit ibn Qurra and the father of Ibrahim ibn Sinan. He wrote the mathematical treatise *On the elements of geometry*, a commentary on Archimedes' *On triangles*, and several other astronomical and political treatises. He studied medicine, the science of Euclid, the *[[Almagest]]*, astronomy, the theories of meteorological phenomena, logic and metaphysics.

### Ibrahim ibn Sinan
Although Islamic mathematicians are most famed for their work on algebra, number theory and numeral systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy. Ibrahim ibn Sinan ibn Thabit ibn Qurra (b. 908), son of Sinan ibn Thabit and grandson of Thabit ibn Qurra, introduced a method of integration more general than that of Archimedes, and was a leading figure in a revival and continuation of Greek higher geometry in the Islamic world. He studied optics and investigated the optical properties of mirrors made from conic sections. 

Ibrahim ibn Sinan, like his grandfather, also studied curves required in the construction of sundials, for the purposes of astronomy, time-keeping and geography, which provided motivations for geometrical and trigonometrical research.

### Abu'l-Hasan al-Uqlidisi
The Indian methods of arithmetic with the Indo-Arabic numerals were originally used with a dust board similar to a blackboard. Arab mathematician Abu'l-Hasan al-Uqlidisi (b. 920) showed how to modify the Indian methods of arithmetic for pen and paper use.

### Abul Wáfa
Persian mathematician Abu'l-Wáfa (940-998) invented the tangent function. The Indo-Arabic system of calculating allowed the extraction of roots by Abu'l-Wáfa.

Abu'l-Wáfa applied spherical geometry to astronomy and also used formulas involving sine and tangent.

### Abu Bakr al-Karaji
Algebra was further developed by Persian mathematician Abu Bakr al-Karaji (953-1029) in his treatise *al-Fakhri*, where he extends the methodology to incorporate integral powers and integral roots of unknown quantities.

Al-Karaji is seen by many as 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, x^2, x^3, ... and \ 1/x, 1/x^2, 1/x^3, ... and to give rules for products of any two of these. He started a school of algebra which flourished for several hundreds of years.

The discovery of the binomial theorem for integer exponents by al-Karaji was a major factor in the development of numerical analysis based on the decimal system.

### Ibn Al-Haitham
Ibn Al-Haitham (b. 965), also known as Alhazen, in his work on number theory, seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form \ 2^{k-1}(2^k - 1) where \ 2^k - 1 is prime.

Al-Haytham is also the first person that we know to state Wilson's theorem, namely that if \ p is prime then \ 1+(p-1)! is divisible by \ p. It is unclear whether he knew how to prove this result. It is called Wilson's theorem because of a comment made by Edward Waring in 1770 that John Wilson had noticed the result. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Joseph Louis Lagrange gave the first proof in 1771 and it should be noticed that it is more than 750 years after al-Haytham before number theory surpasses this achievement of Islamic mathematics.

Al-Haytham also studied optics and investigated the optical properties of mirrors made from conic sections.

### Abu Nasr Mansur
Abu Nasr Mansur ibn Ali ibn Iraq (970-1036) applied spherical geometry to astronomy and also used formulas involving sine and tangent. He is well known for discovering the sine law.

### Abu Sahl al-Kuhi
Persian mathematician Abu Sahl Waijan ibn Rustam al-Quhi (10th century), also known as Abu Sahl al-Kuhi or just Kuhi, was a leading figure in a revival and continuation of Greek higher geometry in the Islamic world. He studied optics and investigated the optical properties of mirrors made from conic sections. He also did some important work on the centers of gravity.

### Al-Biruni
Persian mathematician al-Biruni (b. 973) used the sine formula in both astronomy and in the calculation of longitudes and latitudes of many cities. Both astronomy and geography motivated al-Biruni's extensive studies of projecting a hemisphere onto the plane.

### Al-Baghdadi
Arab mathematician al-Baghdadi (b. 980) looked at a slight variant of Thabit ibn Qurra's theorem of amicable numbers. There were three different types of arithmetic used around this period and, by the end of the 10th century, authors such as al-Baghdadi were writing texts comparing the three numeral systems: Finger-reckoning arithmetic (a system derived from counting on the fingers with the numerals written entirely in words), the sexagesimal numeral system (developed by the Babylonians), and the Indo-Arabic numerals. This third system of calculating allowed most of the advances in numerical methods. Al-Baghdadi also contributed to improvements in the Indo-Arabic decimal system.

### Omar Khayyam
The Persian poet Omar Khayyam (b. 1048) was also a mathematician, and wrote *Discussions of the Difficulties in Euclid*, a book about flaws in Euclid's ''Elements''. He gave a geometric solution to cubic equations, one of the most original discoveries in Islamic mathematics. He was also very influential in calendar reform. He also wrote influential work on Euclid's parallel postulate.

Omar Khayyam gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work:

"*If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared.*"

The Indo-Arabic system of calculating also allowed the extraction of roots by Omar Khayyam. Omar Khayyam also combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means.

### Al-Samawal
Moroccan mathematician Al-Samawal (b. 1130) was an important member of al-Karaji's school of algebra. Al-Samawal was the first to give the new topic of algebra a precise description when he wrote that it was concerned "*with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.*" 

### Sharaf al-Din al-Tusi
Persian mathematician Sharaf al-Din al-Tusi (b. 1135), although almost exactly the same age as al-Samawal, did not follow the general development that came through al-Karaji's school of algebra but rather followed Khayyam's application of algebra to geometry. He wrote a treatise on cubic equations, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.

### Nasir al-Din al-Tusi
Spherical trigonometry was largely developed by Muslims, and systematized (along with plane trigonometry) by Persian Shi'a mathematician Nasir al-Din al-Tusi (1201–1274). He also wrote influential work on Euclid's parallel postulate.

Nasir al-Din al-Tusi, like many other Muslim mathematicians, based his theoretical astronomy on Ptolemy's work, but al-Tusi made the most significant development in the Ptolemaic planetary system until the development of the Nicolaus Copernicus. One of these developments is the Tusi-couple, which was later used in the Copernican model.

### Ibn Al-Banna
Moroccan mathematician ibn al-Banna (b. 1256) used symbols in algebra, though symbols were used by other Islamic mathematicians at least a century before this.

### Al-Farisi
Persian mathematician Al-Farisi (b. 1260) gave a new proof of Thabit ibn Qurra's theorem of amicable numbers, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17,296 and 18,416 which have been attributed to Leonhard Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself. Apart from number theory, his other major contribution to mathematics was on light.

### Ghiyath al-Kashi
Persian mathematician Ghiyath al-Kashi (1380-1429) contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π, which he computed to 16 decimal place of accuracy. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Kashi also developed an algorithm for calculating ''n''th roots, which was a special case of the methods given many centuries later by Ruffini and Horner.

Al-Kashi also produced tables of trigonometric functions as part of his studies of astronomy. His sine tables were correct to 4 sexagesimal digits, which corresponds to approximately 8 decimal places of accuracy. The construction of astronomical instruments such as the astrolabe, invented by Mohammad al-Fazari, was also a speciality of Muslim mathematicians.

### Ulugh Beg
Timurid mathematician Ulugh Beg (1393 or 1394 – 1449), also ruler of the Timurid Empire, produced tables of trigonometric functions as part of his studies of astronomy. His sine and tangent tables were correct to 8 decimal places of accuracy.

### Al-Qalasadi
Moorish mathematician Abu'l Hasan ibn Ali al Qalasadi (b. 1412) used symbols in algebra, though symbols were used by other Islamic mathematicians much earlier.

In the time of the Ottoman Empire (from 15th century onwards) the development of Islamic mathematics became stagnant. This parallels the stagnation of mathematics when the Romans conquerored the Hellenistic world.

### Muhammad Baqir Yazdi
In the 17th century, Muhammad Baqir Yazdi gave the pair of amicable numbers 9,363,584 and 9,437,056 many years before Euler's contribution to amicable numbers.

## Terjemahan
Many Arabic texts on Islamic mathematics were translated into Latin and had an important role in the evolution of later European mathematics. A list of translations, from Greek and Sanskrit to Arabic, and from Arabic to Latin, is given below.

### Bahasa Yunani ke Bahasa Arab
The following mathematical Greek texts on Hellenistic mathematics were translated into Arabic, and subsequently into Latin:
*Euclid's *Data*, *Optics*, *Phaenomena* and *On Divisions*.
*Euclid's *[[Euclid's Elements|Elements]]* by al-Hajjaj (c. 8th century)
*Revision of Euclid's *Elements* by Thabit ibn Qurra.
*Apollonius' *Conics* by Thabit ibn Qurra.
*Ptolemy's *[[Almagest]]* by Thabit ibn Qurra.
*Archimedes' *Sphere and Cylinder* and *Measurement of the Circle* by Thabit ibn Qurra.
*Archimedes' *On triangles* by Sinan ibn Thabit.
*Diophantus' *[[Arithmetica]]* by Abu'l-Wáfa.
*Menelaus of Alexandria' *Sphaerica*.
*Theodosius of Bithynia's *Spherics*.
*Diocles' treatise on mirrors.
*Pappus of Alexandria' work on mechanics.

### Sanskrit to Arabic
The following mathematical Sanskrit texts on Indian mathematics were translated into Arabic, and subsequently into Latin:
**The [[Sindhind]]* by Ibrahim al-Fazari, Muhammad al-Fazari and Yaqub ibn Tāriq (c. 8th century).
**[[Surya Siddhanta]]* by al-Fazari.
*Brahmagupta's *[[Brahmasphutasiddhanta|Brahma Sphuta Siddhanta]]* by al-Fazari.
*Brahmagupta's *Khandakhayaka*.
*Aryabhata's *Aryabhatiya*.
*Aryabhata's *Arya Siddhanta*.
*Varahamihira's *Pancha Siddhanta*.
*Bhaskara I's *Lagu Bhaskariya*.
*Bhaskara II's *Lilavati* (to Persian rather than Arabic).

## Lihat pula
* Ahli sains Muslim

## Pautan luar
* J. J. O'Connor and E. F. Robertson. ''Arabic mathematics : forgotten brilliance?''.
* History of Trigonometry 
* http://www.math.uu.nl/people/hogend/Islamath.html 
* http://www.usc.edu/dept/MSA/introduction/woi_knowledge.html#28