Islamic Mathematics. Aybars Hallik. 23 October, 2017

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1 Islamic Mathematics Aybars Hallik 23 October, 2017 Islamic Mathematics is the term used to refer to the mathematics done in the Islamic world between the 8th and 13th centuries CE. Mathematics from the medieval Middle East is very important to the mathematics we use today. While Europe endured its Dark Ages, the Middle East preserved and expanded the arithmetic, geometry, trigonometry, and astronomy from the ancient Greek philosophers, such as Euclid. The most important contribution may be the invention of algebra, which originated in Baghdad in the House of Wisdom. House of Wisdom The House of Wisdom was primarily a library and a place for translation and research. Scholars would work here in translating Greek and Hindu treatises to Arabic, and also conducted their own research and wrote original treatises. The House of Wisdom was established in the early 9th century, by Caliph alrashid. His son, Caliph al-ma mun, was the ruler who made the House of Wisdom so important. Al-Ma mun had a dream in which Aristotle appeared to him; after this dream, alma mun wanted to translate as many Greek manuscripts as he could! He commissioned scholars to begin translating Greek, Hindu, Syriac-Persian, and Hebrew texts into Arabic. Most of these texts dealt with philosophy or mathematics and science. The Use of Mathematics In order to observe holy days on the Islamic calendar (times determined by the phases of the moon), astronomers initially used Ptolemy's method to calculate the place of the moon and stars. Islamic months do not begin at the astronomical new moon, instead they begin when the thin crescent moon is first sighted in the western evening sky. The Qur'an says: They ask you about the waxing and waning phases of the crescent moons, say they are to mark fixed times for mankind and Hajj. This led Muslims to find the phases of the moon in the sky, leading to new mathematical calculations. Predicting just when the crescent moon would become visible was a test for the Islamic mathematical astronomers. To predict the first visibility of the moon, it was essential to express its motion according to the horizon, and this problem demands pretty complicated spherical geometry. However, finding the direction of Mecca and knowing the specific times for prayer motivated the Muslims to study and develop knowledge of spherical geometry.

2 Islamic Scholars & Their Contributions 1 ) Al-Khwārizmī ( CE) Muhammad ibn Mūsā al-khwārizmī is probably the most famous Muslim mathematician. He lived about CE. AlKhwārizmī was born in Qutrubull, an area near Baghdad between the Tigris and Euphrates rives, but was brought to work at the House of Wisdom by the Caliph al-ma mun. He popularized a number of mathematical concepts, including the use of HinduArabic numbers and the number zero, algebra, and the use of geometry to demonstrate and prove algebraic results. Many of his works deal with astronomy, but he also wrote about the Jewish calendar, arithmetic, and algebra. Arithmetic Hindu-Arabic Numbers Arabic-language Numbers Al- Khwārizmī wrote a very important treatise on Hindu-Arabic numerals, which made the use of these numbers popular. The introduction of the number zero was especially important for mathematics, and the number 0 was used for about 250 years throughout the Islamic world before Europe ever heard of it! He also introduced the Hindu concept of decimal positioning notation to the Arab and European worlds, which we still use today! Algebraic Operations Al-Khwārizmī wrote a treatise entitled Kitab aljabr wa l-muqabalah. The treatise actually had a very practical reason behind it: the longest chapter of the treatise teaches people how to apply algebra to Islamic inheritance laws! The words al-jabr and al-muqabalah were operations used by Al- Khwārizmī, much like addition, subtraction, multiplication, and division. Al-jabr means something like restoration or completion, and was the operation used to add equal terms to both sides of an equation to get rid of a negative term. For example, with the equation al-khwārizmī uses al-jabr to add to both sides of the equation, getting the result:

3 He can then complete the problem by division Though we now know x = 0 & 8, Al-Khwārizmī never allows a variable to equal zero. Al-muqabalah means something like balancing, and was the operation used to subtract equal terms from both sides of an equation. For example, al-khwārizmī has the equation so he uses al-muqabalah to subtract 29 from each side, getting the result: From here, al-khwārizmī can then complete the problem: As you can see, al-muqabalah and al-jabr were operations defined by al-khwārizmī which we still use today, though we don t call them the same thing! His operation al-jabr, adding equal amounts to both sides of the equation, is where our word algebra comes from! 6 Chapters of Al-jabr Squares equal its roots Squares equal a number Roots equal a number Squares & roots equal a number Squares & numbers equal roots Roots & numbers equal squares 2 ) Thābit ibn Qurra al-harrānī ( CE) Thābit ibn Qurra followed al-khwārizmī's general solutions; however, al-khwārizmī presents his general proofs in conjunction with particular equations, whereas ibn Qurra presents his demonstrations in general. At this point, ibn Qurra had full access to Euclid's Elements, and freely used Euclid's theorems in his algebraic proofs. Ibn Qurra also correctly solved the quadratic equation x^2 + px = q (Berggren). He follows his demonstrations with general proofs, following Euclid's examples of the definition-theorem-proof model.

4 3 ) Al-Battani ( CE) Muhammad Ibn Jabir Ibn Sinan Abu Abdullah, the father of trigonometry, was born in Battan, Mesopotamia and died in Damascus in 929 CE. An Arab prince and governor of Syria, he is considered to be the greatest Muslim astronomer and mathematician. Al-Battani raised trigonometry to higher levels and computed the first table of cotangents. 4 ) Al-Biruni ( CE) Al-Biruni was among those who laid the foundation for modern trigonometry. He was a philosopher, geographer, astronomer, physicist and mathematician. Six hundred years before Galileo, Al-Biruni discussed the theory of the earth rotating about its own axis.al-biruni carried out geodesic measurements and determined the earth's circumference in a most ingenious way. With the aid of mathematics, he enabled the direction of the Qibla to be determined from anywhere in the world. In the domain of trigonometry, the theory of the functions; sine, cosine, and tangent was developed by Muslim scholars of the tenth century. Muslim scholars worked diligently in the development of plane and spherical trigonometry. The, trigonometry of Muslims is based on Ptolemy's theorem but is superior in two important respects: it employs the sine where Ptolemy used the chord and is in algebraic instead of geometric form. 5 ) Omar Khayyam ( CE) Khayyam s most famous works include his highly influential mathematical treatise called Treatise on Demonstration of Problems of Algebra which he completed in This treatise highlighted the basic algebraic principles that were ultimately shifted to Europe. He laid the foundation of the Pascal s triangle with his work on triangular array of binomial coefficients. In 1077 another major work was written by Khayyam namely Sharh ma ashkala min musadarat kitab Uqlidis meaning Explanations of the Difficulties in the Postulates of Euclid. It was published in English as On the Difficulties of Euclid s Definitions. In this book he contributed to non-euclidean geometry even though this was not his original plan. It is said that Omar Khayyam was originally trying to prove the parallels postulate when he proven the properties of figures in the non-euclidean geometry. His geometrical work consisted of his efforts on the theory of proportion and geometrical algebra topics such as cubic equations. Khayyam was the first mathematican to consider the Saccheri quadrilateral in the 11th century. It was mentioned in his book the Explanations of the difficulties in the postulates of Euclid. It wasn t until 6 centuries later when another mathematician, Giordano Vitale made further advances on Khayyam s theory. Other books by Khayyam include his book named Problems of Arithmetic, a book on music and algebra. Other Contributions Al-Hajjaj ibn Yusuf ibn Matar ( ) Translated Euclid s Elements into Arabic.

5 Al- Abbas ibn Sa id al-jawhari ( ) Mathematician who worked at House of Wisdom; most important work: Commentary on Euclid s Element (contained 50 additional propositions and attempted proof of the parallel postulate) Abd al-hamid ibn Turk (830) Wrote a work on algebra (only 1 chapter of quadratic equations survived) Ya qub ibn Ishaq al-kindi ( ) Contributions to mathematics include many works on arithmetic and geometry Banu Musa ( ) Three brothers in Baghdad ; most famous mathematical treatise : The Book of the Measurement of Plane and Spherical Figures ; The eldest, Ja far Muhammad (c.800) specialized in mechanics and wrote on mechanics ; The youngest, al-hasan (c.810) specialized in geometry and wrote The elongated circular figure. Ikhwan al-safa (First half of 10th century) Group wrote series 50+ letters on science, philosophy and theology. The first letter is on arithmetic and number theory, the second letter on geometry. Labana Cordoba (Spain, ca. 10th century) Islamic female mathematicians & secretary of the Umayyad Caliph al-hakem II; could solve the most complex geometrical and algebraic problems known in her time. Al-Hassar (ca.1100s) Developed the modern mathematical notation for fractions and the digits he uses for the ghubar numerals also closely resembles modern Western Arabic numerals. Ibn al-yasamin (ca.1100s) First to develop a mathematical notation for algebra

6 Abu al-hasan ibn Ali al-qalasadi ( ) Last major medieval Arab mathematician; Pioneer of symbolic algebra References [1] The Center for South Asian and Middle Eastern Studies, University of Illinois at UrbanaChampaign < [2] Islamic Mathematics - The Story of Mathematics < [3]Omar Khayyam < [4]Algebra Historical Development <

ISLAMIC MATHEMATICS. Buket Çökelek. 23 October 2017

ISLAMIC MATHEMATICS. Buket Çökelek. 23 October 2017 ISLAMIC MATHEMATICS Buket Çökelek 23 October 2017 Islamic Mathematics is the term used to refer to the mathematics done in the Islamic world between the 8th and 13th centuries CE. Mathematics from the