History of Mathematics: The Medieval Era PDF

History of Mathematics: The Medieval Era Islamic World: The Golden Age Prof. Dr. Deniz Karlı Department of Mathematics IŞIK UNIVERSITY Version #: 28–11–2023 Scientific progress in Islamic World starts at around 750 AD and continues through the Golden Age of Islam until 1250 AD. Abbasid Caliphs establ...

History of Mathematics: The Medieval Era Islamic World: The Golden Age Prof. Dr. Deniz Karlı Department of Mathematics IŞIK UNIVERSITY Version #: 28–11–2023 Scientific progress in Islamic World starts at around 750 AD and continues through the Golden Age of Islam until 1250 AD. Abbasid Caliphs established a very large library, called House of Wisdom, in the late 8th century. A big movement of translation of Greek, Indian, Persian (and many other languages) books into Arabic began in House of Wisdom. In two centuries, a very large collection was formed, which was a large contributing factor to the growth of scientific knowledge during the golden age of Islamic science. Mathematics in this era was a synthesis of Greek, Indian and Mesopotamian influences. Number systems, arithmetic, trigonometry, and algebra were under the influence of Mesopotamian and Indian Mathematics whereas Geometry was under Greek influence. There are nearly 50 Mathematician who lived in this era and whose work survived until today. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 4 Muhammad ibn Musa al-Khwarizmi (780 - 850 AD) Al-Khwarizmi was a Persian mathematician who produced vastly influential works in mathematics, astronomy, and geography. He became a librarian in the House of Wisdom after 810. There he wrote 4 books on: Geography, Astronomy, Arithmetic and Algebra. His most famous book ”The Compendious Book on Calculation by Completion and Balancing” was translated into Latin in 1440, and was used as a textbook until 1600’s. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 5 The Compendious Book on Calculation by Completion and Balancing This book contains examples and applications to a wide range of problems in trade, surveying and legal inheritance. Al-Khwarizmi presented the first systematic solution of linear and quadratic equations. Al-Khwarizmi classifies a second degree equation ax2 + bx + c = 0 into 6 classes and shows an algorithmic approach for solution in each case. These cases are: (where b and c are positive integers) * squares equal roots (ax2 = bx) * squares and roots equal number (ax2 + bx = c) * squares equal number (ax2 = c) * squares and number equal roots (ax2 + c = bx) * roots equal number (bx = c) * roots and number equal squares (bx + c = ax2 ) Today this method is called the algorithmic approach. The word ”Algorithm” comes from the (Latinized) name ”Al-Khwarizmi”. Algebra became an area in Mathematics after this book. The name ”Algebra” originated from this book. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı The Book on Arithmetic: Algorithmo de Numero Indorum Al-Khwarizmi’s second most influential work was on the subject of arithmetic, which survived in Latin translations but lost in the original Arabic. In this book, he codified the various Indian numerals, introduced the decimal positional number system to the Western world. He explains how to use modern number representation (0,1,2,...) in arithmetic. However, 0 is not a number here, rather it is a symbol to fill blanks. Zero is used in India in 876 for the first time. 6 IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 7 Omar Khayyam (1048 - 1131 AD) Omar Khayyam was a mathematician, astronomer, philosopher, and poet. He was born in Neyshabur, in northeastern Persia. Khayyam entered the service of Malik-Shah in 1074 and commissioned to set up an observatory in Isfahan and lead a group of scientists in carrying out precise astronomical observations aimed at the revision of the Persian calendar. He designed the Jalali calendar, a solar calendar with a very precise 33-year cycle. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı Omar Khayyam Khayyam computed the length of a year as 365.24219858156 years based on observation and measurement. Today, 1 year is assumed to be 365.242190 years where last digit is modified every 70-80 years. He classified 3rd degree polynomials (ax3 + bx2 + cx + e = 0) and found their roots using geometric approach by means of conic sections. Khayyam also observed that a cubic (3rd degree polynomial may have more than 1 root.) 8 IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 9 Sharaf al-Din al-Tusi (1135 - 1213 AD) Al-Tusi was a Persian mathematician and astronomer. Very little known about his life. He was born in Tus, Iran. He taught Mathematics in Damascus, Aleppo, Baghdad and Mosul. By some others, he was assumed to be outstanding in geometry and the mathematical sciences, having no equal in his time. Al-Tusi has been credited with proposing the idea of a function, however his approach being not very explicit. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 10 Sharaf al-Din al-Tusi Al-Tusi worked on finding roots of cubic polynomials. He divided equations of degree at most three into 25 different types. First al-Tusi discusses twelve types of equation of degree at most two. He then looks at eight types of cubic equation which always have a positive solution, then five types which may have no positive solution. In his work, al-Tusi, remakably, uses a method which we call discriminant, D, of polynomial today. He analyses cases where D > 0, D = 0, and D < 0. He states that for a cubic equation of the form x3 − cx = e to have solution in a certain interval the value e must be between between maximum and minimum of x3 − cx. He finds max and min by finding derivative! al-Tusi’s use of the derivative of a function, without of course saying so, is very interesting. For some mathematicians, this is the first invention of derivative. However it was not understood at his time. Derivative was properly defined 400 years later. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 11 Nasir al-Din al-Tusi (1201 - 1274 AD) He is a very famous Persian mathematician, architect, philosopher, physician, scientist, and theologian. One of the greatest minds of Islamic Golden Age. Nasir al-Din al-Tusi is a student of a student of Sharaf al-Din al-Tusi. He was the first to write a work on trigonometry independently of astronomy. So trigonometry became an independent area of Mathematics rather than a tool for Astronomy. He prepared Zij-i Ilkhani, a Zij book with astronomical tables of planetary movements. N. Al-Tusi became the most cited scientist of Islamic Age by western world. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 12 Jamshid al-Kashi (1380 - 1429 AD) Al-Kashi was born in 1380, in Kashan, in central Iran. Al-Kashi’s Theorem In 1409, Ulugh Beg, a Timurid sultan, as well as an astronomer and mathematician, founded an institute in Samarkand which soon became a prominent university. Students from all over the Middle East and beyond, flocked to this academy. Consequently, Ulugh Beg gathered many great mathematicians and scientists of the Middle East. Around 60 best scientist of this time were teaching and doing research here. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı Jamshid al-Kashi (1380 - 1429 AD) Al-Kashi produced a Zij entitled the Khaqani Zij, which was based on Nasir al-Din al-Tusi’s earlier Zij-i Ilkhani. In this Zij, he calculated sinus values of angles from 1 to 90 degrees where each degree is divided into 60 minutes. This means there are 60 · 90 = 5400 entries in this table. Each value is calculated up to 8th decimal. He also calculated the number π up to 16th decimal by inserting a polygon with 805,306,368 edges inside a circle. This record was not broken for 200 years. six a sina.ee 13