Jamshid al-Kashi: One of the Best Mathematicians in the Islamic World

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Introduction:
“You can learn more from solving one problem in many different ways than you can from solving many different problems, each in only one way.” Islamic civilization in the middle ages, like all of Europe, had a dichotomy between theoretical and practical mathematics. Practical mathematics was the common subject, “whereas theoretical and argumentative mathematics were reserved for specialists” (Abedljaouad, 2006, p. 629). Between the eighth and the fifteenth centuries, Islamic civilization produced a series of remarkable mathematicians. Among them was Ghiyath al-Din Jamshid Mas’ud al-Kashi. Following this dichotomy, al-Kashi designed his book for use by students who were looking to apply mathematics in their professions. The book does not contain any theoretical proof for any problem, but it does contain methods for solution and correctness verification, such as performing the opposite operation to check a result, and the method of casting out nines to check whether the product, quotient, or root is correct. Objective of the study:

1.Life History of Ghiyath Ai-Din Jamshid Mas'ud Al-Kashi
2.Contribution in Mathematics
3.Multiples Algorithm and Multiple Solutions
4.Law of Cosines
5.Fixed Point Iteration Method
6.Calculation of PI

1.Life History of Jamshid al-Kashi
Al-Kashi was one of the best mathematicians in the Islamic world. He was born in 1380, in Kashan, in central Iran. This region was controlled by Tamurlane, better known as Timur, who was more interested in invading other areas than taking care of what he had. Due to this, al-Kashi lived in poverty during his childhood and the beginning years of his adulthood. He was born in Kashan which lies in a desert at the eastern foot of the Central Iranian Range. At the time that al-Kashi was growing up Timur (often known as Tamburlaine) was conquering large regions. He had proclaimed himself sovereign and restorer of the Mongol empire at Samarkand in 1370 and, in 1383, Timur began his conquests in Persia with the capture of Herat. Timur died in 1405 and his empire was divided between his two sons, one of whom was Shah Rokh. While Timur was undertaking his military campaigns, conditions were very difficult with widespread poverty. al-Kashi lived in poverty, like so many others at this time, and devoted himself to astronomy and mathematics while moving from town to town. Conditions improved markedly when Shah Rokh took over after his father's death. He brought economic prosperity to the region and strongly supported artistic and intellectual life. With the changing atmosphere, al-Kashi's life also improved markedly. The first event in al-Kashi's life which we can date accurately is his observation of an eclipse of the moon which he made in Kashan on 2 June 1406. It is reasonable to assume that al-Kashi remained in Kashan where he worked on astronomical texts. He was certainly in his home town on 1 March 1407 when he completed Sullam Al-sama the text of which has survived. The full title of the work means The Stairway of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes (of the heavenly bodies). At this time it was necessary for scientists to obtain patronage from their kings, princes or rulers. Al-Kashi played this card to his advantage and brought himself into favour in the new era where patronage of the arts and sciences became popular. His Compendium of the Science of Astronomy written during 1410-11 was dedicated to one of the descendants of the ruling Timurid dynasty. Samarkand, in Uzbekistan, is one of the oldest cities of Central Asia. The city became the capital of Timur's empire and Shah Rokh made his own son, Ulugh Beg, ruler of the city. Ulugh Beg, himself a great scientist, began to build the city into a great cultural centre. It was to Ulugh Beg that Al-Kashi dedicated his important book of astronomical tables Khaqani Zij which was based on the tables of Nasir...
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