Jamshid al-Kashi

Master of the number pi · 1380–1429

Jamshid al-Kashi

Al-Kashi (Persian: الكاشي), or by his full name Ghiyath al-Din Jamshid ibn Mas'ud al-Kashi (Arabic: غياث الدين جمشید بن مسعود بن محمد الكاشي, b. 1380, Kashan – d. 1429, Samarkand), was a physician, mathematician, and astronomer born in Kashan in the latter half of the 14th century.

Life story

His dates of birth and death are not known with certainty. He completed his education in Kashan, went to Samarkand at the invitation of Ulugh Beg, and continued his work there. Al-Kashi, who conducted studies in mathematics and astronomy, was the first person to use the decimal system in arithmetic. By recomputing the tables in the zij titled the Ilkhanic Zij, which contained the observations made at the Maragha Observatory, he wrote his work titled the Khaqani Zij, thereby completing the Ilkhanic Zij; and in his work titled Sullam al-Sama he discussed the problem of the distances of the celestial bodies.

Ghiyath al-Din Jamshid al-Kashi's most important work is the book titled The Key of Arithmetic (Miftah al-Hisab), in which he set out in all its aspects the mathematical knowledge of the medieval Islamic world; in one section of this work he examined decimal fractions from a theoretical standpoint and showed with examples how arithmetical operations such as addition, subtraction, multiplication, and division are performed with these fractions; the knowledge he provided here would later be used by Taqi al-Din (Arabic: تقي الدين محمد بن معروف الشامي السعدي, Taqī al-Dīn Abū Bakr Muhammad ibn Qādhī Ma'rūf ibn Ahmad al-Shāmī al-'Asadī al-Rāsid), one of the famous Ottoman mathematicians and astronomers of the 16th century, and would be developed by being applied to trigonometry and astronomy.

In order to determine sin 1° in accordance with method, Ghiyath al-Din Jamshid al-Kashi found the following solution, which was later frequently used by the French mathematician François Viète in the 16th century.

{\displaystyle \ sin3\phi =3sin\phi -4sin^{3}\phi }

The number Pi

Using the method of nested polygons that Archimedes had proposed for calculating the number pi, Ghiyath al-Din Jamshid went as far as the 14th place after the decimal point and became the person of his time who had calculated the number pi most accurately. Until that day, the best result had been reached by Zu Chongzhi, up to the 6th place. This record remained with al-Kashi for as long as 180 years. After 180 years it was broken by Adriaan van Roomen, again with the same method, at 15 (or 17) places after the decimal point.

Sources

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