**Mathematical developments in ancient India: **Mathematics in India and literary works of Kalidasa, Sudraka and other writers flourished almost at the same time. Aryabhata led the development of Mathematics in India. After Aryabhata, mathematics in India was developed by Brahmagupt and Bhaskaracharya.

**Aryabhata (476—550AD):**He was 5^{th}century Indian mathematician who wrote Aryabhatiya. Aryabhatiya is a Sanskrit text having four sections. They are Gitikapada, Ganitapada, Kalakriyapada and Golapada. Aryabhatiya contains arithmetic, algebra and trigonometry. Aryabhata worked on approximation of Pi, gave a formula for the area of triangle and concept of sine. He used the word ardhajya for sine. Aryabhata's works were influenced by development in the mathematical system in Greece, Babylon and Rome.**Brahmagupt (598****‐****668 AD):**His work is known as Brahm Sputa Siddhantika. He first used negative numbers and operations on zero. His Brahm Sputa Siddhantika played a key role in familiarizing Arabs with a mathematical system in India.**Bhaskaracharya (12**His book is Siddanta Shiromani. He started the Chakrawat method for the solution of algebraic equations. European mathematicians rediscovered the Chakrawat method in the 18^{th}century):^{th}century and described it as an inverse cycle. Lilavati section of Siddanta Shiromani was converted into English.

**Contributions of mathematical developments in India to world:**

Mathematical developments in India contributed to the progress of mathematics in other areas of the world. Mathematical understanding in India reached to Greece and Rome. Gautama became the head of the official Chinese Board of Astronomy in the 8^{th} century. He was an Indian mathematician. Mathematical developments in India also spread to Arabs. From Arabs, mathematical developments in India reached Europe and contributed to mathematical progress between the 8^{th} and 11^{th} centuries. Arabs transmitted the decimal system to Europe. Decimal system was developed in India.

**Trigonometry in ancient India and external influence: **In the first century BC, trigonometry was applied to astronomy in the Surya Siddhanta, a Sanskrit text in Indian astronomy. Greek mathematicians like Euclid and Archimedes worked on trigonometry in the 3rd century BC. Hipparchus in Asia Minor worked on trigonometry in 4^{th} century. Trigonometry in the Surya Siddhanta was more advanced than the influences brought by Alexander and Greek settlers.

**Development of modern trigonometric word sine from Aryabhata’s jya:** In the 5th century, Aryabhata used the word jya-ardha for the concept of sine. Short form of jya-ardha was jya. In Arabic translations, jya was replaced by Arabic word jaib. In Latin translation, the Arabic word jaib was translated into Latin word sinus. So, Aryabhata’s concept of jya led to modern trigonometric word sine through Arabic and Latin translations.

**Mains Question:**

**Comment on mathematical developments in ancient India and their spread to other areas of the world. How Aryabhata’s jya got converted to modern trigonometric word sine?**

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