Indian mathematicians have made substantial contributions to the development of mathematical concepts across different historical periods. Their work spans a wide range of mathematical fields, including arithmetic, algebra, geometry, trigonometry, and even early glimpses of calculus.

1. Number System and Zero:

• Indian Numerals: The decimal place-value system, including the concept of zero (shunya), originated in ancient India. This numeral system, transmitted to the West through Arab scholars, forms the basis of the modern numerical system used globally.
• Aryabhata I (476 CE - 550 CE): Aryabhata introduced the symbol for zero and developed rules for arithmetic operations using zero in his work "Aryabhatiya."

2. Algebraic Concepts:

• Brahmagupta (598 CE - 668 CE): Brahmagupta made significant contributions to algebra, providing rules for solving linear and quadratic equations, including the Brahmagupta-Fibonacci identity.
• Aryabhata I: Aryabhata formulated methods for solving linear and quadratic indeterminate equations, a precursor to Diophantine equations.

3. Geometry:

• Baudhayana (800 BCE): Baudhayana's "Baudhayana Sulba Sutra" contains geometric constructions and concepts, including a statement of the Pythagorean theorem.
• Bhaskara I (600 CE - 680 CE): Bhaskara I provided geometric methods for finding the area of various shapes and introduced the concept of chakravala, an algorithm for solving indeterminate quadratic equations.

4. Trigonometry:

• Aryabhata I: Aryabhata's "Aryabhatiya" included trigonometric concepts and methods for computing sine and cosine values, laying the foundation for later developments.
• Madhava of Sangamagrama (14th century): Madhava of the Kerala School developed infinite series expansions for trigonometric functions, anticipating concepts of calculus.

5. Calculus Precursors:

• Madhava, Nilakantha Somayaji, and the Kerala School (14th - 16th centuries): The Kerala School mathematicians made significant strides in calculus-like concepts. They developed infinite series expansions for trigonometric functions and introduced the notion of differentiation and integration.

6. Astronomy and Mathematical Techniques:

• Aryabhata I: Aryabhata's work in "Aryabhatiya" included mathematical techniques for astronomy, such as calculating the positions of planets and eclipses.
• Brahmagupta: Brahmagupta's "Brahmasphutasiddhanta" contains mathematical techniques applied to astronomy, including methods for calculating longitudes.

7. Legacy and Transmission:

• Transmission to the Islamic World: Indian mathematical concepts, including the numeral system and algebraic ideas, were transmitted to the Islamic world through translations of Indian texts. Arab scholars played a crucial role in preserving and disseminating these mathematical ideas.
• Influence on European Mathematics: Indian mathematical concepts, particularly the numeral system and zero, were introduced to Europe through translations of Arabic texts. This influence laid the groundwork for the development of European mathematics during the medieval period.

8. Sulba Sutras and Geometry:

• Ancient Sulba Sutras: The Sulba Sutras contained geometric rules for constructing altars, showcasing advanced knowledge of geometry, including the Pythagorean theorem.

9. Notable Mathematicians and Works:

• Aryabhata II (920 CE - 1000 CE): Aryabhata II contributed to the field of algebra and provided methods for solving quadratic equations with multiple variables.
• Nilakantha Somayaji (1444 CE - 1544 CE): Nilakantha Somayaji made contributions to infinite series and trigonometry, particularly with his work on the "Tantrasangraha."

The contributions of Indian mathematicians have left an enduring impact on the global development of mathematical concepts. Through their innovative ideas and practical applications, these mathematicians laid the groundwork for subsequent mathematical advancements, influencing not only the mathematical traditions of the Indian subcontinent but also contributing significantly to the broader world of mathematics.