Which of the following Indian mathematicians are known as founders of ‘numerical analysis’?
(i)Ramanujan $\quad$ (ii) Bhaskaracharya
(iii)Varahmihir $\quad$ (iv) Aryabhatta

Question

Accepted Answer

Option 1, which includes Ramanujan and Varahmihir, is identified as the correct answer for the question regarding the founders of 'numerical analysis' among Indian mathematicians. Here is a detailed explanation supporting this choice:

1. **Ramanujan**: Srinivasa Ramanujan is one of the most renowned Indian mathematicians, known for his substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions. His work laid the groundwork for various areas of mathematics, including numerical analysis. Ramanujan's intuitive grasp of numbers and his ability to derive results that were later proven using rigorous mathematical proofs highlight his foundational role in numerical analysis.

2. **Varahmihir**: Although primarily known for his contributions to astronomy and astrology, Varahmihir's work also encompassed mathematical principles that are fundamental to numerical analysis. His texts, such as the "Pancha-Siddhantika," include mathematical calculations and methods that are essential for numerical computations, thereby establishing his role in the early development of numerical analysis.

Now, let's examine why the other options are incorrect:

- **Option 2 (ii) and (iv)**: Bhaskaracharya (Bhaskara I and Bhaskara II) and Aryabhatta are indeed significant figures in Indian mathematics. Bhaskaracharya made notable contributions to calculus and algebra, while Aryabhatta is known for his work in astronomy and mathematics, including the approximation of pi. However, their primary contributions are not directly aligned with the foundational aspects of numerical analysis as highlighted in the context of this question.

- **Option 3 (ii) and (iii)**: This option includes Bhaskaracharya and Varahmihir. While Varahmihir's contributions to numerical methods are acknowledged, Bhaskaracharya's work, although extensive in other areas, does not specifically focus on the foundational aspects of numerical analysis as required by the question.

- **Option 4 (i) and (iv)**: This option pairs Ramanujan with Aryabhatta. While Ramanujan's contributions to numerical analysis are significant, Aryabhatta's work, although pioneering in many respects, does not specifically emphasize numerical analysis in the same way as Varahmihir's contributions do.

In conclusion, Option 1 is the correct answer because it accurately identifies Ramanujan and Varahmihir as key figures whose work laid the groundwork for the development of numerical analysis in Indian mathematics.