Concept:Numerical analysis is a branch of mathematics that focuses on designing and applying algorithms to obtain approximate solutions for problems that cannot be solved exactly. The question asks which two Indian mathematicians are recognised as pioneers or founders of this field.
Explanation:Step 1: Identify the mathematicians who contributed to early numerical methods and approximations. Bhaskaracharya (Bhaskara II) made major advances in algebra and calculus, including early ideas of differential calculus. Aryabhatta developed important numerical methods and approximations, such as his work on the approximation of pi and the sine function. Both are considered founders of numerical analysis.
Step 2: Eliminate the others. Ramanujan was a brilliant mathematician, but his primary contributions were in number theory, infinite series, and modular forms—not in founding numerical analysis. Varahamihira is best known for his work in astronomy and astrology, not for founding numerical analysis.
Conclusion: Therefore, the correct pair is Bhaskaracharya (ii) and Aryabhatta (iv).
Answer:Option B: (ii) and (iv)