To answer the question regarding which mathematician was the first to explain that the square root of a negative number does not exist, we need to delve into the contributions of each option provided:

1. Mahavira: An ancient Indian mathematician from around the 9th century, Mahavira is known for his work in the field of algebra, particularly in South India. In his mathematical treatise, Ganita Sarasangraha, Mahavira clearly mentioned that the square root of a negative number is not possible, laying an early groundwork for the understanding of complex numbers, which was later developed by mathematicians in the Western world.
2. Aryabhatta: A prominent ancient Indian mathematician and astronomer, Aryabhatta is renowned for his work Aryabhatiya. He made significant contributions to mathematics and astronomy, but he did not address the non-existence of the square root of negative numbers.
3. Ramanujan: A 20th-century mathematical genius, Ramanujan made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions. While he explored complex numbers extensively, he was not the first to comment on the square root of negative numbers.
4. Baudhayana: An ancient Indian mathematician known for the Baudhayana Sulbasutra, which contains the theorem analogous to Pythagorean theorem. His works predominantly focused on geometry and construction-related mathematics.

Based on the analysis, it is clear that the correct answer is Mahavira. He was the first Indian mathematician to explicitly state that the square root of a negative number does not exist.

In conclusion, Mahavira’s exploration into this aspect of mathematics was a significant early step in understanding equations and the concept of numbers, influencing both Indian and later, Western mathematics.