Final answer:
The crucial advances made by Indian mathematicians that are key to modern computers include the development of the decimal system and the concept of zero, as well as accurate calculations of π (pi) and the understanding of negative numbers and algebra.
Step-by-step explanation:
Two fundamental breakthroughs by Indian mathematicians that are crucial for the operation of today's computers are the invention of the decimal system and the concept of zero. The decimal system, including the numeral notations that we use today, was developed in the first centuries CE, and it allowed for a more simplified representation of numbers compared to previous numeral systems. Additionally, the concept of zero, both as a placeholder and as a digit, was considered a significant leap in number theory and is integral to the binary system, which serves as the foundation of computer operations and programming.
Another critical advancement was the understanding and calculations of the value of π (pi) and its use in calculating the circumference of the Earth and astronomical phenomena such as lunar and solar eclipses. This level of precision in mathematical computation and understanding of geometrical concepts are essential in the development and functioning of computers, particularly in the realm of computer graphics, simulations, and scientific computations. Moreover, Indian mathematicians like Brahmagupta had pioneered the use of multiplication, division, and negative numbers, advancing the broader field of algebra that underpins computer algorithms.