The achievements of the Gupta Empire in mathematics included the invention of the numeral zero and the development of the decimal system. These innovations were crucial because they provided a more efficient way to represent numbers and perform calculations, making higher-level mathematics more manageable and systematic. The introduction of zero as a placeholder allowed for the representation of large numbers and facilitated operations such as subtraction and addition, which are foundational for advanced mathematical concepts. Therefore, the correct response would be:
They invented the numeral zero and the decimal system of numbers.