A hyperdual number is an extension of the real numbers that includes two infinitesimal components. It is expressed in the form a + b \epsilon + c \eta , where a and b are real numbers, and \epsilon and \eta are infinitesimals that satisfy the properties \epsilon^2 = 0 and \eta^2 = 0 . This structure allows for the representation of derivatives and other mathematical concepts in a more flexible way. Hyperdual numbers are particularly useful in automatic differentiation, a technique used in computational mathematics and machine learning. By incorporating hyperdual numbers, one can efficiently compute derivatives of functions without the need for traditional numerical methods. This makes them valuable in optimization problems and simulations, where precise gradient information is essential for performance.