Hyperreal numbers are an extension of the real number system that includes infinitesimal and infinite quantities. They allow for a rigorous way to handle concepts of calculus, such as limits and continuity, by providing numbers that are smaller than any positive real number or larger than any real number. Developed in the 1960s by mathematician Abraham Robinson, hyperreal numbers are used in non-standard analysis. This framework helps mathematicians and scientists explore mathematical concepts in a more intuitive way, making it easier to understand and apply calculus in various fields, including physics and engineering.