Non-standard analysis is a branch of mathematical analysis that involves using non-standard models of the real number system to carry out rigorous mathematical reasoning. Synonyms for non-standard analysis include hyperreal analysis, non-Archimedean analysis, and non-standard calculus. These terms describe the same approach to mathematical analysis and are often used interchangeably. Hyperreal analysis emphasizes the concept of infinitesimals, while non-Archimedean analysis highlights the property that there is no bound on the distance between two non-zero numbers. Non-standard calculus emphasizes the use of non-standard models of the real number system to carry out calculus operations. No matter which term is used, non-standard analysis provides a powerful tool for mathematical research and applications.