History and applications
The notion of an infinitesimal essentially goes back to Archimedes, but became popular as a means to explain calculus. An infinitesimal was thought of as an infinitely small but non-zero quantity. Bishop Berkeley (1685–1753) described them as the `ghosts of vanished quantities' and was opposed to their use.
Unfortunately, such quantities do not exist in the real number system, although the concept may be useful for discovering facts that can then be made precise using limits.
The real number system is an example of an Archimedean system: given any real number \(\alpha\), there is an integer \(n\) such that \(n\alpha >1\). This precludes the existence of infinitesimals in the real number system.
Non-archimedean systems can be defined, which contain elements which do not have this property. Indeed, all of calculus can be done using a system of mathematics known as non-standard analysis, which contains both infinitesimals and infinite numbers.