👉 Contrary math, also known as non-standard analysis, challenges the conventional real number system by introducing infinitesimals—numbers smaller than any positive real number but not zero. Unlike standard calculus, which relies on limits and rigorous epsilon-delta definitions to handle infinitesimals indirectly, contrary math allows direct manipulation of these infinitesimal quantities. For example, instead of proving a function \( f(x) \) is continuous at \( x = a \) by showing that for any positive \( \epsilon \), there exists a \( \delta \) such that \( |x - a| < \delta \) implies \( |f(x) - f(a)| < \epsilon \), contrary math might define a "continuity" condition directly using infinitesimals, where \( f(a + \epsilon) - f(a) \) is infinitesimal for any infinitesimal \( \epsilon \). This approach simplifies proofs but requires accepting the existence of non-standard numbers, which are not part of the standard real line. While powerful for certain applications, it diverges from the foundational rigor of standard analysis, making its results context-dependent and less universally applicable.
