Fixed point arithmetic

In mathematics, a fixed point is a value that remains unchanged under the application of a given function. Formally, for a function f mapping a set to itself, a fixed point is an element x such that f(x) = x. Fixed points appear across many areas of mathematics, including algebra, analysis, topology, and applied fields. Examples include solutions to equations, equilibrium states in dynamical systems, and steady states in iterative processes. Important results related to fixed points include the Banach Fixed-Point Theorem, which guarantees the existence and uniqueness of fixed points for contraction mappings in complete metric spaces, and Brouwer’s Fixed-Point Theorem, which states that any continuous function from a compact convex set to itself has at least one fixed point.