Realvalued

Real-valued describes objects whose outputs are real numbers. In mathematics, a real-valued function is a function f from a domain X to the real numbers, written as f: X -> R. A real-valued sequence has each term in R, and a real-valued random variable is a measurable function from a probability space to R. The term is used to distinguish these objects from complex-valued ones, which map into the complex numbers, or from vector-valued objects that take values in higher-dimensional spaces.

Real-valued objects appear across many areas. Examples include real-valued functions such as f(x) = x^2, which maps real inputs to nonnegative real outputs, and trigonometric functions like sin(x) with real-valued outputs for real x. Real-valued functions are studied for properties such as continuity, differentiability, integrability, and measurability, as well as convergence concepts like pointwise and uniform convergence. In probability, real-valued random variables allow descriptions of expectation, variance, and distribution functions.

In applied contexts, real-valued signals or data are those that take real numbers as values, in contrast to complex-valued signals that incorporate imaginary components. Real-valued optimization and numerical computation often focus on problems where all quantities of interest are real numbers, with real-valued coefficients and constraints.

The term real-valued emphasizes the codomain or range lying in the real number system, and it is routinely used to contrast with complex-valued, vector-valued, or abstract-valued counterparts in analysis, probability, and applied mathematics.