Epsilon

Epsilon, written as the Greek letter ε (uppercase Ε), is the fifth letter of the Greek alphabet. It represents a vowel sound in Greek and is widely used in science and mathematics as a symbol. The name and form trace back to the Phoenician letter He.

In mathematics, epsilon commonly denotes a small positive quantity. It is central to the epsilon-delta definition of a limit, where for every ε > 0 there exists a δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ. It also serves as a generic bound on error, tolerance, or perturbation.

Typography and notation distinguish several variants, such as varepsilon. In computer science and numerical analysis, 'machine epsilon' (the difference between 1 and the next representable floating-point number) measures the precision of a given system and guides numerical stability checks.

In physics and engineering, ε denotes permittivity. In electromagnetism, the vacuum permittivity ε0 is a fundamental constant, and the relative permittivity εr expresses how a material polarizes in an electric field. The symbol also appears in complex permittivity, modeling lossy dielectric media.

Beyond mathematics and physics, epsilon appears as an abstract error term or tolerance in statistics, optimization, and dynamical systems. Across disciplines, it is used to denote small quantities, bounds, or material properties in a concise, conventional way.