Irn

IRn is a notation that appears in some mathematical texts to denote n-dimensional Euclidean space. In standard notation, this space is written as R^n or ℝ^n, where ℝ represents the real numbers. When written without the superscript in plain text, IRn may be used, but this is less common in modern typography.

IR^n consists of all ordered n-tuples (x1, x2, ..., xn) with each xi real. It is a vector space under componentwise addition and scalar multiplication. It carries the Euclidean metric d(x, y) = sqrt(sum over i of (xi - yi)^2) and the associated Euclidean topology. The space is convex, connected, and contractible, making it a fundamental object in analysis, geometry, and physics. It serves as the standard model for n-dimensional space in many theoretical and applied contexts.

Notation and usage can vary. The preferred symbol for the real numbers is usually a bold or blackboard bold R (ℝ); in plain-text environments, IR^n or IRn may appear, though this can lead to ambiguity with other uses of the letters. When precision is important, ℝ^n is recommended to avoid confusion with alternate meanings of IRn in different texts or software.

Other uses of the string IRn may occur outside mathematics, such as product names, identifiers, or domain-specific acronyms. In such cases, the intended meaning depends on the specific field and context. If IRn appears without explanation, consult surrounding material or definitions to determine whether it refers to Euclidean space or to an unrelated term.