The concept establishes Ordered Pairs (x, y) as the formal algebraic representation of points within a Two-Dimensional Euclidean Space defined by two orthogonal coordinate axes. This theory relies on the Axiom of Orthogonality to define unit distance and metric properties, ensuring that spatial location is uniquely determined by real-valued components independent of geometric visualization or arrow-based vector notation. It functions as a foundational structural definition in analytic geometry and calculus, serving strictly as an algebraic mapping mechanism between one-dimensional scalars and two-dimensional loci without invoking vector addition or magnitude operations.
An ordered pair is formally defined within the Cartesian coordinate system as a set $(x, y)$ representing a unique solution to linear equations and serving as the geometric representation on a two-di…