realaxeln

Realaxeln, or the real axis, is a fundamental concept in mathematics. It refers to the set of all real numbers when represented in a standard coordinate system. In the complex plane, the real axis is the horizontal axis consisting of points z = x + iy with y = 0; equivalently the subset {x ∈ R} embedded as real numbers. Real numbers can be identified with points on this axis via x ↔ (x,0). The axis is unbounded and one-dimensional, serving as a reference line for many constructions.

Geometrically, the real axis is the x-axis in the Cartesian plane. It is used to plot real-valued functions y = f(x) and to locate real roots, where f(x) = 0. In the complex plane, the real axis often serves as a boundary or line of symmetry in the Argand diagram; many theorems relate behavior on or across the real axis, such as the Schwarz reflection principle.

The concept extends to the extended real line by adding a point at infinity, making it a compact space. In applications, the real axis appears when considering real-valued signals, the real parts of complex quantities, or the region of convergence in certain transforms, where the real component of a complex variable has particular significance.