Orthant

In mathematics, an orthant is a generalization of a quadrant (in two dimensions) and an octant (in three dimensions) to n dimensions. It is one of the 2ⁿ regions into which n-dimensional Euclidean space (ℝⁿ) is divided by its coordinate hyperplanes.

More formally, an orthant is a set of the form {(x₁, ..., x_n) ∈ ℝⁿ | ε₁x₁ ≥ 0, ..., ε_nx_n ≥ 0}, where each ε_i is either +1 or -1. Each choice of the n signs ε_i specifies a unique orthant. Therefore, in n dimensions, there are 2ⁿ orthants.

Each orthant is the Cartesian product of n half-lines, where each half-line is either the non-negative or non-positive real numbers.

The positive orthant (sometimes called the first orthant) is the orthant where all coordinates are non-negative; that is, where all ε_i are +1. This corresponds to the familiar first quadrant in 2D and the first octant in 3D.

Orthants are important in various areas of mathematics, including optimization, linear programming, and convex analysis. They provide a way to partition space into well-defined regions based on the signs of the coordinates.