👉 In mathematics, distances measure the "distance" between points in various spaces, such as Euclidean spaces, metric spaces, or more abstract structures. In a Euclidean space like R^n (n-dimensional real numbers), the distance between two points \( \mathbf{a} = (a_1, a_2,..., a_n) \) and \( \mathbf{b} = (b_1, b_2,..., b_n) \) is calculated using the Euclidean distance formula: \( d(\mathbf{a}, \mathbf{b}) = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2 +... + (a_n - b_n)^2} \). This formula generalizes to higher dimensions and is based on the Pythagorean theorem. In more abstract spaces, like metric spaces, a distance function (or metric) \( d: X \times X \rightarrow \mathbb{R} \) must satisfy certain properties, including non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. These properties ensure that distances behave intuitively, such as being zero for identical points, symmetric under point ordering, and lessening as the distance increases.
