távolságokkal

Távolságokkal is a Hungarian term formed from the noun távolság (distance) in the instrumental plural, used to indicate doing something by means of distances or in relation to distances. This form appears in sentences such as “távolságokkal mérve” or “távolságokkal számolva,” meaning that measurement or calculation is performed with the distances in question or using distances as a reference.

In mathematics, a distance is a function that assigns a nonnegative real number to every pair of points in a set, representing their separation. A distance function must satisfy non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. If such a function exists on a set X, (X, d) is called a metric space. Distances can be defined in many ways: Euclidean distance in R^n, Manhattan (L1) distance, Chebyshev (L∞), and Minkowski distances (Lp). In graph theory, distances are often defined as the length of the shortest path between vertices.

Distances also play a central role in data analysis and computation: they provide a notion of similarity, enable clustering, k-nearest neighbors, and multidimensional scaling. In geometry and physics, distances define shapes, trajectories, and spacetime separations (for example, spatial distances, light-years).

In everyday language, távolságok refers to the physical space between objects or places, measured in units such as meters or kilometers. The instrumental form távolságokkal is a grammatical tool for expressing that a measurement, comparison, or operation is performed with or by reference to distances.

See also: metric, distance, geometry, graph theory, clustering.