skalaren

Skalaren, in mathematical and physical terminology, denotes a scalar quantity—a quantity described by a single magnitude and a unit, with no associated direction. This contrasts with vectors, which have both magnitude and direction. Scalars are invariant under coordinate transformations: changing the observer’s frame does not alter their value.

Common examples include temperature, mass, energy, time, and distance (the magnitude of a displacement). Scalar quantities obey the ordinary rules of arithmetic: they add, subtract, multiply, and divide as real or complex numbers, and they scale when multiplied or divided by numbers. They form a field in the mathematical sense, enabling algebraic operations independent of geometry.

In physics, scalars can be extended to scalar fields: a temperature field assigns a scalar value to every point in space. The gradient of a scalar field yields a vector field; the Laplacian is a scalar operator acting on a scalar field. Scalars may carry units; whether a quantity is dimensionless or dimensional affects interpretation, but dimensional analysis applies like for other quantities.

Relation to measurement and modeling: a scalar is fully specified by its numerical value and units, with no orientation information required. In data science and engineering, scalars contrast with vectors and tensors, which encode directional or multi-dimensional information. In mathematics and programming, scalars are typically treated as elements of a field, enabling straightforward algebraic operations with other scalars and with higher-dimensional objects such as vectors and matrices.