Scalar

A scalar is a single number, normally a real number or complex number. The word “scalar” is normally used to separate single numbers from more complicated algebraic structures like vectors, matrices, or tensors.

Formally, scalars need to be part of a field, so they have the normal arithmetic operations of addition, subtraction, multiplication, and division. In an algebraic structure like a vector space, this field (called the underlying field) also has an operation called scalar multiplication with elements of the space. A scalar k can be multiplied to a vector \mathbf{v} or a matrix A, resulting in the vector k\mathbf{v} and the matrix kA, respectively. For vectors, scalar multiplication produces a new vector of different length in the same or opposite direction of the original vector.

In physics

A scalar quantity is something that can be measured using a scalar (with a matching unit of measure). For example, consider the length of a rod: the whole measurement can be described as something like “2 metres” or “3 centimetres”. Since only a single number and unit are needed to completely describe the measurement, it is a scalar quantity.

A special type of scalar quantity is the magnitude of a vector. In three-dimensional space, velocity is not a scalar quantity because it needs to measure both how fast something is moving and what direction it is moving in. The single number for “how fast something is moving”, the magnitude of the velocity vector, is a scalar quantity called the speed. Sometimes (like velocity and speed, or displacement and distance) the vector and scalar magnitude have different names. In other cases, like force, they have the same name and notation shows what type of quantity to use (e.g. \vec{F} or \bold{F} for vector force versus F for scalar force.