The concept of a **scalar** is used in mathematics, physics, and computing. The concept used in physics is a more concrete version of the same idea that goes by that name in mathematics.

In mathematics, the meaning of *scalar* depends on the context; it can refer to real numbers or complex numbers or rational numbers, or to members of some other specified field. Generally, when a vector space over the field *F* is studied, then *F* is called the *field of scalars*.

In physics a *scalar* is a quantity that can be described by a single number (either dimensionless, or in terms of some physical quantity). Scalar quantities have magnitude, but not a direction and should thus be distinguished from vectors. More formally, a scalar is a quantity that is invariant under coordinate rotations (or Lorentz transformations, for relativity). A scalar is formally a tensor of rank zero.

Examples of (non-relativistic) scalar quantities include:

In computer programming languages, the invariance under changes in coordinate systems ceases to be in evidence, and scalars are defined as variables that can hold only one value at a time, as distinct from arrays which are variables that can hold many values at the same time.

A related concept is a **pseudoscalar**, which is invariant under proper rotations but (like a pseudovector) flips sign under improper rotations. One example is the scalar triple product (see vector), and thus volume. (Another example, if it existed, would be magnetic charge.)

The word *scalar* derives from the English word "scale" for a range of numbers, which in turn is derived from *scala* (Latin for "ladder"). According to a citation in the *Oxford English Dictionary* the first usage of the term (by W. R. Hamilton in 1846) described it as:

- "The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers from negative to positive infinity; we shall call it therefore the scalar part."

Hamilton's usage actually describes his quaternion-based notation, which (in modern terms) represented scalars by the real part of the quaternion and vectors by the other three parts. (This notation eventually proved unpopular.)