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.)