# Scalar

Scalar is a concept that has meaning in mathematics, physics, and computing. A simple definition is that a scalar is a quantity which only specifies magnitude (i.e. a numerical value and unit), unlike a vector which has both magnitude and direction. For example, speed (180 km/h) is a scalar, while velocity (180 km/h north) is a vector. While this is a useful definition, it is not quite complete. A scalar is more completely defined as a magnitude which does not change under a change of coordinate system. In the above example, suppose the velocity vector has two components (e.g, 180 km/h north and 0 km/h east). Each component has a magnitude, yet they are not scalars, because they change when the coordinate system used to calculate them changes. (e.g to 180/√2 km/h northwest and 180/√2 km/h northeast.) Similar considerations hold for the mathematical definition. It is only for the computer definition that "scalar" simply means a single number.

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 rotations by a scalar, the real part of the quaternion, and vectors by the other three parts. Quaternions are widely used in spacecraft attitude determination and control, because they are not subject to the singularities of Euler angles and have only four components, while a rotation matrix has nine components to represent only three angles.

## In physics

In physics, a scalar is a physical quantity which assumes a single value which is independent of the coordinate system being used to describe the physical system. In this sense it is a "real" quantity and not an artifact of the coordinate system. For example, the distance between two points in space is a scalar. It does not depend on one's choice of coordinate system.

A physical quantity is expressed as the product of a numerical value and a physical unit, not just a number. It does not depend on the unit distance (1 km is the same as 1000 m), although the number depends on the unit. Thus distance does not depend on the length of the base vectors of the coordinate system. Also, other changes of the coordinate system may affect the formula for computing the scalar (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal), but not the scalar itself. In this sense, physical distance deviates from the definition of metric in not being just a real number; however it satisfies all other properties. The same applies for other physical quantities which are not dimensionless.

A scalar field is a scalar-valued function of position, again independent of the coordinate system.

A vector is a physical entity which has a magnitude which is a scalar, but in addition, in contrast with a scalar, has a direction. The components of a vector as such are not scalars, since they change with a change of coordinate system; a scalar field may however for one choice of the coordinate system be equal to a particular component.

Examples of scalar quantities:

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 the signed volume. Another example is magnetic charge (as it is mathematically defined, regardless of whether it exists physically).

## 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 and members of F are called scalars.

More generally, a scalar for a module over a ring, is simply an element of the ring. This happens in manifold theory, where the tangent bundle forms a module over the algebra of real functions on the manifold. Since spacetime is supposed to be a manifold, the physical and mathematical concepts agree.

A scalar is a tensor of rank zero.

## In computing

In computing scalar refers to variables that can hold only one value at a time, as distinct from arrays, list or other containers which are variables that can hold many values at the same time.