Scalar
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Scalar is a concept that has meaning in mathematics, physics, and computing.
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.)
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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. This is in contrast to, for example, a vector, which is a physical entity which can be described by from one to an infinite number of numerical values (called "components"), each of which is dependent upon the particular coordinate system being used. In physics, a vector is, like a scalar, a "real" object with properties which are independent of the coordinate system used to describe it. It follows that a change in coordinate systems must alter the components of a vector in a very particular way, so as to conserve those aspects of the vector which are specific to the vector itself, and not the coordinate system that describes it. In this sense, the components of a vector are not scalars, since they change with a change of coordinate system.
Examples of scalar quantities:
- electric charge and charge density (the latter nonrelativistically; in relativity it must be combined with current density to comprise a 4-vector)
- relativistic distance
- mass and mass density (the latter nonrelativistically; in relativity it must be made part of the energy tensor in combination with momentum density)
- speed, but not velocity or momentum
- temperature
- energy and energy density (the latter nonrelativistically)
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, in a sense, volume. Volume is normally considered a non-negative scalar, however, so an absolute value is taken. The use of the triple scalar product to calculate volume is, anyway, an artifice that works in three dimensions, but not in more dimensions. (Another example, if it exists, would be magnetic charge.)
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 a member of said 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, we see that 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.
See also
de:Skalar (Mathematik) fr:Scalaire io:Skalaro id:Skalar he:סקלר ja:スカラー pl:Skalar sv:Skalär