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# Scalar

The concept of a scalar is used in mathematics and physics. 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. A scalar is formally a tensor of rank zero.

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 vectorss. More formally, a scalar is a quantity that is invariant under coordinate rotations (or Lorentz transformations, for relativity).

Examples of (non-relativistic) scalar quantities include:

The word scalar is derived from scala Latin for "ladder" and means "resembling a 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."

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). (Another example, if it existed, would be magnetic charge.)