Quantity#Quantity in mathematics

Variables are represented mathematically as a real multiple of a unit, as part of a unit produced by a real vector space. The multiplication of the unit x with a real number r is also called scalar multiplication and is written as rx. The choice of the unit is characteristic of the type of size, for example for everyday variables such lengths with units of meters (m), mass with units of grams ( g) or money values ​​with the unit Euros ( € ). The largest scope is the physics with a plurality of physical quantities.

History

Sizes were defined implicitly in the ancient world already by Eudoxus of Cnidus. Its size doctrine survives in Euclid's Elements. He generalized the Pythagorean doctrine of numbers in it so that even irrational proportions are included. Its axioms and rules of calculation, which he applied in proofs that guarantee an embedding of ancient sizes in a modern size range. The Eudoxischen size axioms include, among others already called the Archimedean axiom. In ancient sciences various sizes were common even before Euclid, about length, area and volume in the geometry, the time in the physics of Aristotle, also the duration and interval size in the music theory of Aristoxenus. About Euclid's Elements of the concept of magnitude then became canonical status until the end of the 19th century. Even Peano stood in the Euclidean tradition and spoke of sizes ( quantitates ) instead of positive real numbers.

In the mathematics of the 20th century, the concept of magnitude but was displaced by the concept of real number, which is an abstraction of the concept of size, because it neglects the respective unit. In modern physics, sizes play with units but still an important role. There, the term was but tailored to physical quantities; this is both a generalization, which also includes complex values ​​with direction ( vectors) with, but on the other hand, a limitation that forces from other areas not considered.