In engineering analysis, physical quantities such as the length of a beam or the velocity of a body are routinely modeled by variables in equations with numbers as values. While human engineers can interpret these numbers as physical quantities by inferring dimension and units from context, the representation of quantities as numbers leaves implicit other relevant information about physical quantities in engineering models, such as physical dimension and unit of measure. Furthermore, there are many classes of models where the magnitude of a physical quantity is not a simple real number - a vector or higher-order tensor for instance. Our goal here is to extend standard mathematics to include unit and dimension semantics.
In this theory, we attempt to define the basic concepts associated with physical quantities. A quantity is a hypothetically measurable amount of something. We refer to those things whose amounts are described by physical-quantities as physical-dimensions (following the terminology used in most introductory Physics texts). Time, length, mass, and energy are examples of physical-dimensions. Comparability is inherently tied to the concept of quantities. Quantities are described in terms of reference quantities called units-of-measure. A meter is an example of an unit-of-measure for quantities of the length physical-dimension.
The physical-quantities theory defines the basic vocabulary for describing physical quantities in a general form, making explicit the relationships between magnitudes of various orders, units of measure and physical dimensions. It defines the general class physical-quantity and a set of algebraic operators that are total over all physical quantities. Specializations of the physical-quantity class and the operators are defined in other theories (which use this theory).
The theory also describes specific language for physical units such as meters, inches, and pounds, and physical dimensions such as length, time, and mass. The theory provides representational vocabulary to compose units and dimensions from basis sets and to describe the basic relationships between units and physical dimensions. This theory helps support the consistent use of units in expressions relating physical quantities, and it also supports conversion of units needed in calculations.
Quantity-Spaces Scalar-Quantities Standard-Dimensions
Orthogonal-Dimension-Set Physical-Dimension Physical-Quantity Constant-Quantity Dimensionless-Quantity Unit-Of-Measure Function-Quantity Zero-Quantity Relation-Extended-To-Quantities System-Of-Units
< Compatible-Quantities Dimension-Composable-From
* + - / Base-Units Expt Magnitude Magnitude-In-System-Of-Units Quantity.Dimension Recip Standard-Unit Summation
The following constants were used from included theories:
All constants that were mentioned were defined.