Quantity calculus is the formal method for describing the mathematical relations between ''abstract''

physical quantities A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For exam ...

. (Here the term ''calculus'' should be understood in its broader sense of "a system of computation", rather than in the sense of

differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...

and

integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

.) Its roots can be traced to Fourier's concept of

dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...

(1822). The basic

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

of quantity calculus is

Maxwell's Maxwell's, last known as Maxwell's Tavern, was a bar/restaurant and music club in Hoboken, New Jersey. Over several decades the venue attracted a wide variety of acts looking for a change from the New York City concert spaces across the river. Ma ...

description of a physical quantity as the

product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

of a "numerical value" and a "reference quantity" (i.e. a "unit quantity" or a "

unit of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multi ...

"). De Boer summarized the multiplication, division, addition, association and commutation rules of quantity calculus and proposed that a full

axiomatization In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains ...

has yet to be completed. Such axiomatization needs to begin from a definition of ''quantity'' in terms of ''physical dimension'' (see

) which is indeed a more fundamental concept than of ''unit'' or ''unit quantity'' or ''

''. Measurements are expressed as products of a numeric value with a unit symbol, e.g. "12.7 m". Unlike algebra, the unit symbol represents a measurable quantity such as a meter, not an algebraic variable. So, the use of a unit symbol, as well as the use of a dimension symbol during

, as an algebraic variable leads to an associated logical subtlety. A careful distinction needs to be made between abstract quantities and measurable quantities. The multiplication and division rules of quantity calculus are applied to

SI base unit The SI base units are the standard units of measurement defined by the International System of Units (SI) for the seven base quantities of what is now known as the International System of Quantities: they are notably a basic set from which al ...

s (which are measurable quantities) to define

SI derived unit SI derived units are units of measurement derived from the seven base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate p ...

s, including

dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

derived units, such as the

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...

(rad) and

steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radian ...

(sr) which are useful for clarity, although they are both algebraically equal to 1. Thus there is some disagreement about whether it is meaningful to multiply or divide units. Emerson suggests that if the units of a quantity are algebraically simplified, they then are no longer units of that quantity. Johansson proposes that there are logical flaws in the application of quantity calculus, and that the so-called dimensionless quantities should be understood as "unitless quantities". How to use quantity calculus for unit conversion and keeping track of units in algebraic manipulations is explained in the handbook on Quantities, Units and Symbols in Physical Chemistry.

References

Further reading