Quantity calculus
Encyclopedia
Quantity calculus is the formal method for describing the mathematical relations between abstract physical quantities
. Despite the name, it is more analogous to a system of algebra
than calculus
in the mathematical sense of the term. Measurements are expressed as products of a numeric value with a unit symbol, e.g. "12.7 m". Unlike algebra, the unit symbol represents an actual quantity such as a meter, not an algebraic variable
.
The basic axiom
of quantity calculus can, for most purposes, be taken to be Maxwell's
description of a physical quantity as the product
of a "numerical value" and a "unit of measurement", although the roots can be traced to Fourier's
concept of dimensional analysis
(1822) and a full axiomatization has yet to be completed.
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
s (which are measurable quantities
) to define SI derived unit
s, but if the units are algebraically simplified, it has been suggested that they then are no longer units of that quantity.
Physical quantity
A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...
. Despite the name, it is more analogous to a system of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
than calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
in the mathematical sense of the term. Measurements are expressed as products of a numeric value with a unit symbol, e.g. "12.7 m". Unlike algebra, the unit symbol represents an actual quantity such as a meter, not an algebraic variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
.
The basic axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
of quantity calculus can, for most purposes, be taken to be Maxwell's
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
description of a physical quantity as the product
Product (mathematics)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
of a "numerical value" and a "unit of measurement", although the roots can be traced to Fourier's
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...
concept of dimensional analysis
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
(1822) and a full axiomatization has yet to be completed.
A careful distinction needs to be made between abstract quantities and measurable quantities
Measured quantity
In a physical setting a measurement instrument may be gauged to measuring values of a specific physical quantity. In such a context the specific physical quantity is called a measured quantity....
. The multiplication and division rules of quantity calculus are applied to SI base unit
SI base unit
The International System of Units defines seven units of measure as a basic set from which all other SI units are derived. These SI base units and their physical quantities are:* metre for length...
s (which are measurable quantities
Measured quantity
In a physical setting a measurement instrument may be gauged to measuring values of a specific physical quantity. In such a context the specific physical quantity is called a measured quantity....
) to define SI derived unit
SI derived unit
The International System of Units specifies a set of seven base units from which all other units of measurement are formed, by products of the powers of base units. These other units are called SI derived units, for example, the SI derived unit of area is square metre , and of density is...
s, but if the units are algebraically simplified, it has been suggested that they then are no longer units of that quantity.
Further reading
- International Organization for StandardizationInternational Organization for StandardizationThe International Organization for Standardization , widely known as ISO, is an international standard-setting body composed of representatives from various national standards organizations. Founded on February 23, 1947, the organization promulgates worldwide proprietary, industrial and commercial...
. ISO 80000-1:2009 Quantities and Units. Part 1 - General.. ISO. Geneva