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

physical quantities A physical quantity (or simply quantity) is a 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 '' ...

. Its roots can be traced to Fourier's concept of dimensional analysis (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 description of a physical quantity as the product of a "numerical value" and a "reference quantity" (i.e. a "unit quantity" or a "

unit of measurement A unit of measurement, or unit of measure, is a definite magnitude (mathematics), 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 qua ...

"). De Boer summarized the multiplication, division, addition, association and commutation rules of quantity calculus and proposed that a full axiomatization has yet to be completed. 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 metre, not an algebraic variable i.e. the unit symbol does not satisfy the axioms of arithmetic. 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 SI 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 appropriat ...

s, including

dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...

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. It is defined such that one radian is the angle subtended at ...

(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. A solid angle in the fo ...

(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 ''

Quantities, Units and Symbols in Physical Chemistry ''Quantities, Units and Symbols in Physical Chemistry'', also known as the ''Green Book'', is a compilation of terms and symbols widely used in the field of physical chemistry. It also includes a table of physical constants, tables listing the pro ...

''.

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Further reading