2Physical quantities

2.1Quantity, unit, quantity value

A quantity is either a multitude, if it is discrete, or a magnitude,15 if it is continuous (as already mentioned in the introduction, this fact was familiar even to the ancient Greeks and stated by Aristotle in his Categories). A multitude can be counted, e. g. a countable set of entities, and a magnitude can be measured. Physical quantities, such as length or mass, are thus magnitudes.

There are two different meanings of the term quantity. One refers to an abstract metrological concept (e. g. duration, length, mass, temperature, etc.), the other to a concrete realization of this concept by a characteristic of a certain physical object or phenomenon (e. g. the period of oscillation of a pendulum, the length of a copper wire, the propagation velocity of water waves, the mass of the proton, etc.). Both meanings are based on the term quantitative characteristic.

A characteristic is a recognizable attribute that makes it possible to distinguish one object or abstract context from others. Characteristics also play a special role in the sciences for the classification of objects and phenomena according to certain criteria (classification scheme, taxonomy), whereby they are arranged in categories or classes. In the natural sciences, we are particularly interested in quantitative characteristics. Their characteristic values are countable or form a continuum.

Characteristic values of a quantitative characteristic can always be compared with each other, i. e. in principle it can always be determined whether two characteristic values are the same or whether one is larger (or smaller) than the other. The property of total comparability is shared by the characteristic values of a quantitative characteristic with the numbers, but without being numbers for that reason.

On the basis of the term quantitative characteristic, we can now define the term quantity as

Quantity. A quantitative characteristic is called quantity if there exists a ratio16 for each pair of its characteristic values to which a real number is assigned.

This definition of the term quantity initially entails a restriction to scalars. But there are quantities that are vectors or tensors. In these cases, however, their components can be treated as scalar quantities. We can therefore restrict the following reasoning to scalar quantities.

Real numbers can be assigned to the characteristic values of a quantity in a one-to-one correspondence by assigning the number one to exactly one characteristic value. All others are then multiples or fractions of this special characteristic value. We therefore define the term unit as

Unit. The characteristic value of a quantity to which the number one is assigned by agreement is called the unit of the quantity.

and the term quantity value as

Quantity value. The representation of a characteristic value of a quantity as a product of a number and a unit is called quantity value, i. e.

|Q|={Q}[Q]

applies, where |Q| denotes the quantity value of the quantity Q and [Q] its unit. The number {Q} is called the numerical value of the quantity.

The unit of a quantity is denoted by its agreed unit symbol and the preceding numeral for the number one, the digit 1. If the quantity is a pure number, then its unit is the number one, and it is only represented by the digit 1. In a product or quotient of units or quantity values, the digit 1 may be omitted, but care must always be taken that unit symbols are not allowed to stand alone.

Examples:

3.6 N (Quantity value of a force in the SI) 1 N (Unit of force in the SI) 300 m2 (Quantity value of an area in the SI) 1 m2 (Unit of area in the SI) 120 N /cm2 (Quantity value of a pressure in the SI) 1 N /m2 (Unit of pressure in the SI)

2.2Systems of quantities

A system of quantities consists of a set of quantities and relations between them. Some of these quantities, which are considered to be pairwise independent of each other and of all other quantities, form the set of base quantities for the entire system. All other quantities of the system of quantities are then derived quantities, defined in terms of the base quantities and expressed algebraically by products of powers of the base quantities.

To each quantity a dimension with the same name is uniquely assigned in a one-to-one correspondence. The representation of a quantity as a product of powers of the base quantities is therefore usually called the dimensional product (or physical dimension) of the quantity with respect to the chosen base quantities or base dimensions. The integers used as exponents of the base quantities or base dimensions are called dimensional exponents.

In the International System of Quantities (ISQ) there are seven base quantities and accordingly seven base dimensions, each sharing the same name. The base dimensions are denoted by upright, sans-serif capital letters (see table 2.1).

Tab. 2.1: Names and symbols of the base dimensions in the ISQ.

Name of quantity and dimension Symbol of dimension (time) duration ┼ T length L mass M electric current I thermodynamic temperature Ø amount of substance N luminous intensity J

┼ Time is not strictly a quantity (nor is physical space). In physics, in fact, only (time) duration is used, i. e. a time interval, which is a quantity comparable to the length of a distance.

In the ISQ, dimensional products are written in the form

dimQ= T a L ß M ? I d Ø e N ? J ? ,

where dim Q denotes the physical dimension of the quantity Q and a , ß, ?, d, e, ? ? denote the integer dimensional exponents.

Quantities whose dimensional exponents are all equal to zero are called quantities of dimension number. These quantities (e. g. angular measure, refractive index, etc.) are numbers. Their quantity values are expressed only by numerical values. Their unit, the number one, is usually omitted, but can be expressed by a supplementary unit if necessary (see section 3.4).

The dimension number cannot be a base dimension in any system of quantities (for details see [1]). It is denoted by the letter Z. In the ISQ therefore the following applies

Z:= T 0 L 0 M 0 I 0 Ø 0 N 0 J 0 .

Thus the relation dim Q = Z expresses the fact that the quantity Q is a number and consequently for its unit [Q] = 1 is valid.

A system of units can be unambiguously assigned to each system of quantities by defining a base unit for each base quantity and expressing all other units of the system of units by products of powers of the base units so that the relations existing between the quantities correspond to those between the units. The dimensional exponents of the base units are identical to those of the corresponding base quantities and base dimensions.

2.3Symbols for physical quantities

Symbols for physical quantities shall consist of single capital or small italic letters of the Latin or Greek alphabet with or without supplementary symbols (subscripts or superscripts, accents, underlining or overlining, etc.). The only exception to this rule are the two-letter symbols used to represent characteristic numbers (see section 6.14). If such a two-letter symbol appears as a factor in a product, it shall be separated from the other symbols by a dot (multiplication sign), a small space or a bracket. It is treated as a single symbol and can be raised to a positive or negative power without the use of a bracket.

Abbreviations (i. e. abbreviated forms of names or expressions, such as e. g. pdf for probability density function) may be used in the text, provided they have been previously introduced, but they shall not be used in physical equations. Abbreviations in the text shall be written in standard Latin script.

Symbols for physical quantities and numerical variables shall always be written in an italic font. This also applies to indices denoting quantities or numbers, while explanatory indices shall be written in standard (upright) script. Numbers are always written...