# Physical size

Vernier caliper to measure the length
Beam balance for measuring mass by comparing its weight with that of known weights
Stopwatch for measuring time , unit of measurement: second
Ammeter for measuring the current strength , unit of measurement: ampere
Thermometer for measuring the temperature , unit of measurement: degrees Celsius

A physical quantity is a quantitatively determinable property of a process or state on an object of physics . Examples of such quantities are length, mass, time, and current. Every special value of a physical quantity (size value) is specified as the product of a numerical value (also a measure ) and a unit of measurement . Vector quantities are specified by size value and direction.

The term physical quantity in today's understanding was introduced by Julius Wallot and slowly gained acceptance from 1930. This led to a conceptually clear distinction between size equations, numerical value equations and tailored size equations (see numerical value equation ). A quantity equation is the mathematical representation of a physical law that describes the states of a physical system and their changes. It represents the relationship between different physical quantities that applies, with a symbol for each of these quantities . Size equations apply regardless of the units of measure selected.

Those physical quantities that are defined as the basis of a size system are called base quantities .

## Basics

A comparison of two things always requires a criterion on the basis of which the comparison is made ( tertium comparationis ). This must be a characteristic (or quality ) common to both things. A feature is called a physical quantity if it has a value so that the ratio of two feature values is a real number factor (ratio quantity ) . A comparison based on a size can thus be quantified. The comparison process to determine the numerical factor is called measurement . The measurability of a feature, i. H. the specification of a clear and reproducible measurement specification for a comparison is equivalent to the definition of a physical quantity.

All characteristics of an object fall into two classes, physical quantities and all others. The physics deals exclusively with the former class. It establishes general relationships between size values, i.e. relationships that apply to all wearers of this size. In this context, all objects that have the observed size as a feature are called carriers . Physical relationships are therefore independent of the specific nature of a carrier.

The following sections deal with individual terms that are used in connection with physical quantities.

### dimension

If the quotient of two quantities of different physical quantities is a real number , then it is physical quantities of the same dimension. In every equation between physical quantities, both sides must be of the same dimension ( dimension consideration ).

The term dimension should be viewed in connection with a system of sizes. The dimension represents the respective physical quantity qualitatively in the quantity system. The dimension of a derived physical quantity is defined as the power product of dimensions of the base quantities. This power product is based on the underlying size equations; Possible numerical factors, mathematical operations such as scalar or vector product, differential quotient, integral, level of the tensors belonging to the quantities are not taken into account. In this way, a qualitative dependence of the derived quantity on the base quantities can be shown.

Example:

In the International System of Quantities (ISQ), the derived physical quantity is mechanical work as

${\ displaystyle W: = \ int {\ vec {F}} \, \ mathrm {d} {\ vec {r}}}$

Are defined. The dimension of the mechanical work can be derived from the dimensions of the quantities involved in this size equation.

${\ displaystyle \ mathrm {dim} \ W \ equiv \ mathrm {dim} \ {\ vec {F}} \ cdot \ mathrm {dim} \ {\ vec {r}} \ equiv {\ mathsf {MLT ^ {- 2}}} \ cdot {\ mathsf {L}} \ equiv {\ mathsf {ML ^ {2} T ^ {- 2}}}}$

### Size type

The size type is used to subdivide the set of sizes of a given dimension. According to the International Dictionary of Metrology (VIM), 3rd edition 2010, the type of size or type of a size is the "aspect that is common to comparable sizes", and a note states: "The subdivision of the generic term 'size' according to the The type of size is [...] arbitrary ”. Sizes of the same kind can be linked in a meaningful way through addition and subtraction. In addition, the order relationships “larger”, “smaller” and “equal” apply to quantities of the same type .

For example, the width, height, and length of a cuboid, the diameter of a pipe, the wingspan of a bird, and the wavelength are all quantities of the " length " type; they can be compared to the length of a folding rule . It is up to the user whether the amount of precipitation , given as volume / area, is considered to be the same, although it can also be easily measured with a meter. The consumption figures for motor vehicles in “liters per 100 kilometers” will, however, hardly be ascribed to the size type area, although it has the dimension of an area.

### Size value

The value of a physical quantity (quantity value) is generally accepted as the product of a number and the physical unit that is assigned to the type of quantity in question. The ratio of two values ​​of the same size is a real number.

This was presented more cautiously within the German set of standards in the first edition "Notation of physical equations" of the DIN  1313 standard from November 1931: The formula symbols occurring in the physical equations can be calculated as if they were the physical "quantities", i.e. H. named numbers meant. They are then expediently understood as symbolic “products” of the numerical values ​​(dimensions) and the units according to the equation

Physical quantity = numerical value “times” the unit.

A 10-fold difference between values ​​of the same size is called an order of magnitude .  So orders of magnitude correspond to a factor of . ${\ displaystyle n}$${\ displaystyle 10 ^ {n}}$

There are a number of sizes whose size values ​​are invariably fixed. These are called natural constants , universal constants or physical constants (examples: speed of light in a vacuum, elementary charge , Planck's constant , fine structure constant ).

### Numerical value and unit

It is useful to determine the ratio of a size value to the value of a similar, fixed and well-defined comparison variable. The comparative value is referred to as the unit of measurement or unit for short , the measured ratio as a measure or numerical value. The size value can then be represented as the product of numerical value and unit (see also section Notation ). Depending on the definition of the quantity, the numerical value is a real number - for some quantities limited to non-negative values ​​- or complex ; for some quantities of the dimension number such as B. some quantum numbers it is always an integer.

The definition of a unit is subject to human arbitrariness. One possibility consists in the choice of a certain object - a so-called normal  - as the carrier of the size, whose size value serves as a unit. A calculated size value can also be selected, for which a suitable physical relationship with other size values ​​must be known (see also section size equations ). A third possibility is to use the value of a physical constant as a unit, if one exists for the desired size.

In theory, it is sufficient to define a single unit for a type of quantity. For historical reasons, however, a large number of different units for the same type of size have often formed. Like all similar size values, they only differ by a pure numerical factor.

### Scalars, vectors and tensors

Certain physical quantities are oriented in physical space , so the quantity value depends on the direction of measurement. For example, the speed of a vehicle is typically directed along a road; the measured speed perpendicular to this is zero - it is a vector quantity. The mechanical stress in a workpiece depends strongly on the considered cutting surface - there is more than one direction to be considered, so a tensor (second level) is necessary to describe it.

A tensor -th level can be described in the Cartesian coordinate system with elements and has certain simple properties for coordinate translation or transformation. Accordingly, it can describe a certain class of physical quantities: ${\ displaystyle n}$${\ displaystyle 3 ^ {n}}$

• A tensor of 0th order is a scalar. It describes a quantity that is independent of direction and is only determined by its size value (as a number).
• A first order tensor is determined by three components. Every vector is a 1st order tensor.
• A second order tensor is determined by nine components. It is usually represented by a 3 × 3 matrix. With “tensor” without addition, a tensor of the 2nd level is usually meant.
 Scalar Mass ; temperature Pseudoscalar Helicity ; Magnetic river vector Force ; shift Pseudo vector Torque ; Angular acceleration 2nd order tensor Inertia tensor ; Strain tensor 3rd order tensor Piezoelectric tensor 4th order tensor Elasticity tensor

### Invariances

Physics is supposed to describe the observed nature, regardless of a special mathematical representation. Therefore, a physical quantity must always be invariant (unchangeable) under coordinate transformations. Just as the system of their size values ​​is independent of the unit, the respective directions are also independent of the choice of the coordinate system.

Under point reflection, tensors have a characteristic behavior for their level. A scalar-valued size of an object does not change if this object is mirrored at a point. On the other hand, a vector-valued quantity, such as the speed , points in the opposite direction after the point reflection. Some quantities behave like tensors with rotation and displacement , but deviate from this under point reflection. Such variables are called pseudotensors . In the case of pseudoscalars , the size value changes its sign. In the case of pseudo vectors such as angular momentum , the direction does not reverse due to a point reflection of the object .

## Notation

The following explanations are based on the national and international regulations of standardization organizations and specialist societies [z. B. DIN 1338 , EN ISO 80000-1 , recommendations of the International Union of Pure and Applied Physics (IUPAP)].

### Formula and unit symbols

In mathematical equations, a physical quantity is assigned a character, the formula symbol . This is basically arbitrary, but there are conventions (e.g. SI , DIN 1304 , ÖNORM A 6438, ÖNORM A 6401 etc.) for the designation of certain sizes. Often the first letter of the Latin name of a quantity is used as a formula symbol. Letters from the Greek alphabet are also often used. A formula symbol usually only consists of a single letter, which can be provided with one or more indices for further differentiation .

There are fixed characters for units, the unit symbols . They usually consist of one or more Latin letters or, less often, a special character such as B. a degree symbol or Greek letter like the Ω (large omega) for the unit ohm . For units named after people, the first letter of the unit symbol is usually capitalized.

 {\ displaystyle {\ begin {aligned} U & = 20 \, \ mathrm {V} \\\ left \ {U \ right \} & = 20 \\\ left [U \ right] _ {\ text {SI}} & = \ mathrm {V} \ end {aligned}}} Specification of a voltage of 20  volts . Above: size value middle: numerical value below: unit

A size value is always given as the product of numerical value and unit. If you only want to specify the numerical value, put the symbol in curly brackets. If you only want to specify the unit, put the symbol in square brackets. Formally, a size value can be written as follows:

${\ displaystyle G = \ left \ {G \ right \} \; \ left [G \ right]}$

This can be well understood using the example of atomic mass . The mass of an atom can be measured in atomic mass units ${\ displaystyle m}$

${\ displaystyle m = A_ {u} \; {\ text {u}}}$.

${\ displaystyle A_ {u}}$is the numerical value { } and the atomic mass unit is the unit [ ] of the physical quantity . ${\ displaystyle m}$ ${\ displaystyle {\ text {u}}}$${\ displaystyle m}$${\ displaystyle m}$

Since the numerical value depends on the selected unit of measurement, the sole representation of the symbol in curly brackets is not clear. This is why the representation “ G / [ G ]” (e.g. “ m / kg”) or “ G in [ G ]” (e.g. “ m in kg”) is common for labeling tables and coordinate axes . The representation of units in square brackets (e.g. " m  [kg]") or in round brackets (e.g. " m  (kg)") does not correspond to the DIN 1313 standard and is used in the recommendations for the system of units SI not recommended.

Since the units used depend on the system of units , the system of units must also be specified:

{\ displaystyle {\ begin {aligned} \ left [U \ right] _ {\ text {SI}} & = \ mathrm {V} \\\ left [U \ right] _ {\ text {CGS-ESU}} & = \ mathrm {StatV} \ end {aligned}}}

### formatting

The formatting is regulated by DIN 1338 . Accordingly, the formula symbol is written in italics , while the unit symbol is written in an upright font to distinguish it from formula symbols. For example, “ m ” denotes the symbol for the quantity “ mass ” and “m” the symbol for the unit of measurement “ meter ”.

A space is written between the dimension and the unit symbol. An exception to this rule are the degree symbols , which are written directly behind the dimension figure without a space (“an angle of 180 °”), provided no further unit symbols follow (“the outside temperature is 23 ° C”). A narrow space is recommended in the font , which should also be protected against a line break so that the numerical value and unit are not separated.

In formulas, vectors are often identified by a special notation. There are different conventions. Vector arrows above the letter ( ), bold type ( ) or dashes below the symbol ( ) are common. For tensors of higher levels, capital letters in sans serif ( ), Fraktur letters ( ) or double underlining ( ) are used. Which spelling is chosen also depends on whether the writing is done by hand or by machine, as features such as bold print or serifs cannot be reliably reproduced with handwriting. ${\ displaystyle {\ vec {a}}}$${\ displaystyle {\ boldsymbol {a}}}$${\ displaystyle {\ underline {a}}}$${\ displaystyle {\ mathsf {A}}}$${\ displaystyle {\ mathfrak {A}}}$${\ displaystyle {\ underline {\ underline {A}}}}$

There are different traditions, depending on the language and subject, of using upright writing and italics in connection with formulas. In more modern specialist literature, however, the convention has prevailed to put not only size symbols, but everything that can be changed in italics; Unit symbols, element symbols , explanations, etc., on the other hand, are set upright. Formula symbols and variable indices appear in italics. Example:

"The total mass of the car is:${\ displaystyle m _ {\ text {ges}}}$
${\ displaystyle m _ {\ text {ges}} \, = \, m _ {\ text {A}} + \ sum _ {i} m_ {i} \, = 1500 \, \ mathrm {kg}}$
Here is the mass of the structure and the mass of other components. "${\ displaystyle m _ {\ text {A}}}$${\ displaystyle m_ {i}}$

### Incorrect sizes

 ${\ displaystyle l = (10 {,} 0072 \ pm 0 {,} 0023) \, \ mathrm {m}}$ ${\ displaystyle l = 10 {,} 0072 (23) \, \ mathrm {m}}$ ${\ displaystyle l \ approx {10 {,} 00 \ mathbf {7}} \, \ mathrm {m}}$ Specification of an incorrect measured variable (the last numerical value is only useful with this accuracy)

In the case of incorrect values, the numerical value is given with its measurement uncertainty or - depending on the circumstances - with its error limits , see also measurement deviation . The identification is mostly done by a "±" after the erroneous numerical value, followed by the error value (brackets are required if a unit follows so that it relates to both values). But also short forms such as bracketed error information or bold type of the uncertain number of the numerical value are common.

The number of uncertain decimal places to be specified for the numerical value depends on the error value. If this begins with a 1 or 2, two digits are noted, otherwise only one. If necessary, the numerical value should be rounded off as usual , see DIN 1333 ; however, an error limit is always rounded up.

### Examples of marking additional information

Additional designations or information may not appear or be added to the size value of a physical quantity (i.e. neither in the unit nor in the numerical value), as this would be nonsensical; they may only be expressed in the naming or designation of the physical quantity, i.e. in the formula symbol .

For example, you can add a subscript to the commonly used symbol for the frequency in the correct notation to indicate that a rotational frequency ( rotational speed) is meant: ${\ displaystyle f}$${\ displaystyle \ mathrm {U}}$

${\ displaystyle \ left [f _ {\ text {U}} \ right] = \ mathrm {s} ^ {- 1}}$ (pronounced "The unit of the (rotational) frequency is 1 per second.")
${\ displaystyle f _ {\ text {U, Motor}} = 2000 \, \ mathrm {min} ^ {- 1}}$ ("The speed of the motor is 2000 per minute.")

You can also use your own, clearly defined formula symbol. To z. For example, to dispense with the double index in the above example in favor of an easier reading, one could introduce the possibly more memorable symbol for "the rotational frequency, the number of revolutions" and write: ${\ displaystyle U}$

${\ displaystyle U _ {\ text {Motor}} = 2000 \, \ mathrm {min} ^ {- 1}}$ ("The speed of the motor is 2000 per minute.")

Without further explanation, you could usually z. Belly

${\ displaystyle h _ {\ text {Auto}} = 1 {,} 5 \, \ mathrm {m}, \ b _ {\ text {Auto}} = 2 {,} 2 \, \ mathrm {m}}$ ("The height of the car is 1.5 meters, the width of the car is 2.2 meters.")

because the symbols for the two special cases of height and width of a length measure are common.

In practice, there is not always a clear distinction between the value or unit of a physical quantity on the one hand and mere additional information on the other, so that there is a mix-up. The listed number of revolutions is a common example of this. “Revolution” is not a unit there, but merely describes the process that causes the frequency in more detail. Not permitted, but frequently occurring, is therefore about

${\ displaystyle f _ {\ text {Motor}} = 2000 \, \ mathrm {U} / \ mathrm {min}}$ ("The speed of the motor is 2000 revolutions per minute").

Further examples of frequently occurring incorrect spelling or speaking are:

Wrong: or "The flux density is 1000 neutrons per square centimeter and second."${\ displaystyle j = 1000 \, \ mathrm {n} \, \ mathrm {cm} ^ {- 2} \ mathrm {s} ^ {- 1}}$
Correct: or "The neutron flux density is 1000 per square centimeter and second."${\ displaystyle j _ {\ mathrm {n}} = 1000 \, \ mathrm {cm} ^ {- 2} \ mathrm {s} ^ {- 1}}$
Wrong: or "... a concentration of 20 nanograms of lead per cubic meter"${\ displaystyle n = 20 \, \ mathrm {ng} {\ text {lead}} / \ mathrm {m} ^ {3}}$
Correct: or "The lead mass concentration is 20 nanograms per cubic meter."${\ displaystyle n _ {\ text {Pb}} = 20 \, \ mathrm {ng} / \ mathrm {m} ^ {3}}$
Wrong: or "The unit of the magnetic field strength is ampere- turns per meter."${\ displaystyle \ left [H \ right] = \ mathrm {Aw} / \ mathrm {m}}$
Correct: or "The unit of the magnetic field strength is amperes per meter."${\ displaystyle \ left [H \ right] = \ mathrm {A} / \ mathrm {m}}$

### Size equations

 ${\ displaystyle {\ vec {F}} = m {\ vec {a}}}$ Equation of magnitude that represents the regularity between force , mass and acceleration of a body. ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle m}$ ${\ displaystyle {\ vec {a}}}$ Example: = 75 kg, = 10 m / s 2 .${\ displaystyle m}$${\ displaystyle a}$ ${\ displaystyle F}$= 750 N = 750 kg m / s 2 = with 1 N (= 1  Newton ) = 1 kg m / s 2${\ displaystyle m \ cdot a,}$

The representation of natural laws and technical relationships in mathematical equations is called size equations . The symbols of a quantity equation have the meaning of physical quantities, unless they are intended as symbols for mathematical functions or operators. Quantity equations apply regardless of the choice of units. Nevertheless, it can happen that the equations are written differently in different systems of units. For example, in some systems of units the speed of light in a vacuum has by definition the value . As a result, the constant factors and are omitted in many equations . The famous equation would become in such a system of units without the statement of the equation changing. ${\ displaystyle c = 1}$${\ displaystyle c}$${\ displaystyle c ^ {2}}$ ${\ displaystyle E = mc ^ {2}}$${\ displaystyle E = m}$

Size equations link various physical quantities and their values ​​with one another. For evaluation you have to replace the formula symbols with the product of numerical value and unit. The units used are irrelevant.

### Arithmetic operations

Not all arithmetic operations that would be possible with pure numbers are useful for physical quantities. It has been shown that a small number of arithmetic operations is sufficient to describe all known natural occurrences.

 ${\ displaystyle 15 \, \ mathrm {s} -3 \, \ mathrm {m}}$ ${\ displaystyle 5 \, \ mathrm {m} +10 \, \ mathrm {kg}}$ ${\ displaystyle \ log \ left ({299 \, 792 \, 458 \, {\ frac {\ mathrm {m}} {\ mathrm {s}}}} \ right)}$ ${\ displaystyle \ sin (5 \, \ mathrm {A})}$ Nonsensical arithmetic operations
• Addition and subtraction are only possible between sizes of the same size type. The dimension and thus also the unit of the size (s) remain unchanged, the dimensions are added or subtracted.
E.g.: ${\ displaystyle l_ {1} + l_ {2} = 2 \, \ mathrm {m} +3 \, \ mathrm {m} = 5 \, \ mathrm {m}}$
However, this only works if the two quantities are measured in the same unit. If this is not the case, both must be converted to the same unit before addition or subtraction.
E.g.: ${\ displaystyle l_ {1} + l_ {2} = 2 \, \ mathrm {km} +300 \, \ mathrm {m} = 2000 \, \ mathrm {m} +300 \, \ mathrm {m} = 2300 \, \ mathrm {m}}$
• Multiplication and division are possible without restrictions. The two quantities are multiplied by multiplying their dimensions and forming the product of the units. The same applies to division. The result usually belongs to a different type of quantity than the two factors, unless one of the factors only has the dimension number.
E.g.: ${\ displaystyle M = r \ cdot F = 2 \, \ mathrm {m} \ cdot 3 \, \ mathrm {N} = 6 \, \ mathrm {Nm}}$
E.g.: ${\ displaystyle v = {\ frac {s} {t}} = {\ frac {3 \, \ mathrm {m}} {2 \, \ mathrm {s}}} = 1 {,} 5 \, {\ frac {\ mathrm {m}} {\ mathrm {s}}}}$
• Powers can therefore also be formed. This applies to positive integer as well as negative and fractional exponents (i.e. also to fractions and roots).
E.g.: ${\ displaystyle V = a ^ {3} = (2 \, \ mathrm {m}) ^ {3} = 8 \, \ mathrm {m} ^ {3}}$
E.g.: ${\ displaystyle f = T ^ {- 1} = (2 \, \ mathrm {s}) ^ {- 1} = 0 {,} 5 \, \ mathrm {s} ^ {- 1}}$
If a quantity is raised to the power of which the unit contains a prefix for decimal parts and multiples, the exponent must be applied to the entire unit (i.e. to the product of the prefactor and unit). For example, a square kilometer is not about 1000 square meters, but
${\ displaystyle 1 \, \ mathrm {km} ^ {2} = 1 \ cdot 1000 ^ {2} \ cdot \ mathrm {m} ^ {2} = 1 \, 000 \, 000 \, \ mathrm {m} ^ {2}}$.
• Transcendental functions such as , , , , etc. are only defined for pure numbers as an argument. They can therefore only be applied to quantities of the dimension number . The function value also has the dimension number.${\ displaystyle \ exp}$${\ displaystyle \ log}$${\ displaystyle \ sin}$${\ displaystyle \ cos}$${\ displaystyle \ tanh}$
E.g.: ${\ displaystyle \ sin {\ frac {\ pi} {2}} = 1}$
E.g.: ${\ displaystyle v = \ int _ {t_ {1}} ^ {t_ {2}} a \ cdot \ mathrm {d} t = \ int _ {0} ^ {2 \, \ mathrm {s}} 3 \ , {\ frac {\ mathrm {m}} {{\ mathrm {s}} ^ {2}}} \ cdot \ mathrm {d} t = 6 \, {\ frac {\ mathrm {m}} {\ mathrm {s}}}}$

A situation is wrongly represented if these arithmetic operations were to be carried out in a nonsensical manner. The corresponding control is carried out in the dimensional analysis in order to check the existence of an as yet unknown law.

### Numerical equations

 ${\ displaystyle \ mathrm {WCT} = 13 {,} 12 + 0 {,} 6215 \, T-11 {,} 37 \, v ^ {0 {,} 16} +0 {,} 3965 \, T \ , v ^ {0 {,} 16}}$ With WCT: = wind chill temperature in degrees Celsius ${\ displaystyle T}$ : = Air temperature in degrees Celsius ${\ displaystyle v}$ : = Wind speed in kilometers per hour
Numerical equation for calculating the wind chill effect

In numerical value equations, the symbols only have the meaning of numerical values, i.e. H. of dimensions with respect to certain units of measurement . A numerical equation is only valid if the units selected for it are used. Using size values ​​in other units usually results in errors. It is therefore advisable to always perform calculations with size equations and only evaluate them numerically in the last step.

Formulas in historical texts, “ rules of thumb ” and empirical formulas are often given in the form of numerical value equations. In some cases the symbols for the units to be used are included in the equation. The sometimes found use of square brackets around the unit symbols , such as instead of , does not conform to the standard: DIN 1313: 1998-12, Chapter 4.3 provides formula symbols in curly brackets or the division of the sizes by the desired unit of measure for the representation of dimensions . With the latter z. B. the numerical value equation above into the tailored size equation${\ displaystyle \ mathrm {[V]}}$${\ displaystyle \ mathrm {V}}$

${\ displaystyle {\ frac {\ mathrm {WCT}} {^ {\ circ} \ mathrm {C}}} = 13 {,} 12 + 0 {,} 6215 \, {\ frac {T} {^ {\ circ} \ mathrm {C}}} - 11 {,} 37 \, \ left ({\ frac {v} {\ mathrm {km / h}}} \ right) ^ {0 {,} 16} +0 { ,} 3965 \, {\ frac {T} {^ {\ circ} \ mathrm {C}}} \, \ left ({\ frac {v} {\ mathrm {km / h}}} \ right) ^ { 0 {,} 16},}$

where the formula symbols now stand for the physical quantities themselves:

WCT: = wind chill temperature
${\ displaystyle T}$ : = Air temperature
${\ displaystyle v}$ : = Wind speed

## Systems of sizes and units

### Size systems

Every field of knowledge in technology and natural sciences uses a limited set of physical quantities that are linked to one another via natural laws . If one selects a few basic quantities from these quantities , so that all others in the area under consideration can be represented as power products of the basic quantities, then all quantities together form a quantity system, provided that no basic quantity can be represented from the other basic quantities . The quantities that can be represented from the basic quantities are called derived quantities, the respective power product of their dimensions is called the dimensional product . Which sizes you choose for the base is basically arbitrary and mostly happens from a practical point of view. The number of basic sizes determines the degree of the size system. For example, the international size system with its seven basic sizes is a size system of the seventh degree.

### International system of units

You need a unit for each size in order to be able to specify size values. Therefore, every system of sizes corresponds to a system of units of the same degree, which is composed analogously of mutually independent basic units and the units derived from these that can be represented. The derived units are represented by the products of powers from the base units - in contrast to size systems, possibly supplemented by a number factor. The system of units is said to be coherent if all units can be formed without this additional factor. In such systems, all size equations can be understood as numerical value equations and accordingly quickly evaluated.

The International System of Units (SI) used worldwide is a coherent system of units of the seventh degree, which is based on the International System of Units; however, the International Size System was developed later than the SI. The SI also defines standardized prefixes for units of measurement , but the multiples or parts of an SI unit formed in this way are not themselves part of the actual system of units, as this would contradict coherence. For example, a fictitious system of units that includes the base units centimeter ( ) and second ( ) as well as the derived unit meter per second ( ) is not coherent: Because of this, a number factor ( ) is required to form this system. ${\ displaystyle \ mathrm {cm}}$${\ displaystyle \ mathrm {s}}$${\ displaystyle \ mathrm {m / s}}$${\ displaystyle 1 \, \ mathrm {\ tfrac {m} {s}} = 100 \, \ mathrm {cm \ cdot s ^ {- 1}}}$${\ displaystyle 100}$

(For further competing systems of units, see below in the section Systems of measurement used in practice . )

## Special sizes

### Quotients and proportions

The quotient of two quantities is a new quantity. Such a quantity is called a ratio (or size ratio) if the output quantities are of the same type of quantity, otherwise as a quotient quantity . The quotient  value is more generally defined in DIN standard 1313 from December 1998; thereafter it is only required that the fraction of numerator size and denominator size is constant. From April 1978 to November 1998, on the other hand, in the standard edition of April 1978, DIN recommended the term size quotient more specifically only for fractions of two sizes of different dimensions and of a size ratio (a ratio size) only required that the output sizes have the same dimensions, but not necessarily are of the same size type. (For example, the electrical current strength and the magnetic flux are of the same dimension, but of different sizes.)

Quotient sizes are often described imprecisely in everyday language. For example, a definition of the driving speed as “distance covered per unit of time” or “distance covered per time elapsed” or “distance per time” is incorrect, because the speed does not have the dimension of a distance (length). Correct would be “the distance covered in a period of time divided by this period”. The abbreviated expression mentioned is common and is sufficient to give a clear concept of the respective quotient size, but the exact definition as a quotient should also always be given.

 ${\ displaystyle v = {\ frac {V} {m}}}$ "Specific volume" ${\ displaystyle \ rho = {\ frac {m} {V}}}$ "Mass density"
Designation of related quantities

If two quantities refer to a property of the same object, the quotient quantity is also called a related quantity. The denominator size is the reference size, while the numerator size is the focus of the naming. In particular, a related quantity is referred to as ...

• ... specifically , if it refers to the mass. (Unit: e.g. "... per gram" )
• ... molar , when it refers to the amount of substance. (Unit: e.g. "... per mole" )
• ... - density , if it relates to the volume (or as - area density to the area or as - length density to the length). (Unit: e.g. "... per liter", "... per square kilometer" or "... per centimeter" )
• ... rate or speed if it relates to a period of time. (Unit: e.g. "... per hour" )

Ratios always have the unit one . They can therefore appear as arguments of transcendent functions according to the above calculation rules . The name of a ratio usually contains an adjective such as relative or normalized or it ends with -number or -wert. Examples are the Reynolds number and the drag coefficient .

 ${\ displaystyle {\ begin {array} {lll} 1 \, {} ^ {0 \!} \! / \! _ {0} & = & 0 {,} 01 \\ 1 \, {} ^ {0 \ !} \! / \! _ {00} & = & 0 {,} 001 \\ 1 \, \ mathrm {ppm} & = & 0 {,} 000 \, 001 \ end {array}}}$
Special ratio units

Only in rare cases do different proportions belong to the same type of size; sometimes the unit symbols are therefore not truncated for better separation when specifying their size value. Ratios are often given in the units % , or ppm .

Ratio units have a special position if they are the ratio of equal units. These are always 1 and therefore idempotent , i.e. that is, they can be multiplied by themselves any number of times without changing their value. Some idempotent ratio units have special names, such as the radian (rad) unit of angle . In coherent systems of units, the ratio units are always 1, i.e. idempotent. With idempotent ratio units, the numerical values ​​can simply be multiplied. Example: From the information that 30% of the earth's surface is land area and Asia represents 30% of the land area, it does not follow that 900% of the earth's surface is covered by the continent Asia, because% is not idempotent, i.e.% 2 is not the same as%. If one says, however, that 0.3 of the earth's surface is land area and Asia takes 0.3 of the land area, one can conclude that Asia makes up 0.09 of the earth's surface, because the idempotent unit 1 is used here.

### Field and performance sizes

 {\ displaystyle {\ begin {aligned} F ^ {2} \ sim P & \ Leftrightarrow {\ frac {F_ {1} ^ {2}} {F_ {2} ^ {2}}} = {\ frac {P_ { 1}} {P_ {2}}} \\\ ln \ left ({\ frac {F_ {1}} {F_ {2}}} \ right) \, \ mathrm {Np} & = {\ frac {1 } {2}} \ ln \ left ({\ frac {P_ {1}} {P_ {2}}} \ right) \, \ mathrm {Np} \\ 20 \ lg \ left ({\ frac {F_ { 1}} {F_ {2}}} \ right) \, \ mathrm {dB} & = 10 \ lg \ left ({\ frac {P_ {1}} {P_ {2}}} \ right) \, \ mathrm {dB} \ end {aligned}}} Relationship between field sizes and performance sizes . ${\ displaystyle F}$${\ displaystyle P}$

Field sizes are used to describe physical fields . In linear systems, thesquare of a field size isproportional to its energetic state, which is recorded by a power size. Without having to know the exact regularity, it follows immediately that the ratio of two power quantities is equal to the square of the ratio of the associated field quantities. It is irrelevant whether both performancevariablesdirectly represent performance or related variables such as energy , intensity or power density .

The logarithmic ratios are of particular interest in many technical areas . Such quantities are called level or measure . If the natural logarithm is used for the formation , this is indicated by the unit Neper (Np) , if it is the decadic logarithm , the Bel (B) or more often its tenth, the decibel (dB) is used.

### State and process variables

In thermodynamics in particular , a distinction is made between state variables and process variables .

State variables are physical variables that represent a property of a system state. A further distinction is made between extensive and intensive sizes . Extensive quantities such as mass and amount of substance double their value when the system is doubled, while intensive quantities such as temperature and pressure remain constant. The distinction between the substance's own and the system's own state variables is also common.

Process variables, on the other hand, describe a process, namely the transition between system states. These include, in particular, the variables “ work ” ( ) and “ warmth ” ( ). In order to express their character as pure process variables, they are given in many places exclusively as differentials, with no , but an or đ in front of them. ${\ displaystyle W}$${\ displaystyle Q}$${\ displaystyle \ mathrm {d}}$${\ displaystyle \ delta}$

## Systems of measurement used in practice

Different measurement systems are used:

Mainly used by theorists and in the USA, with three basic sizes, in which all lengths are given in centimeters and electrical voltages in powers of the basic units cm, g (= gram) and s (= second)
System introduced in practical electrical engineering with four basic units, forerunner of the international system of units, contains the meter (= m), kilogram (= kg) and second (= s) as well as the ampere (= A) as a unit of current strength; the volt (= V) as a voltage unit results from the defined equality of the electrical and mechanical energy units watt-second and newton meter (1 Ws = 1 V A s = 1 N m = 1 kg m 2 s −2 )
all quantities are given in powers of just one unit, the energy unit eV , e.g. B. lengths as reciprocal energies, more precisely: in units of . The natural constants ( speed of light in a vacuum) and (reduced Planck's constant ) are replaced by one.${\ displaystyle \ hbar c / \ mathrm {eV}}$${\ displaystyle c}$${\ displaystyle \ hbar}$

In the various systems of measurement see natural laws, e.g. B. Maxwell's equations , formally different from; but, as mentioned, the laws of physics are invariant to such changes. In particular, one can convert from one system of measurement to another at any time, even if the relationships used can be complicated.

## literature

General

• Horst Teichmann: Physical applications of vector and tensor calculus. (= BI university paperbacks. 39). 3. Edition. Bibliographisches Institut, Mannheim u. a. 1973, ISBN 3-411-00039-2 (especially on the paragraph on scalars, vectors and tensors).
• Friedrich Kohlrausch: General information about measurements and their evaluation . In: Volkmar Kose, Siegfried Wagner (Ed.): Practical Physics . 24. rework. and exp. Edition. tape 3 . BG Teubner, Stuttgart 1996, ISBN 3-519-23000-3 , 9.1 Systems of terms and units, p. 3–19 ( ptb.de [PDF; 3.9 MB ; accessed on November 24, 2018] published by the Physikalisch-Technische Bundesanstalt).
• H. Fischer, H. Kaul: Mathematics for Physicists. Volume 1, 7th edition, Vieweg u. Teubner 2011, ISBN 978-3-8348-1220-9 .
• Hans Dieter Baehr: Physical quantities and their units - an introduction for students, natural scientists and engineers. (= Study books natural science and technology. Volume 19) Bertelsmann Universitätsverlag, Düsseldorf 1974, ISBN 3-571-19233-8 .
• Hans Rupp: Physical quantities, formulas, laws and definitions. 2nd edition, Oldenbourg Schulbuchverlag, 1995, ISBN 3-486-87093-9 .
• Paul A. Tipler: Physics. 3rd corrected reprint of the 1st edition 1994, Spektrum Akademischer Verlag, Heidelberg / Berlin 2000, ISBN 3-86025-122-8 .

Specifically for the physical size type

• Physical sizes and units ( PDF ) - detailed description of the formatting and specification of size values ​​in physical tests (210 kB).

## References and footnotes

1. Julius Wallot , who has made a great contribution to the theory of size, writes: Instead of “numerical value” one also says “measure”. I cannot consider this use of language appropriate. In French, “mesure” is common (also “valeur numérique”), in English “numerical value” (also “numerical measure” and “numerical magnitude”). "Dimensions in mm" is written on technical drawings and the numbers written on individual routes are called "dimensions". Above all, however, the (...) definition of the numerical value does not necessarily have anything to do with measure and measurement; these two words did not appear at all in a logical connection with the concept of numerical value. The German word “numerical value” is also easy to understand for foreigners. (Julius Wallot: size equations, units and dimensions . 2nd edition. Verlag Joh. Ambros. Barth, Leipzig 1957, page 50)
2. ^ R. Pitka et al .: Physics . Harri Deutsch, Frankfurt am Main. 2009, ISBN 978-3-8171-1852-6 , pp. 1 and 27 ( limited preview in Google Book search).
3. ^ Julius Wallot: Equations of quantities, units and dimensions . 2nd Edition. Publishers Joh. Ambros. Barth, Leipzig 1957.
4. DIN 1313 December 1998: sizes.
5. An exception are the common units for temperature , which also differ by a constant additive term. The reason lies in the different definition of the zero point.
6. ^ H. Goldstein, CP Poole Jr., JL Safko Sr .: Classical Mechanics. 3rd edition, Wiley-VCH, 2012, ISBN 978-3-527-66207-4 , Section 5.2: Tensors.
7. Pseudoscalars are scalars that reverse their sign when reflecting space . Example: the determinant (so-called late product) from 3 vectors.${\ displaystyle {\ vec {r}} \ to - {\ vec {r}}}$
8. Pseudovectors are vectors that do not reverse their sign when mirroring space . Example: the vector product of 2 vectors.${\ displaystyle {\ vec {r}} \ to - {\ vec {r}}}$
9. The inertia tensor conveys the relationship between the pseudovectors torque and angular acceleration in analogy to mass (or to a tensorial extension) . The force vector is analogous to the pseudo- vector torque, and the law of force = mass × acceleration is analogous to the law of torque = inertia tensor × angular acceleration.
10. The strain tensor describes the strain in a second direction depending on the first direction.
11. ^ Jack R. Vinson, RL Sierakowski: The behavior of structures composed of composite materials. Kluwer Academic, ISBN 1-4020-0904-6 , p. 76 ( limited preview in Google book search).
12. DIN 1313, December 1998: sizes. P. 5.
13. Ambler Thompson, Barry N. Taylor: Guide for the Use of the International System of Units (SI) . In: NIST Special Publication . tape 811 , 2008, p. 15 ( physics.nist.gov [PDF; accessed December 3, 2012]).
14. Note: According to the relevant standards and rules, the term “error” should not be used in this context. The terms " deviation " and " uncertainty " are therefore better (see EN ISO 80000-1, Chapter 7.3.4; " Glossary of metrology "; VIM and GUM )
15. Unfortunately, the German and international standards allow occasional mix-ups, especially for auxiliary units of measurement , e.g. B. "dB (C)"; Here, the "C" is an indication of the measurement method according to which the level measurement is determined, which is specified with the aid of the auxiliary unit of measurement decibel .
16. a b c The additions for neutrons, lead and windings are here in the incorrect formulas arbitrarily sometimes in italics, sometimes not in italics, because a correct spelling is not possible anyway and both possibilities occur. The corresponding correct notations, however, also follow the rules for italics mentioned in the section Spelling .
 This version was added to the list of articles worth reading on November 2, 2006 .