2.2.1.  The nature of the seven defining constants

The nature of the defining constants ranges from fundamental constants of nature to technical constants.

The use of a constant to define a unit disconnects definition from realization. This offers the possibility that completely different or new and superior practical realizations can be developed, as technologies evolve, without the need to change the definition.

A technical constant such as $K_{\text{cd}}$, the luminous efficacy of monochromatic radiation of frequency $540\times {10}^{12}\mathrm{Hz}$ refers to a special application. In principle, it can be chosen freely, such as to include conventional physiological or other weighting factors. In contrast, the use of a fundamental constant of nature, in general, does not allow this choice because it is related to other constants through the equations of physics.

The set of seven defining constants has been chosen to provide a fundamental, stable and universal reference that simultaneously allows for practical realizations with the smallest uncertainties. The technical conventions and specifications also take historical developments into account.

Both the Planck constant $h$ and the speed of light in vacuum $c$ are properly described as fundamental. They determine quantum effects and space-time properties, respectively, and affect all particles and fields equally on all scales and in all environments.

The elementary charge $e$ corresponds to a coupling strength of the electromagnetic force via the fine-structure constant $\alpha =e^2/\left(2c{\epsilon }_{0}h\right)$ where ${\epsilon }_0$ is the vacuum electric permittivity or electric constant. Some theories predict a variation of $\alpha$ over time. The experimental limits of the maximum possible variation in $\alpha$ are so low, however, that any effect on foreseeable practical measurements can be excluded.

The Boltzmann constant $k$ is a proportionality constant between the quantities temperature (with the unit kelvin) and energy (with the unit joule), whereby the numerical value is obtained from historical specifications of the temperature scale. The temperature of a system scales with the thermal energy, but not necessarily with the internal energy of a system. In statistical physics the Boltzmann constant connects the entropy $S$ with the number $\Omega$ of quantum-mechanically accessible states, $S=k\text{ ln }\Omega$.

The caesium frequency $\mathrm{\Delta }{\nu }_{\text{Cs}}$, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, has the character of an atomic parameter, which may be affected by the environment, such as electromagnetic fields. However, the underlying transition is well understood, stable and a good choice as a reference transition under practical considerations. The choice of an atomic parameter like $\mathrm{\Delta }{\nu }_{\text{Cs}}$ does not disconnect definition and realization in the same way that $h$, $c$, $e$, or $k$ do, but specifies the reference.

The Avogadro constant $N_{\text{A}}$ is a proportionality constant between the quantity amount of substance (with the unit mole) and the quantity number of elementary entities (with the unit one, symbol 1). Thus it has the character of a constant of proportionality similar to the Boltzmann constant $k$.

The luminous efficacy of monochromatic radiation of frequency $540\times {10}^{12}\mathrm{Hz}$, $K_{\text{cd}}$, is a technical constant that gives an exact numerical relationship between the purely physical characteristics of the radiant power stimulating the human eye ($W$) and its photobiological response defined by the luminous flux due to the spectral responsivity of a standard observer ($\mathrm{lm}$) at a frequency of $540\times {10}^{12}\text{ hertz}$.

2.3.1.  Base units

The base units of the SI are listed in Table 2.

Table 2.  SI base units

Base quantity		Base unit
Name	Typical symbol	Name	Symbol
time	$t$	second	$s$
length	$l,x,r$, etc.	metre	$m$
mass	$m$	kilogram	$\mathrm{kg}$
electric current	$I,i$	ampere	$A$
thermodynamic temperature	$T$	kelvin	$K$
amount of substance	$n$	mole	$\mathrm{mol}$
luminous intensity	$I_{\text{v}}$	candela	$\mathrm{cd}$
Note:  The symbols for quantities are generally single letters of the Latin or Greek alphabets, printed in an italic font, and are recommendations. The symbols for units are printed in an upright (roman) font and are mandatory, see Chapter 5.

Starting from the definition of the SI in terms of fixed numerical values of the defining constants, definitions of each of the seven base units are deduced by using, as appropriate, one or more of these defining constants to give the following set of definitions:

The second

The second, symbol $s$, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency, $\mathrm{\Delta }{\nu }_{\text{Cs}}$, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit $\mathrm{Hz}$, which is equal to $s^{-1}$.

This definition implies the exact relation $\mathrm{\Delta }{\nu }_{\text{Cs}}=\mathrm{9 192 631 770}\mathrm{Hz}$. Inverting this relation gives an expression for the unit second in terms of the defining constant $\mathrm{\Delta }{\nu }_{\text{Cs}}$:

The effect of this definition is that the second is equal to the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the unperturbed ground state of the ¹³³Cs atom.

The reference to an unperturbed atom is intended to make it clear that the definition of the SI second is based on an isolated caesium atom that is unperturbed by any external field, such as ambient black-body radiation.

The second, so defined, is the unit of proper time in the sense of the general theory of relativity. To allow the provision of a coordinated time scale, the signals of different primary clocks in different locations are combined, which have to be corrected for relativistic caesium frequency shifts (see Section 2.3.6).

The CIPM has adopted various secondary representations of the second, based on a selected number of spectral lines of atoms, ions or molecules. The unperturbed frequencies of these lines can be determined with a relative uncertainty not lower than that of the realization of the second based on the ¹³³Cs hyperfine transition frequency, but some can be reproduced with superior stability.

The metre

The metre, symbol $m$, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum, $c$, to be 299 792 458 when expressed in the unit $m\cdot s^{-1}$, where the second is defined in terms of the caesium frequency $\mathrm{\Delta }{\nu }_{\text{Cs}}$.

This definition implies the exact relation $c=\mathrm{299 792 458}m\cdot s^{-1}$. Inverting this relation gives an exact expression for the metre in terms of the defining constants $c$ and $\mathrm{\Delta }{\nu }_{\text{Cs}}$:

The effect of this definition is that one metre is the length of the path travelled by light in vacuum during a time interval with duration of $1/\mathrm{299 792 458}$ of a second.

The kilogram

The kilogram, symbol $\mathrm{kg}$, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant, $h$, to be $\mathrm{6.626 070 15}\times {10}^{-34}$ when expressed in the unit $J\cdot s$, which is equal to $\mathrm{kg}\cdot m^2\cdot s^{-1}$, where the metre and the second are defined in terms of $c$ and $\mathrm{\Delta }{\nu }_{\text{Cs}}$.

This definition implies the exact relation $h=\mathrm{6.626 070 15}\times {10}^{-34}\mathrm{kg}\cdot m^2\cdot s^{-1}$. Inverting this relation gives an exact expression for the kilogram in terms of the three defining constants $h$, $\mathrm{\Delta }{\nu }_{\text{Cs}}$ and $c$:

which is equal to

The effect of this definition is to define the unit $\mathrm{kg}\cdot m^2\cdot s^{-1}$ (the unit of both the physical quantities action and angular momentum). Together with the definitions of the second and the metre this leads to a definition of the unit of mass expressed in terms of the Planck constant $h$.

The previous definition of the kilogram fixed the value of the mass of the international prototype of the kilogram, $m\left(\text{𝒦}\right)$, to be equal to one kilogram exactly and the value of the Planck constant $h$ had to be determined by experiment. The present definition fixes the numerical value of $h$ exactly and the mass of the prototype has now to be determined by experiment.

The number chosen for the numerical value of the Planck constant in this definition is such that at the time of its adoption, the kilogram was equal to the mass of the international prototype, $m\left(\text{𝒦}\right)=1\mathrm{kg}$, with a relative standard uncertainty of $1\times {10}^{-8}$, which was the standard uncertainty of the combined best estimates of the value of the Planck constant at that time.

Note that with the present definition, primary realizations can be established, in principle, at any point in the mass scale.

The ampere

The ampere, symbol $A$, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge, $e$, to be $\mathrm{1.602 176 634}\times {10}^{-19}$ when expressed in the unit $C$, which is equal to $A\cdot s$, where the second is defined in terms of $\mathrm{\Delta }{\nu }_{\text{Cs}}$.

This definition implies the exact relation $e=\mathrm{1.602 176 634}\times {10}^{-19}A\cdot s$. Inverting this relation gives an exact expression for the unit ampere in terms of the defining constants $e$ and $\mathrm{\Delta }{\nu }_{\text{Cs}}$:

which is equal to

The effect of this definition is that one ampere is the electric current corresponding to the flow of $1/(\mathrm{1.602 176 634}\times {10}^{-19})$ elementary charges per second.

The previous definition of the ampere was based on the force between two current carrying conductors and had the effect of fixing the value of the vacuum magnetic permeability ${\mu }_0$ (also known as the magnetic constant) to be exactly $4\mathrm{\pi }\times {10}^{-7}H\cdot m^{-1}=4\mathrm{\pi }\times {10}^{-7}N\cdot A^{-2}$, where $H$ and $N$ denote the coherent derived units henry and newton, respectively. The new definition of the ampere fixes the value of $e$ instead of ${\mu }_0$. As a result, ${\mu }_0$ must be determined experimentally.

It also follows that since the vacuum electric permittivity ${\mathrm{\epsilon }}_0$ (also known as the electric constant), the characteristic impedance of vacuum $Z_0$, and the admittance of vacuum $Y_0$ are equal to $1/{\mu }_0{c}^2$, ${\mu }_{0}c$, and $1/{\mu }_{0}c$, respectively, the values of ${\epsilon }_0$, $Z_0$, and $Y_0$ must now also be determined experimentally, and are affected by the same relative standard uncertainty as ${\mu }_0$ since $c$ is exactly known. The product ${\epsilon }_0{\mu }_0=1/c^2$ and quotient $Z_0/{\mu }_0=c$ remain exact. At the time of adopting the present definition of the ampere, ${\mu }_0$ was equal to $4\mathrm{\pi }\times {10}^{-7}H\cdot m^{-1}$ with a relative standard uncertainty of $2.3\times {10}^{-10}$.

The kelvin

The kelvin, symbol $K$, is the SI unit of thermodynamic temperature. It is defined by taking the fixed numerical value of the Boltzmann constant, $k$, to be $\mathrm{1.380 649}\times {10}^{-23}$ when expressed in the unit $J\cdot K^{-1}$, which is equal to $\mathrm{kg}\cdot m^2\cdot s^{-2}\cdot K^{-1}$, where the kilogram, metre and second are defined in terms of $h$, $c$ and $\mathrm{\Delta }{\nu }_{\text{Cs}}$.

This definition implies the exact relation $k=\mathrm{1.380 649}\times {10}^{-23}\mathrm{kg}\cdot m^2\cdot s^{-2}\cdot K^{-1}$. Inverting this relation gives an exact expression for the kelvin in terms of the defining constants $k$, $h$ and $\mathrm{\Delta }{\nu }_{\text{Cs}}$:

which is equal to

The effect of this definition is that one kelvin is equal to the change of thermodynamic temperature that results in a change of thermal energy $kT$ by $\mathrm{1.380 649}\times {10}^{-23}J$.

The previous definition of the kelvin set the temperature of the triple point of water, $T_{\text{TPW}}$, to be exactly $273.16K$. Due to the fact that the present definition of the kelvin fixes the numerical value of $k$ instead of $T_{\text{TPW}}$, the latter must now be determined experimentally. At the time of adopting the present definition $T_{\text{TPW}}$ was equal to $273.16K$ with a relative standard uncertainty of $3.7\times {10}^{-7}$ based on measurements of $k$ made prior to the redefinition.

As a result of the way temperature scales used to be defined, it remains common practice to express a thermodynamic temperature, symbol $T$, in terms of its difference from the reference temperature $T_0=273.15K$, close to the ice point. This difference is called the Celsius temperature, symbol $t$, which is defined by the quantity equation

The unit of Celsius temperature is the degree Celsius, symbol $\mathrm{°C}$, which is by definition equal in magnitude to the unit kelvin. A difference or interval of temperature may be expressed in kelvins or in degrees Celsius, the numerical value of the temperature difference being the same in either case. However, the numerical value of a Celsius temperature expressed in degrees Celsius is related to the numerical value of the thermodynamic temperature expressed in kelvins by the relation

(see Section 5.4.1 for an explanation of the notation used here).

The kelvin and the degree Celsius are also units of the International Temperature Scale of 1990 (ITS-90) adopted in https://www.bipm.org/en/committees/ci/cipm/78-1989/resolution-5. Note that the ITS-90 defines two quantities $T_{90}$ and $t_{90}$ which are close approximations to the corresponding thermodynamic temperatures $T$ and $t$.

Note that with the present definition, primary realizations of the kelvin can, in principle, be established at any point of the temperature scale.

The mole

The mole, symbol $\mathrm{mol}$, is the SI unit of amount of substance. One mole contains exactly $\mathrm{6.022 140 76}\times {10}^{23}$ elementary entities. This number is the fixed numerical value of the Avogadro constant, $N_{\text{A}}$, when expressed in the unit ${\mathrm{mol}}^{-1}$ and is called the Avogadro number.

The amount of substance, symbol $n$, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles.

This definition implies the exact relation $N_{\text{A}}=\mathrm{6.022 140 76}\times {10}^{23}{\mathrm{mol}}^{-1}$. Inverting this relation gives an exact expression for the mole in terms of the defining constant $N_{\text{A}}$:

The effect of this definition is that the mole is the amount of substance of a system that contains $\mathrm{6.022 140 76}\times {10}^{23}$ specified elementary entities.

The previous definition of the mole fixed the value of the molar mass of carbon 12, $M$(¹²C), to be exactly $0.012\mathrm{kg}\cdot {\mathrm{mol}}^{-1}$. According to the present definition $M$(¹²C) is no longer known exactly and must be determined experimentally. The value chosen for $N_{\text{A}}$ is such that at the time of adopting the present definition of the mole, $M$(¹²C) was equal to $0.012\mathrm{kg}\cdot {\mathrm{mol}}^{-1}$ with a relative standard uncertainty of $4.5\times {10}^{-10}$.

The molar mass of any atom or molecule $\text{X}$ may still be obtained from its relative atomic mass from the equation

and the molar mass of any atom or molecule $\text{X}$ is also related to the mass of the elementary entity $m\left(\text{X}\right)$ by the relation

In these equations $M_{\text{u}}$ is the molar mass constant, equal to $M$(¹²C)/12 and $m_{\text{u}}$ is the unified atomic mass constant, equal to $m$(¹²C)/12. They are related to the Avogadro constant through the relation

In the name “amount of substance”, the word “substance” will typically be replaced by words to specify the substance concerned in any particular application, for example “amount of hydrogen chloride”, or “amount of benzene”. It is important to give a precise definition of the elementary entity involved (as emphasized in the definition of the mole); this should preferably be done by specifying the molecular chemical formula of the material involved. Although the word “amount” has a more general dictionary definition, the abbreviation of the full name “amount of substance” to “amount” may be used for brevity. This also applies to derived quantities such as “amount-of-substance concentration”, which may simply be called “amount concentration”. In the field of clinical chemistry, the name “amount-of-substance concentration” is generally abbreviated to “substance concentration”.

The candela

The candela, symbol $\mathrm{cd}$, is the SI unit of luminous intensity in a given direction. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency $540\times {10}^{12}\mathrm{Hz}$, $K_{\text{cd}}$, to be 683 when expressed in the unit $\mathrm{lm}\cdot W^{-1}$, which is equal to $\mathrm{cd}\cdot \mathrm{sr}\cdot W^{-1}$, or $\mathrm{cd}\cdot \mathrm{sr}\cdot {\mathrm{kg}}^{-1}\cdot m^{-2}\cdot s^3$, where the kilogram, metre and second are defined in terms of $h$, $c$ and $\mathrm{\Delta }{\nu }_{\text{Cs}}$.

This definition implies the exact relation $K_{\text{cd}}=683\mathrm{cd}\cdot \mathrm{sr}\cdot {\mathrm{kg}}^{-1}\cdot m^{-2}\cdot s^3$ for monochromatic radiation of frequency $\nu =540\times {10}^{12}\mathrm{Hz}$. Inverting this relation gives an exact expression for the candela in terms of the defining constants $K_{\text{cd}}$, $h$ and $\mathrm{\Delta }{\nu }_{\text{Cs}}$:

which is equal to

The effect of this definition is that one candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency $540\times {10}^{12}\mathrm{Hz}$ and has a radiant intensity in that direction of $(1/683)W\cdot {\mathrm{sr}}^{-1}$. The definition of the steradian is given below Table 4.

2.3.2.  Practical realization of SI units

The highest-level experimental methods used for the realization of units using the equations of physics are known as primary methods. The essential characteristic of a primary method is that it allows a quantity to be measured in a particular unit by using only measurements of quantities that do not involve that unit. In the present formulation of the SI, the basis of the definitions is different from that used previously, so that new methods may be used for the practical realization of SI units.

Instead of each definition specifying a particular condition or physical state, which sets a fundamental limit to the accuracy of realization, a user is now free to choose any convenient equation of physics that links the defining constants to the quantity intended to be measured. This is a much more general way of defining the basic units of measurement. It is not limited by today’s science or technology; future developments may lead to different ways of realizing units to a higher accuracy. When defined this way, there is, in principle, no limit to the accuracy with which a unit might be realized. The exception remains the definition of the second, in which the original microwave transition of caesium must remain, for the time being, the basis of the definition. For a more comprehensive explanation of the realization of SI units see Appendix 2.

2.3.3.  Dimensions of quantities

Physical quantities can be organized in a system of dimensions, where the system used is decided by convention. Each of the seven base quantities used in the SI is regarded as having its own dimension. The symbols used for the base quantities and the symbols used to denote their dimension are shown in Table 3.

Table 3.  Base quantities and dimensions used in the SI

Base quantity	Typical symbol for quantity	Symbol for dimension
time	$t$	$\text{𝖳}$
length	$l,x,r$, etc.	$\text{𝖫}$
mass	$m$	$\text{𝖬}$
electric current	$I,i$	$\text{𝖨}$
thermodynamic temperature	$T$	$\text{ϴ}$
amount of substance	$n$	$\text{𝖭}$
luminous intensity	$I_{\text{v}}$	$\text{𝖩}$

All other quantities, with the exception of quantities that are a number of entities, are derived quantities, which may be written in terms of base quantities according to the equations of physics. The dimensions of the derived quantities are written as products of powers of the dimensions of the base quantities using the equations that relate the derived quantities to the base quantities. In general the dimension of any quantity $Q$ is written in the form of a dimensional product,

where the exponents $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, $\zeta$ and $\eta$, which are generally small integers, which can be positive, negative, or zero, are called the dimensional exponents.

There are quantities $Q$ for which the defining equation is such that all of the dimensional exponents in the equation for the dimension of $Q$ are zero. This is true in particular for any quantity that is defined as the ratio of two quantities of the same kind. For example, the refractive index is the ratio of two speeds and the relative permittivity is the ratio of the permittivity of a dielectric medium to that of free space. Such quantities are simply numbers. The associated unit is the unit one, symbol 1, although this is rarely explicitly written (see Section 5.4.7).

There are also some quantities that cannot be described in terms of the seven base quantities of the SI, but are quantities that are a number of entities. Examples are a number of cellular or biomolecular entities, or degeneracy in quantum mechanics. These quantities are also quantities with the unit one.

The unit one is the neutral element of any system of units – necessary and present automatically. There is no requirement to introduce it formally by decision. Formal traceability to the SI can be established through appropriate, validated measurement procedures.

For reasons of history and convention, plane and solid angles are treated within the SI as quantities with the unit one (see Section 2.3.4). The symbols $\mathrm{rad}$ and $\mathrm{sr}$ are written explicitly where appropriate, in order to emphasize that, for radians or steradians, the quantity being considered is, or involves the plane angle or solid angle respectively. For steradians it emphasizes the distinction between units of flux and intensity in radiometry and photometry for example. It is noted that practice in mathematics and some areas of science $\mathrm{rad}$ and $\mathrm{sr}$ are omitted.

It is especially important to have a clear description of any quantity with the unit one (see Section 5.4.7) that is expressed as a ratio of quantities of the same kind (for example length ratios or amount fractions) or as a number of entities (for example number of photons or decays).

2.3.4.  Derived units

Derived units are defined as products of powers of the base units. When the numerical factor of this product is one, the derived units are called coherent derived units. The base and coherent derived units of the SI form a coherent set, designated the set of coherent SI units. The word “coherent” here means that equations between the numerical values of quantities take exactly the same form as the equations between the quantities themselves.

Some of the coherent derived units in the SI are given special names. Table 4 lists 22 SI units with special names. Together with the seven base units (Table 2) they form the core of the set of SI units. All other SI units are combinations of some of these 29 units.

It is important to note that any of the seven base units and 22 SI units with special names can be constructed directly from the seven defining constants. In fact, the units of the seven defining constants include both base and derived units.

The CGPM has adopted a series of prefixes for use in forming the decimal multiples and sub-multiples of the coherent SI units (see Chapter 3). They are convenient for expressing the values of quantities that are much larger than or much smaller than the coherent unit. However, when prefixes are used with SI units, the resulting units are no longer coherent, because the prefix introduces a numerical factor other than one. Prefixes may be used with any of the 29 SI units with special names with the exception of the base unit kilogram, which is further explained in Chapter 3.

Table 4.  The 22 SI units with special names and symbols

Derived quantity	Special name of unit	Symbol	Unit expressed in terms of base unitsa	Unit expressed in terms of other SI units
plane angle	radianb	$\mathrm{rad}$	b	1
solid angle	steradianc	$\mathrm{sr}$	c	1
frequency	hertzd	$\mathrm{Hz}$	$s^{-1}$
force	newton	$N$	$\mathrm{kg}\cdot m\cdot s^{-2}$
pressure, stress	pascal	$\mathrm{Pa}$	$\mathrm{kg}\cdot m^{-1}\cdot s^{-2}$	$N\cdot m^{-2}$
energy, work, amount of heat	joule	$J$	$\mathrm{kg}\cdot m^2\cdot s^{-2}$	$N\cdot m$
power, radiant flux	watt	$W$	$\mathrm{kg}\cdot m^2\cdot s^{-3}$	$J\cdot s^{-1}$
electric charge	coulomb	$C$	$A\cdot s$
electric potential differencee	volt	$V$	$\mathrm{kg}\cdot m^2\cdot s^{-3}\cdot A^{-1}$	$W\cdot A^{-1}$
capacitance	farad	$F$	${\mathrm{kg}}^{-1}\cdot m^{-2}\cdot s^4\cdot A^2$	$C\cdot V^{-1}$
electric resistance	ohm	$\Omega$	$\mathrm{kg}\cdot m^2\cdot s^{-3}\cdot A^{-2}$	$V\cdot A^{-1}$
electric conductance	siemens	$S$	${\mathrm{kg}}^{-1}\cdot m^{-2}\cdot s^3\cdot A^2$	$A\cdot V^{-1}$
magnetic flux	weber	$\mathrm{Wb}$	$\mathrm{kg}\cdot m^2\cdot s^{-2}\cdot A^{-1}$	$V\cdot s$
magnetic flux density	tesla	$T$	$\mathrm{kg}\cdot s^{-2}\cdot A^{-1}$	$\mathrm{Wb}\cdot m^{-2}$
inductance	henry	$H$	$\mathrm{kg}\cdot m^2\cdot s^{-2}\cdot A^{-2}$	$\mathrm{Wb}\cdot A^{-1}$
Celsius temperature	degree Celsiusf	$\mathrm{°C}$	$K$
luminous flux	lumen	$\mathrm{lm}$	$\mathrm{cd}\cdot \mathrm{sr}$g	$\mathrm{cd}\cdot \mathrm{sr}$
illuminance	lux	$\mathrm{lx}$	$\mathrm{cd}\cdot \mathrm{sr}\cdot m^{-2}$	$\mathrm{lm}\cdot m^{-2}$
activity referred to a radionuclidedh	becquerel	$\mathrm{Bq}$	$s^{-1}$
absorbed dose, kerma	gray	$\mathrm{Gy}$	$m^2\cdot s^{-2}$	$J\cdot {\mathrm{kg}}^{-1}$
dose equivalent	sieverti	$\mathrm{Sv}$	$m^2\cdot s^{-2}$	$J\cdot {\mathrm{kg}}^{-1}$
catalytic activity	katal	$\mathrm{kat}$	$\mathrm{mol}\cdot s^{-1}$
a  The order of symbols for base units in this Table is different from that in the 8th edition following a decision by the CCU at its 21st meeting (2013) to return to the original order in https://www.bipm.org/en/committees/cg/cgpm/11-1960/resolution-12 in which newton was written $\mathrm{kg}\cdot m\cdot s^{-2}$, the joule as $\mathrm{kg}\cdot m^2\cdot s^{-2}$ and $J\cdot s$ as $\mathrm{kg}\cdot m^{-2}\cdot s^{-1}$. The intention was to reflect the underlying physics of the corresponding quantity equations although for some more complex derived units this may not be possible. b  The radian is the coherent unit for plane angle. One radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius. This suggests $\mathrm{rad}=\text{m/m}$ but this representation is not intrinsic and may be misleading since angle is not the same kind of quantity as other length ratios. An alternative definition is that a right angle is equal to $\frac{\mathrm{\pi }}{2}\mathrm{rad}$. The radian is also the coherent unit for phase angle. For periodic phenomena, the phase angle increases by $2\mathrm{\pi }\mathrm{rad}$ in one period. c  The steradian is the coherent unit for solid angle. One steradian is the solid angle subtended at the centre of a sphere by an area of the surface that is equal to the squared radius. This suggests $\mathrm{sr}={\text{m}}^2/{\text{m}}^2$, but this representation is not intrinsic and may be misleading since solid angle is not the same kind of quantity as other area ratios. An alternative definition is that a complete sphere subtends $4\mathrm{\pi }\mathrm{sr}$ about its centre. d  The hertz shall only be used for periodic phenomena and the becquerel shall only be used for stochastic processes in activity referred to a radionuclide. e  Electric potential difference is also called “voltage” in many countries, as well as “electric tension” or simply “tension” in some countries. f  The degree Celsius is used to express Celsius temperatures. The numerical value of a temperature difference or temperature interval is the same when expressed in either degrees Celsius or in kelvins. g  In photometry the name steradian and the symbol $\mathrm{sr}$ are usually retained in expressions for units. h  Activity referred to a radionuclide is sometimes incorrectly called radioactivity. i  See https://www.bipm.org/en/committees/ci/cipm/91-2002/resolution-2 on the use of the sievert.

The seven base units and 22 units with special names and symbols may be used in combination to express the units of other derived quantities. Since the number of quantities is without limit, it is not possible to provide a complete list of derived quantities and derived units. Table 5 lists some examples of derived quantities and the corresponding coherent derived units expressed in terms of base units. In addition, Table 6 lists examples of coherent derived units whose names and symbols also include derived units. The complete set of SI units includes both the coherent set and the multiples and sub-multiples formed by using the SI prefixes.

Table 5.  Examples of coherent derived units in the SI expressed in terms of base units

Derived quantity	Typical symbol of quantity	Derived unit expressed in terms of base units
area	$A$	$m^2$
volume	$V$	$m^3$
speed, velocity	$v$	$m\cdot s^{-1}$
acceleration	$a$	$m\cdot s^{-2}$
wavenumber	$\sigma$	$m^{-1}$
density, mass density	$\rho$	$\mathrm{kg}\cdot m^{-3}$
surface density	${\rho }_{\mathrm{A}}$	$\mathrm{kg}\cdot m^{-2}$
specific volume	$v$	$m^3\cdot {\mathrm{kg}}^{-1}$
current density	$j$	$A\cdot m^{-2}$
magnetic field strength	$H$	$A\cdot m^{-1}$
amount of substance concentration	$c$	$\mathrm{mol}\cdot m^{-3}$
mass concentration	$\rho ,\gamma$	$\mathrm{kg}\cdot m^{-3}$
luminance	$L_{\text{v}}$	$\mathrm{cd}\cdot m^{-2}$

Table 6.  Examples of SI coherent derived units whose names and symbols include SI coherent derived units with special names and symbols

Derived quantity	Name of coherent derived unit	Symbol	Derived unit expressed in terms of base units
dynamic viscosity	pascal second	$\mathrm{Pa}\cdot s$	$\mathrm{kg}\cdot m^{-1}\cdot s^{-1}$
moment of force	newton metre	$N\cdot m$	$\mathrm{kg}\cdot m^2\cdot s^{-2}$
surface tension	newton per metre	$N\cdot m^{-1}$	$\mathrm{kg}\cdot s^{-2}$
angular velocity, angular frequency	radian per second	$\mathrm{rad}\cdot s^{-1}$
angular acceleration	radian per second squared	$\mathrm{rad}\cdot s^{-2}$
heat flux density, irradiance	watt per square metre	$W\cdot m^{-2}$	$\mathrm{kg}\cdot s^{-3}$
heat capacity, entropy	joule per kelvin	$J\cdot K^{-1}$	$\mathrm{kg}\cdot m^2\cdot s^{-2}\cdot K^{-1}$
specific heat capacity, specific entropy	joule per kilogram kelvin	$J\cdot K^{-1}\cdot {\mathrm{kg}}^{-1}$	$m^2\cdot s^{-2}\cdot K^{-1}$
specific energy	joule per kilogram	$J\cdot {\mathrm{kg}}^{-1}$	$m^2\cdot s^{-2}$
thermal conductivity	watt per metre kelvin	$W\cdot m^{-1}\cdot K^{-1}$	$\mathrm{kg}\cdot m\cdot s^{-3}\cdot K^{-1}$
energy density	joule per cubic metre	$J\cdot m^{-3}$	$\mathrm{kg}\cdot m^{-1}\cdot s^{-2}$
electric field strength	volt per metre	$V\cdot m^{-1}$	$\mathrm{kg}\cdot m\cdot s^{-3}\cdot A^{-1}$
electric charge density	coulomb per cubic metre	$C\cdot m^{-3}$	$A\cdot s\cdot m^{-3}$
surface charge density	coulomb per square metre	$C\cdot m^{-2}$	$A\cdot s\cdot m^{-2}$
electric flux density, electric displacement	coulomb per square metre	$C\cdot m^{-2}$	$A\cdot s\cdot m^{-2}$
permittivity	farad per metre	$F\cdot m^{-1}$	${\mathrm{kg}}^{-1}\cdot m^{-3}\cdot s^4\cdot A^2$
permeability	henry per metre	$H\cdot m^{-1}$	$\mathrm{kg}\cdot m\cdot s^{-2}\cdot A^{-2}$
molar energy	joule per mole	$J\cdot {\mathrm{mol}}^{-1}$	$\mathrm{kg}\cdot m^2\cdot s^{-2}\cdot {\mathrm{mol}}^{-1}$
molar entropy, molar heat capacity	joule per mole kelvin	$J\cdot K^{-1}\cdot {\mathrm{mol}}^{-1}$	$\mathrm{kg}\cdot m^2\cdot s^{-2}\cdot {\mathrm{mol}}^{-1}\cdot K^{-1}$
exposure ($\text{x}$- and $\mathrm{\gamma }$-rays)	coulomb per kilogram	$C\cdot {\mathrm{kg}}^{-1}$	$A\cdot s\cdot {\mathrm{kg}}^{-1}$
absorbed dose rate	gray per second	$\mathrm{Gy}\cdot s^{-1}$	$m^2\cdot s^{-3}$
radiant intensity	watt per steradian	$W\cdot {\mathrm{sr}}^{-1}$	$\mathrm{kg}\cdot m^2\cdot s^{-3}$
radiance	watt per square metre steradian	$W\cdot {\mathrm{sr}}^{-1}\cdot m^{-2}$	$\mathrm{kg}\cdot s^{-3}$
catalytic activity concentration	katal per cubic metre	$\mathrm{kat}\cdot m^{-3}$	$\mathrm{mol}\cdot s^{-1}\cdot m^{-3}$

It is important to emphasize that each physical quantity has only one coherent SI unit, even though this unit can be expressed in different forms by using some of the special names and symbols.

The converse, however, is not true, because in general several different quantities may share the same SI unit. For example, for the quantity heat capacity as well as for the quantity entropy the SI unit is joule per kelvin. Similarly, for the base quantity electric current as well as the derived quantity magnetomotive force the SI unit is the ampere. It is therefore important not to use the unit alone to specify the quantity. This applies not only to technical texts, but also, for example, to measuring instruments (i.e. the instrument read-out needs to indicate both the unit and the quantity measured).

In practice, with certain quantities, preference is given to the use of certain special unit names to facilitate the distinction between different quantities having the same dimension. When using this freedom, one may recall the process by which this quantity is defined. For example, the quantity torque is the cross product of a position vector and a force vector. The SI unit is newton metre. Even though torque has the same dimension as energy (SI unit joule), the joule is never used for expressing torque.

The SI unit of frequency is hertz, the SI unit of angular velocity and angular frequency is radian per second The SI unit of activity is becquerel, implying counts per second. The use of the different names emphasizes the different nature of the quantities concerned. It is especially important to carefully distinguish frequencies from angular frequencies, because by definition their numerical values differ by a factor¹ of $2\mathrm{\pi }$. Ignoring this fact may cause an error of $2\mathrm{\pi }$. Note that in some countries, frequency values are conventionally expressed using “cycle/s” (“cps”) or “revolutions/s” (“rev/s”) instead of the SI unit $\mathrm{Hz}$, although “cycle”, “cps”, “revolution” and “rev” are not units in the SI. Note also that it is common, although not recommended, to use the term frequency for quantities expressed in rad/s. Because of this, it is recommended that quantities called “frequency”, “angular frequency”, and “angular velocity” always be given explicit units of $\mathrm{Hz}$ or $\mathrm{rad}\cdot s^{-1}$ and not $s^{-1}$.

In the field of ionizing radiation, the SI unit becquerel rather than the reciprocal second is used. The SI units gray and sievert are used for absorbed dose and dose equivalent, respectively, rather than joule per kilogram. The special names becquerel, gray and sievert were specifically introduced because of the dangers to human health that might arise from mistakes involving the units reciprocal second and joule per kilogram, in case the latter units were incorrectly taken to identify the different quantities involved.

Special care must be taken when expressing temperatures or temperature differences, respectively. A temperature difference of $1K$ equals that of $1\mathrm{°C}$, but for an absolute temperature the difference of $273.15K$ must be taken into account. The unit degree Celsius is only coherent when expressing temperature differences.

2.3.5.  Units for quantities that describe biological and physiological effects

Four of the SI units listed in Table 2 and Table 4 include physiological weighting factors: candela, lumen, lux and sievert.

Lumen and lux are derived from the base unit candela. Like the candela, they carry information about human vision. The candela was established as a base unit in 1954, acknowledging the importance of light in daily life. Further information on the units and conventions used for defining photochemical and photobiological quantities is in Appendix 3.

Ionizing radiation deposits energy in irradiated matter. The ratio of deposited energy to mass is termed absorbed dose $D$. As decided by the CIPM in 2002, the quantity dose equivalent $H=QD$ is the product of the absorbed dose $D$ and a numerical quality factor $Q$ that takes into account the biological effectiveness of the radiation and is dependent on the energy and type of radiation.

There are units for quantities that describe biological effects and involve weighting factors, which are not SI units. Two examples are given here:

Sound causes pressure fluctuations in the air, superimposed on the normal atmospheric pressure, that are sensed by the human ear. The sensitivity of the ear depends on the frequency of the sound, but it is not a simple function of either the pressure changes or the frequency. Therefore, frequency-weighted quantities are used in acoustics to approximate the way in which sound is perceived. They are used, for example, for measurements concerning protection against hearing damage. The effect of ultrasonic acoustic waves poses similar concerns in medical diagnosis and therapy.

There is a class of units for quantifying the biological activity of certain substances used in medical diagnosis and therapy that cannot yet be defined in terms of the units of the SI. This lack of definition is because the mechanism of the specific biological effect of these substances is not yet sufficiently well understood for it to be quantifiable in terms of physico-chemical parameters. In view of their importance for human health and safety, the World Health Organization (WHO) has taken responsibility for defining WHO International Units (IU) for the biological activity of such substances.

2.3.6.  SI units in the framework of the general theory of relativity

The practical realization of a unit and the process of comparison require a set of equations within a framework of a theoretical description. In some cases, these equations include relativistic effects.

For frequency standards it is possible to establish comparisons at a distance by means of electromagnetic signals. To interpret the results, the general theory of relativity is required, since it predicts, among other things, a relative frequency shift between standards of about 1 part in ${10}^{16}$ per metre of altitude difference at the surface of the earth. Effects of this magnitude must be corrected for when comparing the best frequency standards.

When practical realizations are compared locally, i.e. in a small space-time domain, effects due to the space-time curvature described by the general theory of relativity can be neglected. When realizations share the same space-time coordinates (for example the same motion and acceleration or gravitational field), relativistic effects may be neglected entirely.

5.4.1.  Value and numerical value of a quantity, and the use of quantity calculus

Symbols for quantities are generally single letters set in an italic font, although they may be qualified by further information in subscripts or superscripts or in brackets. For example, $C$ is the recommended symbol for heat capacity, $c_{\text{m}}$ for molar heat capacity, $c_{\text{m},p}$ for molar heat capacity at constant pressure, and $c_{\text{m},V}$ for molar heat capacity at constant volume.

Recommended names and symbols for quantities are listed in many standard references, such as the ISO/IEC 80000 series Quantities and units, the IUPAP SUNAMCO Red Book Symbols, Units and Nomenclature in Physics and the IUPAC Green Book Quantities, Units and Symbols in Physical Chemistry. However, symbols for quantities are recommendations (in contrast to symbols for units, for which the use of the correct form is mandatory). In certain circumstances authors may wish to use a symbol of their own choice for a quantity, for example to avoid a conflict arising from the use of the same symbol for two different quantities. In such cases, the meaning of the symbol must be clearly stated. However, neither the name of a quantity, nor the symbol used to denote it, should imply any particular choice of unit.

Symbols for units are treated as mathematical entities. In expressing the value of a quantity as the product of a numerical value and a unit, both the numerical value and the unit may be treated by the ordinary rules of algebra. This procedure is described as the use of quantity calculus, or the algebra of quantities. For example, the equation $p=48\mathrm{kPa}$ may equally be written as $p/\mathrm{kPa}=48$. It is common practice to write the quotient of a quantity and a unit in this way for a column heading in a table, so that the entries in the table are simply numbers. For example, a table of the velocity squared versus pressure may be formatted as shown below.

$p/\mathrm{kPa}$	$v^2/\left(m\cdot s^{-2}\right)$
48.73	94 766
72.87	94 771
135.42	94 784

The axes of a graph may also be labelled in this way, so that the tick marks are labelled only with numbers, as in the graph below.

5.4.2.  Quantity symbols and unit symbols

Unit symbols must not be used to provide specific information about the quantity and should never be the sole source of information on the quantity. Units are never qualified by further information about the nature of the quantity; any extra information on the nature of the quantity should be attached to the quantity symbol and not to the unit symbol.

5.4.3.  Formatting the value of a quantity

The numerical value always precedes the unit and a space is always used to separate the unit from the number. Thus the value of the quantity is the product of the number and the unit. The space between the number and the unit is regarded as a multiplication sign (just as a space between units implies multiplication). The only exceptions to this rule are for the unit symbols for degree, minute and second for plane angle, $°$, $\prime$ and $″$, respectively, for which no space is left between the numerical value and the unit symbol.

This rule means that the symbol $\mathrm{°C}$ for the degree Celsius is preceded by a space when one expresses values of Celsius temperature $t$.

Even when the value of a quantity is used as an adjective, a space is left between the numerical value and the unit symbol. Only when the name of the unit is spelled out would the ordinary rules of grammar apply, so that in English a hyphen would be used to separate the number from the unit.

In any expression, only one unit is used. An exception to this rule is in expressing the values of time and of plane angles using non-SI units. However, for plane angles it is generally preferable to divide the degree decimally. It is therefore preferable to write $22.20°$ rather than $22°$ $12\prime$, except in fields such as navigation, cartography, astronomy, and in the measurement of very small angles.

5.4.4.  Formatting numbers, and the decimal marker

The symbol used to separate the integral part of a number from its decimal part is called the decimal marker. Following a decision in https://www.bipm.org/en/committees/cg/cgpm/22-2003/resolution-10, the decimal marker “shall be either the point on the line or the comma on the line.” The decimal marker chosen should be that which is customary in the language and context concerned.

If the number is between +1 and −1, then the decimal marker is always preceded by a zero.

Following https://www.bipm.org/en/committees/cg/cgpm/9-1948/resolution-7 and https://www.bipm.org/en/committees/cg/cgpm/22-2003/resolution-10, for numbers with many digits the digits may be divided into groups of three by a space, in order to facilitate reading. Neither dots nor commas are inserted in the spaces between groups of three. However, when there are only four digits before or after the decimal marker, it is customary not to use a space to isolate a single digit. The practice of grouping digits in this way is a matter of choice; it is not always followed in certain specialized applications such as engineering drawings, financial statements and scripts to be read by a computer.

For numbers in a table, the format used should not vary within one column.

5.4.5.  Expressing the measurement uncertainty in the value of a quantity

The uncertainty associated with an estimated value of a quantity should be evaluated and expressed in accordance with the document JCGM 100:2008 (GUM 1995 with minor corrections), Evaluation of measurement data — Guide to the expression of uncertainty in measurement. The standard uncertainty associated with a quantity $x$ is denoted by $u\left(x\right)$. One convenient way to represent the standard uncertainty is given in the following example:

where $m_{\text{n}}$ is the symbol for the quantity (in this case the mass of a neutron) and the number in parentheses is the numerical value of the standard uncertainty of the estimated value of $m_{\text{n}}$ referred to the last digits of the quoted value; in this case $u\left(m_{\text{n}}\right)=\mathrm{0.000 000 021}\times {10}^{-27}\mathrm{kg}$. If an expanded uncertainty $U\left(x\right)$ is used in place of the standard uncertainty $u\left(x\right)$, then the coverage probability $p$ and the coverage factor $k$ must be stated.