# SI defining constants

All units in the International System of Units (SI) are defined by the seven constants listed in Table 1. These constants are based on unchanging properties of nature. In the SI, each of the seven defining constants has a fixed exact numerical value. The defining constants can be expressed using SI coherent derived units, or with a combination of SI base units.

The defining constants have been chosen such that, when taken together, their units cover all of the units of the SI. Any SI unit is a product of powers of these seven constants and a dimensionless factor.

 Name Symbol Value Unit Expressed in terms of SI base units the Planck constant h 6.626 070 15 × 10−34 J s kg m2 s−1 the speed of light in vacuum c 299 792 458 m s−1 m s−1 the unperturbed ground state hyperfine transition frequency of the caesium 133 atom ΔνCs 9 192 631 770 Hz s−1 the elementary charge e 1.602 176 634 × 10−19 C s A the Boltzmann constant k 1.380 649 × 10−23 J K−1 kg m2 s−2 K−1 the Avogadro constant NA 6.022 140 76 × 1023 mol−1 mol−1 the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz Kcd 683 lm W−1 kg−1 m−2 s3 cd

These definitions specify the exact numerical value of each constant when its value is expressed in the corresponding SI unit. By fixing the exact numerical value the unit becomes defined, since the product of the numerical value {Q} and the unit [Q] has to equal the value Q of the constant, which is invariant: Q = {Q} [Q].

## SI defining constants expressed in terms of base units

From Table 1, it can be seen that each defining constant can be expressed as the product of powers of the SI base units:

kgα mβ sγ Aδ Kε molζ cdη

 Expressed in terms of SI base units 1 2 -1 0 0 0 0 kg m2 s−1 0 1 -1 0 0 0 0 m s−1 0 0 -1 0 0 0 0 s−1 0 0 1 1 0 0 0 s A 1 2 -2 0 -1 0 0 kg m2 s−2 K−1 0 0 0 0 0 -1 0 mol−1 -1 -2 3 0 0 0 1 kg−1 m−2 s3 cd

The exponents required for each defining constant are presented in rows in Table 2. For example, the exponents used for the Planck constant, h, are (1, 2, −1, 0, 0, 0, 0), giving the coherent unit kg1 m2 s−1 A0 K0 mol0 cd0. Exponent values of zero result in a factor of one, and can be discarded, so in this example the coherent unit is kg m2 s−1.

## SI base units expressed in terms of defining constants

The seven SI base units can be defined in terms of the seven defining constants. The exponent values in Table 2 constitute a 7 × 7 matrix. Inverting this matrix gives the exponents required to express each SI base unit in terms of the product of powers of the defining constants:

hα′ cβ′ ΔνCsγ′ eδ′ kε′ NAζ′ Kcdη′

 Expressed in terms of SI defining constants 1 -2 1 0 0 0 0 h c−2 ΔνCs 0 1 -1 0 0 0 0 c ΔνCs−1 0 0 -1 0 0 0 0 ΔνCs−1 0 0 1 1 0 0 0 ΔνCs e 1 0 1 0 -1 0 0 h ΔνCs k−1 0 0 0 0 0 -1 0 NA−1 1 0 2 0 0 0 1 h ΔνCs2 Kcd

The exponents required for each SI base unit are presented in rows in Table 3. For example, the exponents used for the kilogram, kg, are (1, −2, 1, 0, 0, 0, 0), giving the combined constant h1 c−2 ΔνCs1 e0 k0 NA0 Kcd0. Exponent values of zero result in a factor of one, and can be discarded, so in this example the combined constant is h c−2 ΔνCs.

## Scaling factors for the SI base unit definitions

Table 3 excludes scaling factors. The calculations to derive the exact scaling factor for each SI base unit definition are shown below. For each base unit, substituting the values from Table 1 into the left side of the equation, which is a combination of SI defining constants, gives us the right side, which is the value of the combination in the SI.

### kilogram

$\dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{c^2} = \dfrac{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)}{(299 \mspace{4mu} 792 \mspace{4mu} 458)^2} \mspace{4mu} \text{kg}\\ \\ \\ 1 \mspace{4mu} \text{kg} \mspace{26mu} = \dfrac{(299 \mspace{4mu} 792 \mspace{4mu} 458)^2}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)} \mspace{4mu} \dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{c^2}\\ \\ \\ 1 \mspace{4mu} \text{kg} \mspace{26mu} = 1.475 \mspace{4mu} 521 \mspace{4mu} 399 \mspace{4mu} 735 \mspace{4mu} 270 \mspace{4mu} 916 \text{...} \times 10^{40} \mspace{4mu} \dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{c^2}$

### metre

$\dfrac{c}{\Delta \nu _{Cs}} = \dfrac{299 \mspace{4mu} 792 \mspace{4mu} 458}{9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770} \mspace{4mu} \text{m}\\ \\ \\ 1 \mspace{4mu} \text{m} \mspace{16mu} = \dfrac{9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770}{299 \mspace{4mu} 792 \mspace{4mu} 458} \mspace{4mu} \dfrac{c}{\Delta \nu _{Cs}}\\ \\ \\ 1 \mspace{4mu} \text{m} \mspace{16mu} = 30.663 \mspace{4mu} 318 \mspace{4mu} 988 \mspace{4mu} 498 \mspace{4mu} 369 \mspace{4mu} 762 \text{...} \mspace{4mu} \dfrac{c}{\Delta \nu _{Cs}}$

### second

$\dfrac{1}{\Delta \nu _{Cs}} = \dfrac{1}{9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770} \mspace{4mu} \text{s}\\ \\ \\ 1 \mspace{4mu} \text{s} \mspace{22mu} = 9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770 \mspace{4mu} \dfrac{1}{\Delta \nu _{Cs}}$

### ampere

$\Delta \nu _{Cs} \mspace{6mu} e = (9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)(1.602 \mspace{4mu} 176 \mspace{4mu} 634 \times 10^{-19}) \mspace{4mu} \text{A}\\ \\ \\ 1 \mspace{4mu} \text{A} \mspace{28mu} = \dfrac{1}{(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)(1.602 \mspace{4mu} 176 \mspace{4mu} 634 \times 10^{-19})} \mspace{4mu} \Delta \nu _{Cs} \mspace{6mu} e\\ \\ \\ 1 \mspace{4mu} \text{A} \mspace{28mu} = 6.789 \mspace{4mu} 686 \mspace{4mu} 817 \mspace{4mu} 250 \mspace{4mu} 553 \mspace{4mu} 926 \text{...} \times 10^{8} \mspace{4mu} \Delta \nu _{Cs} \mspace{6mu} e$

### kelvin

$\dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{k} = \dfrac{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)}{1.380 \mspace{4mu} 649 \times 10^{-23}} \mspace{4mu} \text{K}\\ \\ \\ 1 \mspace{4mu} \text{K} \mspace{32mu} = \dfrac{1.380 \mspace{4mu} 649 \times 10^{-23}}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770))} \mspace{4mu} \dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{k}\\ \\ \\ 1 \mspace{4mu} \text{K} \mspace{32mu} = 2.266 \mspace{4mu} 665 \mspace{4mu} 264 \mspace{4mu} 601 \mspace{4mu} 104 \mspace{4mu} 867 \text{...} \mspace{4mu} \dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{k}$

### mole

$\dfrac{1}{N_A} \mspace{13mu} = \dfrac{1}{6.022 \mspace{4mu} 140 \mspace{4mu} 76 \times 10^{23}} \mspace{4mu} \text{mol}\\ \\ \\ 1 \mspace{4mu} \text{mol} = 6.022 \mspace{4mu} 140 \mspace{4mu} 76 \times 10^{23} \mspace{4mu} \dfrac{1}{N_A}$

### candela

$h \mspace{6mu} {\Delta \nu _{Cs}}^2 \mspace{6mu} K_{cd} = (6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^2 (683) \mspace{4mu} \text{cd}\\ \\ \\ 1 \mspace{4mu} \text{cd} \mspace{66mu} = \dfrac {1}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^2 (683)} \mspace{4mu} h \mspace{6mu} {\Delta \nu _{Cs}}^2 \mspace{6mu} K_{cd}\\ \\ \\ 1 \mspace{4mu} \text{cd} \mspace{66mu} = 2.614 \mspace{4mu} 830 \mspace{4mu} 482 \mspace{4mu} 285 \mspace{4mu} 615 \mspace{4mu} 686 \text{...} \times 10^{10} \mspace{4mu} h \mspace{6mu} {\Delta \nu _{Cs}}^2 \mspace{6mu} K_{cd}$

## SI base units expressed in terms of defining constants

Table 4 presents the seven SI base units as defined by the product of powers of the SI defining constants.

 Name Symbol Scaling factor Constant kilogram kg 1.475 521 397 352 … × 1040 h c−2 ΔνCs metre m 30.663 318 988 498 … c ΔνCs−1 second s 9 192 631 770 ΔνCs−1 ampere A 6.789 686 817 250 … × 108 ΔνCs e kelvin K 2.266 665 264 601 … h ΔνCs k−1 mole mol 6.022 140 76 × 1023 NA−1 candela cd 2.614 830 482 285 … × 1010 h ΔνCs2 Kcd

Since all seven base units of the SI can be defined in terms of the seven defining constants, it follows that all other SI units can also be defined in terms of the seven defining constants.

The distinction between SI base units and SI derived units remains useful, but is not essential.