## SI units

Introduction

The name Système International d’Unités (SI Units, in short) was adopted at the 11th General Assembly on Weights and Measures in 1960 (11th CGPM, Conférence Générale des Poids et Mesures), but, in spite of its name, it is not in common use in some places and instances, notably in the USA industry.

Physical sciences are based on measurable magnitudes, i.e. magnitudes that admit unambiguous size-comparison between those of the same class. Metrology is the science of measurement. Perhaps the crucial milestone in metrology took place in 1799, under the French Revolution, when scientists proposed to unify all different weights and measures into the decimal metric system.

A physical magnitude needs to be specified by three items: a unit of that dimension, a number indicating the ratio between the magnitude and the unit used, and a quantification of the accuracy or uncertainty in the previous number (measuring uncertainties can be reduced but never eliminated). It is common practice to implicitly include the uncertainty information in the number by limiting its number of significant figures accordingly, reducing the three parameters above to just two: the number and the unity, what might be further reduced to one if the unit were unambiguously understood (not several different units for the same magnitude, not multiples and submultiples).

Types of magnitude are length, time, volume, mass, temperature, etc. Quoting e.g. L=2.5 m refers to a magnitude L two-and-a-half times the length of the unit metre (m). Notice that the decimal sign is a point (.) in English language but a comma (,) for non-English texts; in any case, if there is no significant figure to the left of the decimal point, a zero must always be placed there (e.g. 0.3 and not .3).

Under the SI Units system, seven magnitudes are chosen to form a basic set (metre, kilogramme, second, ampere, kelvin, mole and candela), and all the others (e.g. hertz, newton, volt, radian) are regarded as being derived from this set by appropriate definitions involving only multiplication, division, differentiation or integration (thus, arguments of functions such as sin(x) or log(x) must be nondimensional; e.g. sin(wt), log(p/p0)).

SI base units

The units quoted below are the basic units of the Système International d’Unités.
NOTE: see 2019 redefinition of the SI base units (
Wiki), and non-SI units mentioned there (Wiki).

• Length: The unit of length called the metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second (17th CGPM, 1983).

• Mass: The unit of mass called the kilogramme is the mass of the international prototype which is in the custody of the Bureau International des Poids et Mesures (BIPM) at Sévres, near Paris, France (3rd CGPM, 1901).

• Time: The unit of time called the second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom (13th CGPM, 1967).

• Temperature: The unit of temperature called the kelvin is the fraction 1/273.16 of the temperature of the triple point of water (13th CGPM, 1967).

• Electric current: The unit of electric current called the ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in a vacuum, would produce a force equal to 2´10-7 newton per metre of length between these conductors (9th CGPM, 1948).

• Luminous intensity: The unit of luminous intensity called the candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540´1012 hertz and that has a radiant intensity in that direction of (1/683) watt per steradian (16th CGPM, 1970).

• Amount of substance: The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogramme of carbon 12 (14th CGPM, 1971). When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

SI derived units

All derived units are products or ratios of basic units without numeric factors, most of them are dimensional (but radian, steradian and others are nondimensional) and some have special names (radian, steradian, hertz, newton, pascal, joule, watt, coulomb, volt, farad, ohm, siemens, weber, tesla, henry, degree Celsius, lumen and lux).

All basic units and derived units with special names have special symbols (m, kg, s, A, K, mol, cd, rad, sr, Hz, N, Pa, J, W, C, V, F, W, S, Wb, T, H, ºC, lm and lx).

Symbols for derived units are formed by appropriate combination of the special symbols above (e.g. m/s for velocity).

Multiples and submultiples

Instead of using raw numbers and base units (e.g. 6.380´106 m, 0.628´10-6 m, 0.012 V, 0.1´106 Pa), some multiples or submultiples of the base unit can be used in speech and writing (e.g. 6 380 km, 0.628 µm, 12 mV, 0.1 MPa), although care must be taken not to forget the factors during numerical calculations. In any case, be it as numerical exponents or as multiples of base units, it is recommended to use powers three order of magnitude apart, beginning on the base unit (e.g. 6.380´106 m and not 0.6380´107 m, 0.628´10-6 m and not 6.28´10-7 m). The prefix and value for the accepted multiples are:

Prefix

 Name Symbol Value exa E 1018 peta P 1015 tera T 1012 giga G 109 mega M 106 kilo k 103 milli m 10-3 micro µ 10-6 nano n 10-9 pico p 10-12 fempto f 10-15 atto a 10-18

Prefixes are applied to just one symbol-unit, and thus any exponent applies to the combination prefix-unit (e.g. 5 kN/m2=5´103 N/m2=0.5 N/cm2)

One exception to the use of prefixes occurs with the unit kilogramme, where the prefixes are applied to the unit gramme (e.g. 5 mg=5´10-6 kg).

Unit symbols must be written in normal upright characters without an end period (unless it is the end of the sentence), separated from the numeric quantity by one non-wrapable space, and remain unaltered in the plural (in writing, not in pronunciation).

Derived units which are products of base units must be written with a period in between (e.g. N.m, or better N´m), those which are ratios can be written in one of the three forms shown in the example for speed: m.s-1, m/s or , although the first one is preferable. These rules may be extended to more complex grouping but more than one solidus (/) should never be used in the same expression unless parentheses are used to avoid ambiguity.

Especial names and symbols

 magnitude symbol unit symbol equivalence length l, L metre m mass m kilogramme kg time t second s electric current i ampere A temperature T kelvin K luminous intensity I,Iv candela cd amount of substance n mole mol angle (plane) a,b,g,q,f radian rad solid angle w,W steradian sr frequency n,f hertz Hz s-1 force F newton N kg.m.s-2 pressure p pascal Pa N.m-2 energy E joule J kg.m2.s-2 power P watt W J.s-1 electric charge q,Q coulomb C A.s el. potential difference V,f volt V W.A-1 electric capacitance C farad F A.s.V-1 electric resistance R ohm W V.A-1 electric conductance G siemens S W -1 magnetic flux F weber Wb V.s magnetic flux density B tesla T Wb.m-2 inductance L,M henry H Wb.A-1 degree Celsius t degree Celsius ºC a) luminous flux F lumen lm cd.sr illumination E,i lux lx lm.m-2

a) There are two possibilities: for single temperature values, t, the numerical equivalence is t ºC=(t+273.15) K, whereas for differences in temperature, t, the numerical equivalence is t ºC=t K. For this reason it is impossible to automatically replace degree Celsius by kelvin in a text.

Temporarily accepted units

Other names and symbols that are temporarily accepted, but do not belong to the SI units are:

 magnitude symbol unit equivalence volume V litre 1 l=10-3 m3 mass m gramme 1 g=10-3 kg mass m tonne 1 t=103 kg time t minute 1 min=60 s time t hour 1 h=3 600 s time t day 1 d=86 400 s angle a,b,g,q,f degree 1 º=p /180 rad angle a,b,g,q,f minute 1 '=p /10 800 rad angle a,b,g,q,f second 1 ''=p /648 000 rad energy e electronvolt 1 eV=1.602´10-19 J length L Astronomic unit 1 AU=149.598´109 m length L parsec 1 pc=30.857´1015 m mass m atomic mass unit 1 u=1.661´10-27 kg

Naming and writing variables

Physical equations represent natural laws or define new magnitudes.

They must be written in terms of physical magnitudes, that is, by means of symbols that represent at the same time a unit and a number.

All numeric factors appearing in a physical equation must be nondimensional.

When physical equations have empirical coefficients with dimensions, they must be represented by physical constants (symbols, comprising the unit and the number, made explicit apart).

Symbols for physical magnitudes should preferably be formed by one latin or greek letter, sometimes with subindices and other modifiers and must be printed in italic characters (symbols for units must be printed in normal upright font). A period should not follow to a symbol except for grammatical rules. To help on reading numbers with many figures, a small space may be used to separate groups of three figures at either side of the decimal sign (but never used a comma or a point).

Some nondimensional physical magnitudes are traditionally represented by a two-letter symbol (e.g. Re, Nu) and care must be taken not to take it as a product of two one-letter symbols in physical equations (a small space should be used). Notice that Re is in italics if it refers to the variable called Reynolds number but in upright Re if it refers to the function named real part.

Subindices can be used to distinguish different physical magnitudes with the same letter (e.g. cp, cv) or to represent the same physical magnitude applied to different systems or different times (e.g. m1, m2, for the mass of gas, mH2O). As shown above, literal names and numbers used as subindices are printed upright, but if variables are used to represent them (e.g. mi) they must be printed in italic.

Numbers must be printed in upright characters using a point at the baseline as decimal sign (a comma in non-English texts). The multiplication sign for numbers is a cross (e.g. 6.02´1023) for English texts and a mid-point or a cross for non-English texts (e.g. 6,02× 1023 or 6.02´1023).

Coefficients and factors (e.g. thermal expansion coefficient, a , dV/V=a dT are used to name dimensional magnitudes that appear in physical equations that show proportional dependence of the form A =KB.

Parameters, ratios and numbers (e.g. thermal capacity ratio g =cp/cv, Reynolds number Re=VL/n ) are used to name nondimensional magnitudes.

Fractions (e.g. mole fraction, xi) are used to name nondimensional magnitudes that are lower than unity.

Constants (e.g. Boltzmann constant, k=R/NA) are used to name dimensional magnitudes that do not vary with experimental circumstances. A bad usage occurs in the so called "chemical equilibrium constant", which is dependent on temperature.

Concentrations (e.g. mass concentration of a component B, r B=mB/V) are used to name a dimensional magnitude in a mixture divided by the volume of the mixture.

Densities (e.g. charge density, r =dQ/dV) are used to name a dimensional magnitude in a sufficiently small system divided by the volume of that system..

Flux densities (e.g. thermal flux density, ) are used to name the dimensional magnitude obtained by division of a flux magnitude (one that pierces a control surface) by the area of the surface.

The adjective "specific" (e.g. specific volume, v=V/m) refers to the original magnitude divided by the mass of the system.

The adjective "molar" (e.g. molar volume, v=V/n) refers to the original magnitude divided by the amount of substance in the system.

Miscellaneous units and conversion values

The units listed below are grouped in:

• Length
• Area
• Volume
• Angle
• Solid angle
• Time
• Velocity (linear)
• Velocity (angular)
• Frequency
• Acceleration
• Mass
• Force
• Pressure
• Viscosity (dynamic)
• Viscosity (kinematic)
• Surface tension
• Energy, heat and work
• Power
• Specific energy
• Heat flux density
• Temperature
• Specific heat capacity
• Thermal conductivity
• Thermal conductance
• Thermal diffusivity
• Luminance, brightness, illumination
• Luminous flux
• Luminous intensity
• Illuminance, illumination