### Introduction to units of measurement and the International System of Units (SI units)

Si Units:

The name Systčme International d'Unités (International System of Units) with the international abbreviation SI is a single international language of science and technology first introduced in 1960. SI is a coherent system based on the seven independent physical quantities (base units) and derived quantities (derived units). Note that since 1995 supplementary units have been abandoned and moved into the class of derived SI units.

## Basic SI Units

Physical quantityquantity
symbol
Basic SI
Unit Name
Unit Symbol
lengthl, b, d, h, r, s, etc.metrem
massmkilogram kg
timetseconds
electric currentIampere A
thermodynamic temperatureTkelvinK
amount of substancenmolemol
luminous intensityIvcandelacd
Table 1. SI base units.

Other physical quantities are derived from the basic units. The derived SI units are obtained by the multiplication, division, integration and differentiation of the basic units without the introduction of any numerical factors. The system of units so derived is said to be coherent.

## Supplementary Dimensionless SI Units

QuantityQuantity
symbol
SI Unit NameUnit Symbol Exp. in SI base units
solid angle ω, Ωsteradiansr m2 m-2
Table 2. SI supplementary units. (classification removed, see notes)

## Definitions of the SI Base Units

#### Length: metre (m)

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.

#### Mass: kilogram (kg)

The kilogram is equal to the mass of the international prototype of the kilogram: a piece of platinum-iridium alloy kept at the International Bureau of Weights and Measures (BIPM) in France.

#### Time: second (s)

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 cesium-133 atom.

#### Electric current: ampere (A)

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 vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per metre of length.

#### Thermodynamic temperature: kelvin (K)

The kelvin is 1/273.16 of the thermodynamic temperature of the triple point of water.

The unit kelvin and its symbol K should be used to express both thermodynamic temperature and an interval or a difference of temperature.

In addition to the thermodynamic temperature (symbol T) there is also the Celsius (symbol t) defined by the equation t=T-T0 where T0=273.15 K. Celsius temperature is expressed in degree Celsius (symbol C). The unit 'degree Celsius' is equal to the unit 'kelvin', and a temperature interval or a difference of temperature may also be expressed in degrees Celsius. (The word degree and the sign o must not be used with kelvin or K).

#### Amount of substance: mole (mol)

The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.

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 particle.

In this definition, it is understood that the carbon 12 atoms are unbound, at rest and in their ground state.

### Luminous intensity: candela (cd)

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

### Definition of Supplementary SI Units

At the time of the introduction of the International System, the question of the nature of these supplementary units was left open. Considering that plane angle is generally expressed as the ratio between two lengths and solid angle as the ratio between an area and the square or a length, it was specified that in the International System the quantities plane angle and solid angle should be considered as dimensionless derived quantities. Therefore, the supplementary units radian and steradian are to be regarded as dimensionless derived units which may be used or omitted in the expressions for derived units.

Since October , 1995, the class of supplementary units as a separate class in the SI has been removed. Thus the SI now consists of only two classes of units: base units and derived units, with the radian and steradian, which were the two supplementary units, moved into the class of SI derived units.

## Derived SI Units with Special Names

Physical Quantity Quantity
symbol
SI Unit Unit
Symbol
Expression in SI base units Alternative expressions
frequencyv, fhertz Hz s-1 -
force Fnewton N kg m s-2 J m-1
pressure ppascal Pa kg m-1 s-2N m-2
energy (all forms) E, U, V, W,etc.joule J kg m2 s-2N m = C V = V A s
power Pwatt W kg m2 s-3J s-1 = VA
electric chargeQcoulomb C A s -
electric potential differenceE, φ, ζ, Φ, η, etc. volt V kg m2 s-3 A-1 J A-1 s-1 = J C-1
electrical capacitance Cfarad F A2 s4 kg-1 m-2C V-1
electrical resistance Rohm Ω kg m2 s-3 A-2V A-1
electrical conductance Gsiemens S A2 s3 kg-1 m-2 A V-1 = Ω-1
magnetic flux Φweber Wb kg m2 s-2 A-1 V s = T m2
magnetic induction Btesla T kg s-2 A-1 Wb m-2 = N A-1 m-1
inductance L, Mhenry H kg m2 s-2 A-2 V A-1 s = Wb A-1
luminous flux Φlumen lm cd sr -
illuminationElux lx cd sr m-2 lm m-2
activity (of a radionuclide) Abecquerel Bq s-1-
absorbed dose Dgray Gy m2 s-2J kg-1
dose equivalent Hsievert Sv m2 s-2J kg-1
catalytic activityzkatalkatmol s-1-
Celsius temperaturetdegree Celsius°C K-
plane angle α , β , γ , θ , Φ radian rad m m-1 dimensionless
solid angle ω , Ωsteradian sr m2 m-2 dimensionless
Table 3. SI derived units with special names.

The special names and symbols of the 22 SI derived units with special names and symbols given in table 3 above may themselves be included in the names and symbols of other SI derived units, as shown in table 5.

Note on degree Celsius. The derived unit in Table 3 with the special name degree Celsius and special symbol °C needs comment. 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 T0 = 273.15 K. This temperature difference is called a Celsius temperature, symbol t, and is defined by the quantity equation

t= T- T0.

The unit of Celsius temperature is the degree Celsius, symbol °C. The numerical value of a Celsius temperature t expressed in degrees Celsius is given by

t/°C = T/K - 273.15.

It follows from the definition of t that the numerical value of a given temperature difference or temperature interval will be the same for both degree Celsius and the kelvin.

## SI Derived Units

#### Some Examples Expressed in Terms of SI Base Units

Derived quantityQuantity
symbol
NameExpression in SI base units
areaAsquare metrem2
volumeVcubic metrem3
speed, velocityu, v, cmetre per secondm s-1
accelerationa, g (free fall)metre per second squaredm s-2
moment of inertiaIkilogram square metrekg m2
kinematic viscosityvsquare metre per secondm2 s-1
wave numberσ, φreciprocal metrem-1
mass densityρkilogram per cubic metrekg m-3
specific volumevcubic metre per kilogramm3 kg-1
current densityj, iampere per square metreA m-2
magnetic field strengthHampere per metreA m-1
concentration of substance B:cB, [B]mole per cubic metre mol/m-3
molar massMkilogram per molekg mol-1
molar volumeVmcubic metre per molem3mol-1
luminanceLcandela per square metrecd m-2
mass fractionwkilogram per kilogramdimensionless
Table 4. SI derived units expressed in terms of SI base units.

The table above shows some examples of derived quantities and units expressed in terms of SI base units.

### Dimensionless Quantities

Some quantities are defined as the ratios of two quantities of the same kind, and thus have a dimension expressed by the number one. Examples of such quantities are refractive index, relative permeability, and mass fraction. Other quantities having the unit 1 include "characteristic numbers" like the quantum number and numbers which represent a count, such as a number of molecules and partition function in statistical thermodynamics. All of these quantities are described as being dimensionless, or of dimension one, and have the coherent SI unit 1. Their values are simply expressed as numbers and, in general, the unit 1 is not shown. In a few cases, a special name is given to this unit, mainly to avoid confusion between some compound derived units. This is the case for the radian, steradian and neper.

# SI Derived Units

## Expressed in Terms of SI Derived Units with Special Names

Derived quantityQuantity
symbol
NameExpression in SI base units Alternative SI expressions
angular momentumLjoule secondkg m2 s-1J s
momentumPnewton secondkg m s-1N s
dynamic viscosityηpascal secondkg m-1s-1 Pa s
surface tensionγ, σnewton per metrekg s-2 N m-1 = J m-2
moment of forceυnewton metrekg m2 s-2 N m = J
heat flux density,
Qwatt per square metrekg s-3 W m-2
heat capacity, entropySjoule per kelvinkg m2 s-2 K-1 J K-1 = C V K-1
specific heat capacity,
specific entropy
cjoule per kilogram kelvin m2 s-1 K-1J kg-1 K-1
specific energyEjoule per kilogramm2 s-2 J kg-1
thermal conductivityλwatt per metre kelvin kg m2 s-3 K-1W m-1 K-1
electric conductivityσ, κsiemens per metre A2 s3 kg-1 m-3 S m-1 = Ω-1 m-1
= A V-1 m-1
electric resistivityρohm metre kg m3 A-2 s-3 Ω m = m S-1
= V m A-1
energy densityujoule per cubic metrekg m-1 s-2 J m-3 = N m-2
= C m-3
electric field strengthEvolt per metrekg m s-3 A-1 V m-1
electric charge densityρcoulomb per cubic metreA s m-3 C m-3
electric flux densityσcoulomb per square metre A s m-2C m-2
permittivityεfarad per metreA2 s4 kg-1 m-3F m-1
permeabilityμhenry per metrekg m s-2 A-2 H m-1
molar energyUm, Hm, etc.joule per mole kg m2 s-2 mol-1J mol-1
molar entropy,
molar heat capacity
Sm, Cc,m, Cp,m joule per mole kelvinkg m2 s-2 mol-1 K-1 J mol-1 K-1
exposure
(x and γ rays)
-coulomb per kilogramA s kg-1 C kg-1
absorbed dose rate-gray per secondm2 s-3 Gy s-1 = J kg-1 s-1
catalytic (activity)
concentration
-katal per cubic metre mol m-3 s-1kat m-3
Table 5. SI derived units expressed in terms of SI derived units with special names.

The above table shows some derived quantities and units expressed in terms of SI units with special names. Some derived quantities like moment of force (newton metre) and thermodynamic energy (joule) are both quantities of energy (kg m2 s-2) but are very often expressed differently.

## Non-SI Units Accepted for use with SI

Physical QuantityUnit NameUnit SymbolExp. in SI Units
timeminutemin60 s
timehourh60 min = 3600 s
timedayd24 h = 86400 s
volumelitrel, L1 dm3 = 10-3 m3
masstonnet103 kg
field level,
power level,
sound pressure level,
logarithmic decrement
neper Np1, dimensionless
field level,
power level,
sound pressure level,
attenuation
bel B(1/2) ln 10 (Np)
dimensionless
energyelectronvolteV1 eV = 1.602 18 x 10-19 J
approx
massunified atomic mass unitu1 u = 1.66054 x 10-27 kg
approx
lengthastronomical unitua1 ua = 1.49598 x 1011 m
approx
Table 6. Non-SI Units Accepted for use with SI.

The above table lists non-SI units which are accepted for use with the SI. It includes units which are in common everyday use, in particular the traditional units of time and of angle, together with a few other units which have assumed technical importance. Also included at the bottom of the table are three non-SI units, whose values expressed in SI units must be obtained by experiment and are therefore not known exactly. Their values are given with their combined standard uncertainty, which apply to the last two digits, shown in parentheses. These units are in common use in certain specialised fields.

## Non-SI units temporarily accepted for use with SI

Physical quantityUnit nameUnit symbolExp. in SI units
lengthnautical mile-1852 m
velocityknot- nautical mile per hour = (1852/3600) m s-1
areahectarehahm2 = 104 m2
pressurebarbar0.1 MPa = 100 kPa = 1000 hPa = 105 Pa
lengthångströmÅ0.1 nm = 10-10 m
areabarnb100 fm2 = 10-28 m2
Table 7. Non-SI units temporarily accepted for use with SI.

The above table lists some other non-SI units which are currently accepted for use with the SI to satisfy the needs of commercial, legal and specialised scientific interests. These units should be defined in relation to the SI in every document in which they are used. Their use is not encouraged.

# Derived CGS Units with Special Names

Physical quantityUnit nameUnit symbolExp. in SI units
energyergerg10-7 J
forcedynedyn10-5 N
dynamic viscositypoisePdyn s cm-2 = 0.1 Pa s
kinematic viscositystokesSt cm2 s-1 = 10-4 m2 s-1
magnetic inductiongaussG10-4 T
magnetic field strengthoerstedOe(1000/4π) A m-1
magnetic fluxmaxwellMx10-8 Wb
luminancestilbsb cd cm-2 = 104 cd m-2
illuminationphotph104 lx
acceleration
(due to gravity)
galGal 1 cm s-2 = 10-2 m s-2
Table 8. Derived CGS Units with Special Names.

Some non-SI units are still occasionally used. Some are important for the interpretation of older scientific texts, but their use is not encouraged. The above table shows the relationship between CGS units and the SI, and lists those CGS units that were assigned special names. In the field of mechanics, the CGS system of units was built upon three quantities and the corresponding base units: the centimetre, the gram and the second. In the field of electricity and magnetism, units were expressed in terms of these three base units. Because this can be done in different ways, it led to the establishment of several different systems, for example, the CGS Electrostatic System, the CGS Electromagnetic System and the CGS Gaussian System. In these three last-mentioned systems, the system of quantities and the corresponding system of equations differ from those used with SI units.

## Other Non-SI Units

Physical quantityUnit nameUnit symbolExp. in SI units
activity (of a radionuclide)curieCi3.7 x 1010 Bq
exposure
(x and γ rays)
röntgenR2.58 x 10-4 C kg s-1
dose equivalentremremcSv = 10-2 Sv
length (x-ray wavelength)X unit-1.002 x 10-4 nm
approximately
magnetic inductiongammaγnT = 10-9 T
flux, radio astronomyjanskyJy10-26 W m-2 Hz-1
lengthfermi-fm = 10-15 m
massmetric carat-200 mg = 2 x 10-4 kg
pressuretorrTorr(101 325/760) Pa
pressurestandard atmosphereatm760 mmHg
= 101 325 Pa
pressuremillimeter of mercurymmHg133.322 39 Pa
energythermochemical caloriecalth4.184 J
lengthmicronµ1 µm = 10-6 m
timeyeara365.242199 days
= 31556925.9747 s
forcekilogram-forcekgf9.80665 N
Table 9. Other Non-SI Units.

The table above lists units which are common in older texts and also some are units derived directly from a measurement system like barometric pressure measurement in mmHg. For current texts, it should be noted that if these units are used the advantages of the SI are lost. The relation of these units to the SI should be specified in every document in which they are used.

## Prefixes for SI Units and Derived SI Units

 Prefix-------- Symbol -------------------------------------Factor------------------- --------------- ----------

 yotta Y 1 000 000 000 000 000 000 000 000 = 10 24 (e+24) zetta Z 1 000 000 000 000 000 000 000 = 10 21 (e+21) exa E 1 000 000 000 000 000 000 = 10 18 (e+18) peta P 1 000 000 000 000 000 = 10 15 (e+15) tera T 1 000 000 000 000 = 10 12 (e+12) giga G 1 000 000 000 = 10 9 (e+9) mega M 1 000 000 = 10 6 (e+6) kilo k 1 000 = 10 3 (e+3) hecto h 100 = 10 2 (e+2) deca da 10 = 10 1 (e+1) ------------------ ------------ 1 -------------------------------------------------------------- --------------- ---------- deci d 0.1 = 10 -1 (e-1) centi c 0.01 = 10 -2 (e-2) milli m 0.001 = 10 -3 (e-3) micro µ 0.000 001 = 10 -6 (e-6) nano n 0.000 000 001 = 10 -9 (e-9) pico p 0.000 000 000 001 = 10 -12 (e-12) femto f 0.000 000 000 000 001 = 10 -15 (e-15) atto a 0.000 000 000 000 000 001 = 10 -18 (e-18) zepto z 0.000 000 000 000 000 000 001 = 10 -21 (e-21) yocto y 0.000 000 000 000 000 000 000 001 = 10 -24 (e-24)

Table 11. Prefixes for SI Units and Derived SI Units.

The table lists the prefixes used to denote decimal fractions and multiples of SI units and derived SI units. The factos are 103n except around unity where additional prefixes are permitted to denote 10-2 10-1 100 101 102. Compound prefixes are not permitted (e.g. millimicro). The prefix attaches directly to the name of a unit, and a prefix symbol attaches directly to the symbol for a unit.

### Prefixes and the kilogram

For historical reasons, the name "kilogram" for the SI base unit of mass contains the name "kilo," the SI prefix for 103. Thus, because compound prefixes are unacceptable, symbols for decimal multiples and submultiples of the unit of mass are formed by attaching SI prefix symbols to g, the unit symbol for gram, and the names of such multiples and submultiples are formed by attaching SI prefixes to the name "gram."
Example: 10-3 kg = 1 g (1 gram) but not: 10-3 kg = 1 mkg (1 millikilogram)

## Natural and atomic units

In some cases, the values of quantities are expressed in terms of fundamental constants of nature or so-called natural units. The use of these units occur when it is necessary for the most effective communication of information. In such cases, the specific natural units that are used must be identified. Examples of some physical quantities used as natural units are given in the table. While theoretical results intended primarily for other theorists may be left in natural units, if they are also intended generally, they must also be given in acceptable units.

## Some Fundamental Physical Constants

 Physical Quantity Symbol Value in SI Units speed of light in vacuum c, co 299 792 458 m s-1 elementary charge e 1.602 176 53(14) x 10-19 C Planck constant h 6.626 0693(11) x 10-34 J s Avogadro constant L, NA 6.022 1415(10) x 1023 mol-1 electron mass me 9.109 3826(16) x 10-31 kg proton mass mp 1.672 621 71(29) x 10-27 kg electronvolt eV 1.602 176 53(14) x 10-19 J Faraday constant F 9.648 533 83(83) x 104 C mol-1 Hartree energy Eh 4.359 744 17(75) x 10-18 J Bohr radius ao 5.291 772 108(18) x 10-11 m Bohr magneton µB 9.274 009 49(80) x 10-24 J T-1 nuclear magneton µN 5.050 783 43(43) x 10-27 J T-1 Rydberg constant R∞ 10 973 731.568 527(73) m-1 molar gas constant R 8.314 472(15) J mol-1 K-1 Boltzmann constant k, kB 1.380 650 5(24) x 10-23 J K-1 gravitational constant G 6.6742(10) x 10-11 m3 kg-1 s-2 standard acceleration of gravity gn 9.806 65 m s-2 triple point of water Ttp(H20) 273.16 K zero of Celsius scale T(0oC) 273.15 K molar volume of ideal gas (273.15 K, 100 kPa) Vm 22.710 981(40) x 10-3 m3 mol-1 magnetic constant (permeability of vacuum) µo 4p x 10-7 = 12.566 370 614 x 10-7 N A-2 electric constant (permittivity of vacuum) eo 8.854 187 817 x 10-12 F m-1
Table 10. Some Fundamental Physical Constants.

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