Units

Table of Contents






What are Units?

A unit of measurement, or unit, is a defined magnitude of some quantity or property that is used as a standard for measurement of the same kind of quantity, such that any other quantity of that kind can be expressed as a multiple of the unit of measurement [1]. To be specific, units of measurement are used to quantify physical quantities, which are properties of a material, object, or system that can be measured [2]. For example, length, mass, and duration are all physical quantities. These physical quantities correspond to the metre, kilogram, and second in SI units, or the foot, pound, and second in US customary units respectively. In this way, a length measurement of 10 metres (10 m) really represents a length equal to 10 times the predefined length of the metre [1].


What are Systems of Measurement?

A system of measurement is a collection of units of measurement and rules relating them to each other [1].

The most commonly used system of measurement is the International System of Units, internationally recognized by the acronym SI. The International System of Units is colloquially known as "the metric system", but it is just the most recent standard for a metric system. which is a system of measurement that is based on or follows from the introduction of the original metre in France.  Previous standards for metric systems include the centimetre-gram-second (CGS) system, the metre-tonne-second (MTS) system, the metre-kilogram-second (MKS) system, and the metre-kilogram-second-ampere (MKSA) system, which the International System of Units is based on [3]. 

The next most used systems of measurement are the British imperial system of units and United States customary units. These two systems are often conflated, as they are both derived from the English units of measure, but they differ in some ways [4]. In short, the units for length and time are identical, but there are differences in measuring volume, weight, and mass. Both systems use the international yard, internationally agreed to be exactly 0.9144 metres (meaning intl. foot = 0.3048 m, intl. inch = 0.0254 m), and the SI second [4].

Lastly, natural units are systems of measurement commonly used in physics, but are outside the scope of this article.

The "collection of units" part has been explained, but the "rules relating them" involve interconverting between units.

The most immediate example is in converting between units that measure the same physical quantity. For example, in US customary units, 1 yard = 3 feet, and 1 foot = 12 inches, so it is easy to see that 1 yard = 36 inches. Additionally, SI prefixes mean that 1 kilometre = 1000 metre, and 1 metre = 100 centimetres, so 1 kilometre = 100 000 centimetres.

Next, some units measure derived physical quantities, which are quantities that are calculated using a combination of other physical quantities and measurements. For example, length, mass, and time are sufficient to define force, energy, pressure, and power. Let's look at force and units of force as an example. Newton's second law, F=ma, means that force is equal to mass times acceleration. In SI units, force is measured with the newton (N). One newton can be described as the "force necessary to accelerate one kilogram with an acceleration of 1 metre per second per second", but is better defined as "kgms−2" like the equation describes [5]. This means that 1 newton is equal to the combination of units resulting from multiplying mass (kg) by acceleration (m/s2=ms−2). Notice that no conversion factors are necessary to convert to newtons when the quantities are expressed in newtons, kilograms, metres, and seconds, but be careful as this does not necessarily hold true when using SI prefixes.


The International System of Units

The International System of Units is the current international standard for measurements, and is the modern successor and standard of the metric system [6]. SI is short for the French Système international d'unités [3]. The SI is decimal and metric, meaning that is is based on powers of ten and the metre, respectively [6]. The core of the International System of Units is the seven base units and the seven defining constants. The metre and kilogram were historically defined with physical objects, but since 2019, all seven units are defined using some combination of physical constants [3]. For example, the metre is defined as the length light travels in 1/299 792 458 seconds, and seconds are defined as 9 192 631 770 periods of the microwave radiation that caesium-133 emits [3]. These are not approximate values; they are the definitions of the units. Generally, the definitions of the units are not relevant in any field except metrology.


SI Units

SI Base Units

All units in the SI are based on some combination of products, quotients, and powers of the SI base units, which themselves are derived from fundamental constants.


SI Base Units [3][6]
UnitUnit SymbolQuantity SymbolPhysical QuantityRough Definitions
secondsttimehalf the period of a pendulum 1 m long
metremllength1/40 000 000 the length of the equator
kilogramkgmmassthe mass of a litre (1000 cm3) of cold water
ampereAIelectric currentthe current drawn by two 60 W incandescent light bulbs at 120 V
kelvinKTthermodynamic temperaturea temperature difference of 1 K is equal to a difference of 1°C
molemolnamount of substancethe number of atoms in 12 g of carbon
candelacdIvluminous intensitythe brightness of a medium candle


SI Derived Units

Many calculations require the usage of derived units, some of which have special names as a result of how commonly used they are, or due to historical reasons. However, it is always possible to derive them from SI base units. 


Common SI Derived Units with Special Names [6]
UnitUnit SymbolPhysical QuantityIn SI base unitsIn other SI units
radianradplane anglem/m1 (dimensionless)
hertzHzfrequencys−1
newtonNforce, weightkg⋅m⋅s−2
pascalPapressure, stresskg⋅m-1⋅s−2N/m2
jouleJenergy, work, heatkg⋅m2⋅s−2N⋅m
wattWpower, radiant fluxkg⋅m2⋅s−3J/s
coulombCelectric charges⋅A
voltVvoltage, electromotive forcekg⋅m2⋅s−3⋅A−1W/A, J/C
faradFcapacitancekg−1⋅m−2⋅s4⋅A2C/V
ohmΩresistancekg⋅m2⋅s−3⋅A−2V/A
henryHinductancekg⋅m2⋅s−2⋅A−2V⋅s/A
degree Celsius°Ctemperature relative to 273.15 KK


The majority of derived units do not have special names, but they are predictable based on the units they are based on. The pattern for derived units is using "per" for denoting division or negative powers, concatenation with a hyphen for denoting multiplication, and "squared" or "cubed" for denoting the respective power.


Common Derived Units [6]
NameSymbolPhysical Quantity
square metrem2area
cubic metrem3volume
metre per secondm/sspeed, velocity
metre per second squaredm/s2 = (m/s)/sacceleration
newton-secondN⋅s = kgm/smomentum, impulse
kilogram per metrekg/mlinear density
kilogram per cubic metrekg/m3density
mole per cubic metremol/m3concentration
kilogram per cubic metrekg/m3mass concentration
newton-metreN⋅m

torque, moment of force

radian per secondrad/sangular velocity, angular frequency
radian per second squaredrad/s2angular acceleration
joule per kelvinJ/Kheat capacity, entropy
joule per kilogram-kelvinJ/(kg⋅K) = J/kg/Kspecific heat capacity, specific entropy
joule per cubic metreJ/m3energy density


Accepted Non-SI Units

There are a number of units that are not strictly in the SI, but are nonetheless officially accepted by the SI in combination with SI units due to their prevalence.

Non-SI Units Accepted by SI (Not Including Logarithmic Units) [6]
NameSymbolDefinitionPhysical Quantity
minutemin1 min = 60 stime

hourh1 h = 60 min = 3600 s
dayd1 d = 24 h = 86 400 s
astronomical unitau1 au = 149 597 870 700 mlength
degree°1° = (π/180) rad

plane and

phase angle

minute1′ = (1/60)° = (π/10800) rad
second1″ = (1/60)′ = (π/648000) rad
hectareha1 ha = 1 hm2 = 104 m2

area

litrel, L1 L = 1 dm3 = 1000 cm3 = 0.001 m3volume
tonne (metric ton)t1 t = 1 000 kg = 1 Mgmass
dalton (atomic mass unit)Da, amu1 Da = 1 amu = 1.660539040(20)×10−27 kg
electron volteV1 eV = 1.602176634×10−19 Jenergy


Metric Prefixes

Adding a metric prefix to a unit name creates a multiple or submultiple (division) of the original unit. Prefixes for SI units are decimal, and are based on powers of 10. Though for anything past kilo-, prefixes more accurately described as based on powers of 1000.

Prefix symbols are case-sensitive, as to not confuse milli- (m-) with mega (M-) or pico- (p-) with peta- (P-). Additionally, the symbol for micro- is the Greek letter mu (μ), but 'mc-' or a plain Latin 'u-' are acceptable. Compound prefixes are not allowed, which also applies to the kilogram, which is the only SI base unit with a prefix in it. The gram should be considered to be the base, meaning megagram (Mg) for 1 000 kg is correct, while kilokilogram (kkg) is not.


Common SI Prefixes (Not Including Prefixes Beyond 1015/10-15) [6]
NameSymbolPower of 10Power of 1000DecimalEnglish (short scale)
peta-P-1015100051 000 000 000 000 000quadrillion
tera-T-1012100041 000 000 000 000trillion
giga-G-109100031 000 000 000billion
mega-M-106100021 000 000million
kilo-k-103100011 000thousand
hecto-h-10210002/3100hundred
deca-da-/D-/Da-10110001/310ten


100100001one
deci-d-10-11000-1/30.1tenth
centi-c-10-21000-2/30.01hundredth
milli-m-10-31000-10.001thousandth
micro-μ-/mc-/u-10-61000-20.000 001millionth
nano-n-10-91000-30.000 000 001billionth
pico-p-10-121000-40.000 000 000 001trillionth
femto-f-10-151000-50.000 000 000 000 001quadrillionth

A similar system of binary prefixes are in use for units of information (bits and bytes) based on powers of 1024 (210) to disambiguate between powers of 1000 and powers of 1024 when SI prefixes were used. A gigabyte (GB) can either be 1 000 000 000 (109) bytes, or 1 073 741 824 (230) bytes, but a gibibyte (GiB) is unambiguously 230 bytes as indicated by the binary prefix. Generally, SI prefixes on information units tend to indicate the decimal interpretation, but it is still ambiguous.

Common Binary Prefixes (Not Including Prefixes Beyond 250)
 NameSymbolPower of 2DecimalSI approx.
kibi-Ki-2101024kilo-
mebi-Mi-2201 048 576mega-
gibi-Gi-2301 073 741 824giga-
tebi-Ti-2401 099 511 627 776tera-
pebi-Pi-2501 125 899 906 842 624peta-


Non-SI units

Several non-SI units, including US customary units, are also commonly used in Canada.

Common US Customary Units and Non-SI Units [7]
NameSymbolDivisionsSI equivalentApplicationsDerivationPhysical Quantity
thou, milthou, mil
25.4 μmmanufacturethousandth of an inchlength
inchin., "1000 thou25.4 mm

footft., '12 in.0.304 8 m

yardyd.3 ft.0.914 4 m

milemi.1760 yd.1.609 344 km

nautical mileM, NM, nmi
1 852 mnavigationhistorically, 1 minute of latitude along a line of longitude
parsecpc
~3.0857×1016 mastronomyparallax of one arcsecond
miles per hourmph, mi/h
1.609 344 km/h

speed, velocity
knotkn
1 852 km/hnavigationdefined as 1 nautical mile per hour
square inchsq. in.
6.452 cm2

area
square (survey) footsq. ft., ft2~144 square inches

~0.092 903 41 m2


based on the length of the foot before internationalization
acreacre, ac43 560 sq. ft. (survey)~4046.9 m2

fluid ouncefl. oz.
29.573 529 562 5 mL

volume
pintpt.16 fl. oz.473.176 473 mL

quartqt.2 pt. 0.946 352 946 L

gallongal.4 qt.3.785 411 784 L

ounceoz.
28.349 523 125 g

mass
poundlb.16 oz.453.592 37 g

(short) ton
2000 lb.907.184 74 kg

pound per square inchpsi
~6 895 Pa

pressure, stress
kilowatt-hourkWh
3.6 MJelectric utilities, energy storageamount of electric energy transferred at 1 kW in 1 hourenergy, work, heat

(small) calorie, thermochemical caloriecal
4.184 Jchemistryamount of heat to raise the temperature of 1 g of water by 1°C
kilocalorie, Calorie, food caloriekcal, Cal1000 cal4.184 kJnutritionamount of heat to raise the temperature of 1 kg of water by 1°C
British thermal unitBtu, BTU~0.25 kcal~1.055 kJ

heating, natural gas pricing

amount of heat to raise the temperature of 1 lb. of water by 1°F
horsepowerhp
~745.7 Wengines, motorspower of the average horsepower

British thermal unit per hourBtu/h, BTU/h
~0.293 1 Wheating/cooling
ton of refrigerationTR, TOR(=12 000 BTU/h)~3.5 kWcooling/ refrigerationrate of heat transfer to melt/freeze 1 ton of pure ice at 0°C in 24 hours
ampere-hour, amp hourA⋅h, Ah
3 600 Celectrochemistry, battery capacityamount of electric charge transferred at 1 ampere in 1 hourelectric charge
degree Fahrenheit°F

F = (5 / 9)(C - 32)


referenced on temperature of ice-ammonium chloride solution (0°F) and human body temperature (96°F)temperature


References

[1] “Unit of measurement,” Wikipedia, 16-Feb-2021. [Online]. Available: https://en.wikipedia.org/wiki/Unit_of_measurement. [Accessed: 11-Apr-2021].

[2] P. P. Urone and R. Hinrichs, “Physical Quantities and Units,” College Physics, 23-Jan-2012. [Online]. Available: https://opentextbc.ca/openstaxcollegephysics/chapter/physical-quantities-and-units/. [Accessed: 11-Apr-2021].

[3] “Measurement units” BIPM. [Online]. Available: https://www.bipm.org/en/measurement-units/. [Accessed: 12-Apr-2021].

[4] “Imperial and US customary measurement systems,” Wikipedia, 11-Mar-2021. [Online]. Available: https://en.wikipedia.org/wiki/Imperial_and_US_customary_measurement_systems. [Accessed: 12-Apr-2021].

[5] “Newton,” Encyclopædia Britannica. [Online]. Available: https://www.britannica.com/science/newton-unit-of-measurement. [Accessed: 12-Apr-2021].

[6] “International System of Units,” Wikipedia, 05-Apr-2021. [Online]. Available: https://en.wikipedia.org/wiki/International_System_of_Units. [Accessed: 12-Apr-2021].

[7] “United States customary units,” Wikipedia, 12-Apr-2021. [Online]. Available: https://en.wikipedia.org/wiki/United_States_customary_units. [Accessed: 12-Apr-2021].

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