Table of Contents
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 centimetregramsecond (CGS) system, the metretonnesecond (MTS) system, the metrekilogramsecond (MKS) system, and the metrekilogramsecondampere (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 "1 kg⋅m⋅s^{−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/s^{2}=m⋅s^{−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 caesium133 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]  

Unit  Unit Symbol  Quantity Symbol  Physical Quantity  Rough Definitions 
second  s  t  time  half the period of a pendulum 1 m long 
metre  m  l  length  1/40 000 000 the length of the equator 
kilogram  kg  m  mass  the mass of a litre (1000 cm^{3}) of cold water 
ampere  A  I  electric current  the current drawn by two 60 W incandescent light bulbs at 120 V 
kelvin  K  T  thermodynamic temperature  a temperature difference of 1 K is equal to a difference of 1°C 
mole  mol  n  amount of substance  the number of atoms in 12 g of carbon 
candela  cd  I_{v}  luminous intensity  the 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]  

Unit  Unit Symbol  Physical Quantity  In SI base units  In other SI units 
radian  rad  plane angle  m/m  1 (dimensionless) 
hertz  Hz  frequency  s^{−1}  
newton  N  force, weight  kg⋅m⋅s^{−2}  
pascal  Pa  pressure, stress  kg⋅m^{1}⋅s^{−2}  N/m^{2} 
joule  J  energy, work, heat  kg⋅m^{2}⋅s^{−2}  N⋅m 
watt  W  power, radiant flux  kg⋅m^{2}⋅s^{−3}  J/s 
coulomb  C  electric charge  s⋅A  
volt  V  voltage, electromotive force  kg⋅m^{2}⋅s^{−3}⋅A^{−1}  W/A, J/C 
farad  F  capacitance  kg^{−1}⋅m^{−2}⋅s^{4}⋅A^{2}  C/V 
ohm  Ω  resistance  kg⋅m^{2}⋅s^{−3}⋅A^{−2}  V/A 
henry  H  inductance  kg⋅m^{2}⋅s^{−2}⋅A^{−2}  V⋅s/A 
degree Celsius  °C  temperature relative to 273.15 K  K 
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]  

Name  Symbol  Physical Quantity 
square metre  m^{2}  area 
cubic metre  m^{3}  volume 
metre per second  m/s  speed, velocity 
metre per second squared  m/s^{2} = (m/s)/s  acceleration 
newtonsecond  N⋅s = kg⋅m/s  momentum, impulse 
kilogram per metre  kg/m  linear density 
kilogram per cubic metre  kg/m^{3}  density 
mole per cubic metre  mol/m^{3}  concentration 
kilogram per cubic metre  kg/m^{3}  mass concentration 
newtonmetre  N⋅m  torque, moment of force 
radian per second  rad/s  angular velocity, angular frequency 
radian per second squared  rad/s^{2}  angular acceleration 
joule per kelvin  J/K  heat capacity, entropy 
joule per kilogramkelvin  J/(kg⋅K) = J/kg/K  specific heat capacity, specific entropy 
joule per cubic metre  J/m^{3}  energy density 
Accepted NonSI 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.
NonSI Units Accepted by SI (Not Including Logarithmic Units) [6]  

Name  Symbol  Definition  Physical Quantity 
minute  min  1 min = 60 s  time 
hour  h  1 h = 60 min = 3600 s  
day  d  1 d = 24 h = 86 400 s  
astronomical unit  au  1 au = 149 597 870 700 m  length 
degree  °  1° = (π/180) rad  plane and phase angle 
minute  ′  1′ = (1/60)° = (π/10800) rad  
second  ″  1″ = (1/60)′ = (π/648000) rad  
hectare  ha  1 ha = 1 hm^{2} = 10^{4} m^{2}  area 
litre  l, L  1 L = 1 dm^{3} = 1000 cm^{3} = 0.001 m^{3}  volume 
tonne (metric ton)  t  1 t = 1 000 kg = 1 Mg  mass 
dalton (atomic mass unit)  Da, amu  1 Da = 1 amu = 1.660539040(20)×10^{−27} kg  
electron volt  eV  1 eV = 1.602176634×10^{−19} J  energy 
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 casesensitive, 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 10^{15}/10^{15}) [6]  

Name  Symbol  Power of 10  Power of 1000  Decimal  English (short scale) 
peta  P  10^{15}  1000^{5}  1 000 000 000 000 000  quadrillion 
tera  T  10^{12}  1000^{4}  1 000 000 000 000  trillion 
giga  G  10^{9}  1000^{3}  1 000 000 000  billion 
mega  M  10^{6}  1000^{2}  1 000 000  million 
kilo  k  10^{3}  1000^{1}  1 000  thousand 
hecto  h  10^{2}  1000^{2/3}  100  hundred 
deca  da/D/Da  10^{1}  1000^{1/3}  10  ten 
10^{0}  1000^{0}  1  one  
deci  d  10^{1}  1000^{1/3}  0.1  tenth 
centi  c  10^{2}  1000^{2/3}  0.01  hundredth 
milli  m  10^{3}  1000^{1}  0.001  thousandth 
micro  μ/mc/u  10^{6}  1000^{2}  0.000 001  millionth 
nano  n  10^{9}  1000^{3}  0.000 000 001  billionth 
pico  p  10^{12}  1000^{4}  0.000 000 000 001  trillionth 
femto  f  10^{15}  1000^{5}  0.000 000 000 000 001  quadrillionth 
A similar system of binary prefixes are in use for units of information (bits and bytes) based on powers of 1024 (2^{10}) 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 (10^{9}) bytes, or 1 073 741 824 (2^{30}) bytes, but a gibibyte (GiB) is unambiguously 2^{30} 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 2^{50})  

Name  Symbol  Power of 2  Decimal  SI approx. 
kibi  Ki  2^{10}  1024  kilo 
mebi  Mi  2^{20}  1 048 576  mega 
gibi  Gi  2^{30}  1 073 741 824  giga 
tebi  Ti  2^{40}  1 099 511 627 776  tera 
pebi  Pi  2^{50}  1 125 899 906 842 624  peta 
NonSI units
Several nonSI units, including US customary units, are also commonly used in Canada.
Common US Customary Units and NonSI Units [7]  

Name  Symbol  Divisions  SI equivalent  Applications  Derivation  Physical Quantity 
thou, mil  thou, mil  25.4 μm  manufacture  1 thousandth of an inch  length  
inch  in., "  1000 thou  25.4 mm  
foot  ft., '  12 in.  0.304 8 m  
yard  yd.  3 ft.  0.914 4 m  
mile  mi.  1760 yd.  1.609 344 km  
nautical mile  M, NM, nmi  1 852 m  navigation  historically, 1 minute of latitude along a line of longitude  
parsec  pc  ~3.0857×10^{16} m  astronomy  parallax of one arcsecond  
miles per hour  mph, mi/h  1.609 344 km/h  speed, velocity  
knot  kn  1 852 km/h  navigation  defined as 1 nautical mile per hour  
square inch  sq. in.  6.452 cm^{2}  area  
square (survey) foot  sq. ft., ft^{2}  ~144 square inches  ~0.092 903 41 m^{2}  based on the length of the foot before internationalization  
acre  acre, ac  43 560 sq. ft. (survey)  ~4046.9 m^{2}  
fluid ounce  fl. oz.  29.573 529 562 5 mL  volume  
pint  pt.  16 fl. oz.  473.176 473 mL  
quart  qt.  2 pt.  0.946 352 946 L  
gallon  gal.  4 qt.  3.785 411 784 L  
ounce  oz.  28.349 523 125 g  mass  
pound  lb.  16 oz.  453.592 37 g  
(short) ton  2000 lb.  907.184 74 kg  
pound per square inch  psi  ~6 895 Pa  pressure, stress  
kilowatthour  kWh  3.6 MJ  electric utilities, energy storage  amount of electric energy transferred at 1 kW in 1 hour  energy, work, heat  
(small) calorie, thermochemical calorie  cal  4.184 J  chemistry  amount of heat to raise the temperature of 1 g of water by 1°C  
kilocalorie, Calorie, food calorie  kcal, Cal  1000 cal  4.184 kJ  nutrition  amount of heat to raise the temperature of 1 kg of water by 1°C  
British thermal unit  Btu, 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  
horsepower  hp  ~745.7 W  engines, motors  power of the average horse  power  
British thermal unit per hour  Btu/h, BTU/h  ~0.293 1 W  heating/cooling  
ton of refrigeration  TR, TOR  (=12 000 BTU/h)  ~3.5 kW  cooling/ refrigeration  rate of heat transfer to melt/freeze 1 ton of pure ice at 0°C in 24 hours  
amperehour, amp hour  A⋅h, Ah  3 600 C  electrochemistry, battery capacity  amount of electric charge transferred at 1 ampere in 1 hour  electric charge  
degree Fahrenheit  °F  F = (5 / 9)(C  32)  referenced on temperature of iceammonium chloride solution (0°F) and human body temperature (96°F)  temperature 
References
[1] “Unit of measurement,” Wikipedia, 16Feb2021. [Online]. Available: https://en.wikipedia.org/wiki/Unit_of_measurement. [Accessed: 11Apr2021].
[2] P. P. Urone and R. Hinrichs, “Physical Quantities and Units,” College Physics, 23Jan2012. [Online]. Available: https://opentextbc.ca/openstaxcollegephysics/chapter/physicalquantitiesandunits/. [Accessed: 11Apr2021].
[3] “Measurement units” BIPM. [Online]. Available: https://www.bipm.org/en/measurementunits/. [Accessed: 12Apr2021].
[4] “Imperial and US customary measurement systems,” Wikipedia, 11Mar2021. [Online]. Available: https://en.wikipedia.org/wiki/Imperial_and_US_customary_measurement_systems. [Accessed: 12Apr2021].
[5] “Newton,” Encyclopædia Britannica. [Online]. Available: https://www.britannica.com/science/newtonunitofmeasurement. [Accessed: 12Apr2021].
[6] “International System of Units,” Wikipedia, 05Apr2021. [Online]. Available: https://en.wikipedia.org/wiki/International_System_of_Units. [Accessed: 12Apr2021].
[7] “United States customary units,” Wikipedia, 12Apr2021. [Online]. Available: https://en.wikipedia.org/wiki/United_States_customary_units. [Accessed: 12Apr2021].
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Faculty Advisor: Sanjeev Bedi