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 "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/s²=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 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]
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 cm³) 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/m²
joule	J	energy, work, heat	kg⋅m²⋅s⁻²	N⋅m
watt	W	power, radiant flux	kg⋅m²⋅s⁻³	J/s
coulomb	C	electric charge	s⋅A
volt	V	voltage, electromotive force	kg⋅m²⋅s⁻³⋅A⁻¹	W/A, J/C
farad	F	capacitance	kg⁻¹⋅m⁻²⋅s⁴⋅A²	C/V
ohm	Ω	resistance	kg⋅m²⋅s⁻³⋅A⁻²	V/A
henry	H	inductance	kg⋅m²⋅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	m²	area
cubic metre	m³	volume
metre per second	m/s	speed, velocity
metre per second squared	m/s² = (m/s)/s	acceleration
newton-second	N⋅s = kg⋅m/s	momentum, impulse
kilogram per metre	kg/m	linear density
kilogram per cubic metre	kg/m³	density
mole per cubic metre	mol/m³	concentration
kilogram per cubic metre	kg/m³	mass concentration
newton-metre	N⋅m	torque, moment of force
radian per second	rad/s	angular velocity, angular frequency
radian per second squared	rad/s²	angular acceleration
joule per kelvin	J/K	heat capacity, entropy
joule per kilogram-kelvin	J/(kg⋅K) = J/kg/K	specific heat capacity, specific entropy
joule per cubic metre	J/m³	energy 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]
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 hm² = 10⁴ m²	area
litre	l, L	1 L = 1 dm³ = 1000 cm³ = 0.001 m³	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	mass
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 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]
Name	Symbol	Power of 10	Power of 1000	Decimal	English (short scale)
peta-	P-	10¹⁵	1000⁵	1 000 000 000 000 000	quadrillion
tera-	T-	10¹²	1000⁴	1 000 000 000 000	trillion
giga-	G-	10⁹	1000³	1 000 000 000	billion
mega-	M-	10⁶	1000²	1 000 000	million
kilo-	k-	10³	1000¹	1 000	thousand
hecto-	h-	10²	1000^2/3	100	hundred
deca-	da-/D-/Da-	10¹	1000^1/3	10	ten
		10⁰	1000⁰	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 (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)
Name	Symbol	Power of 2	Decimal	SI approx.
kibi-	Ki-	2¹⁰	1024	kilo-
mebi-	Mi-	2²⁰	1 048 576	mega-
gibi-	Gi-	2³⁰	1 073 741 824	giga-
tebi-	Ti-	2⁴⁰	1 099 511 627 776	tera-
pebi-	Pi-	2⁵⁰	1 125 899 906 842 624	peta-

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]
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		~3.0857×10¹⁶ 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	speed, velocity
square inch	sq. in.		6.452 cm²			area
square (survey) foot	sq. ft., ft²	~144 square inches	~0.092 903 41 m²		based on the length of the foot before internationalization
acre	acre, ac	43 560 sq. ft. (survey)	~4046.9 m²
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
kilowatt-hour	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
ampere-hour, 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 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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