# 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 d*erived *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 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 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 |

newton-second | 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 |

newton-metre | 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 kilogram-kelvin | J/(kg⋅K) = J/kg/K | specific heat capacity, specific entropy |

joule per cubic metre | J/m^{3} | 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 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 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 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- |

### 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 | ~3.0857×10^{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 | 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 | |||

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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