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Units

Units

Oct 28, 2021





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 "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]

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

Iv

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]

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]

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]

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

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]

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-