There have been numerous systems of units of measurement for physical quantities such as mass, length and energy. Some of these systems of units are inconsistent and worse have no repeatable definition.

Early systems, and even modern imperial systems, are based on the lengths of part of the human body. The best example is the foot. Other units have the same name but different values in different countries. A good example is the gallon. An imperial (UK) gallon is the volume of ten pounds of water at \(17^\circ\). A US gallon is defined to be 231 cubic inches, which as now an inch is defined to be exactly 2.54 centimetres, is exactly 3.785411784 litres!

In 1875 the *Metre Convention* standardised the
definition of the metre for length. This led to the
International System of Units (SI). There are now seven base
units. They were originally defined in terms of physical
quantities. Some of these were later refined to be in terms of
specially created physical entities. Most are now defined
exactly. Each unit has a symbol. It usually starts with a lower
case letter unless it is named after a person.

In May 2019 all SI units got redefined. A number of fundamental physical constants, such a Planck’s constant, were given exact values. All SI units are now defined exactly in terms of fundamental physical constants and other SI units. Their values can never change and are not dependent on any physical artifact.

Many units are too large or too small in some environments. The convention is to prefix the unit with a modifier which is a power of \(10\) and is usually a power of \(1,000=10^3\). Some of the multiplier names have curious derivation. The multipliers are:

- Yotta (Y) \(10^{24}\) is derived from the Greek word for eight as it is \(1,000^8\).
- Zetta (Z) \(10^{21}\) was originally Hepta and is derived from the Greek word for seven as it is \(1,000^7\).
- Exa (E) \(10^{18}\) is derived from the Greek word for six as it is \(1,000^6\).
- Peta (P) \(10^{15}\) is derived from the Greek word for five as it is \(1,000^5\).
- Tera (T) \(10^{12}\) is derived from the Greek word for monster.
- Giga (G) \(10^9\) is derived from the Greek word for giant.
- Mega (M) \(10^6\) is derived from the Greek word for great.
- kilo (k) \(10^3\) is derived for the Greek word for thousand.
- hecto (h) \(10^2\) is derived for the Greek word for hundred.
- deca (da) \(10\) is derived for the Greek word for ten, it is not often used.
- deci (d) \(10^{-1}\) is derived from the Latin for tenth.
- centi (c) \(10^{-2}\) is derived from the Latin for hundred.
- milli (m) \(10^{-3}\) is derived from the Latin for thousand.
- micro (\(\mu\)) \(10^{-6}\) derived from the Greek word for small and is the only multiplier which doesn’t us a letter from the Latin alphabet.
- nano (n) \(10^{-9}\) is derived from the Greek for dwarf.
- pico (p) \(10^{-12}\) is derived from the Spanish for peak or beak.
- femto (f) \(10^{-15}\) is derived from the Danish for fifteen.
- atto (a) \(10^{-18}\) is derived from the Danish for eighteen.
- zepto (z) \(10^{-21}\) is derived from the Latin for seven as it is \(1,000^{-7}\).
- yocto (y) \(10^{-24}\) is derived from the Greek word for eight as it is \(1,000^{-8}\).

*The SI base units are a choice of seven well-defined
units which by convention are regarded as dimensionally
independent.*

Every other unit can be derived from one or more base units.

There are many units of time. Some are short, some are very long. Most are based of periods of the Earth and Moon. The week has varied between cultures and has been six, seven and even ten days in duration. The unit of time which has been adopted as an international standard is the second.

The second was originally defined in terms of an Earth day. A day consists of 24 hours. An hour consists of 60 minutes. A minute consists of 60 seconds. Hence a day is \(8,6400\) seconds.

Due to the fact that Earth days vary in length, an exact definition of a second was required.

A Caesium-133 atom has 55 electrons. Of these electrons; 54 of them are tightly bound in their stable orbitals. The outermost electron in the \(6s\) shell is not affected by the others and has two possible spin states. There is a small energy difference between the spin states which emits microwave radiation on transition. The energy difference is small and is called a hyperfine level. The second was tied to the frequency of this radiation in 1967.

*The second, symbol \(s\), is the SI unit of time. It is
defined by taking the fixed numerical value of the caesium
frequency \(\Delta\nu Cs\), the
unperturbed ground-state hyperfine transition frequency of the
caesium-133 atom, to be \(9 192 631
770\) when expressed in the unit \(Hz\), which is equal to \(s^{-1}\).*

There have been many units of length. Many were based on the dimensions of parts of the human body or of other common objects. The SI unit of length is the metre.

The metre, symbol \(m\) is the SI unit of length.

The metre was originally defined to be \(1/10,000,000th\) of the distance between the Equator and the North Pole along the great circle through Paris. It was later redefined in terms of the wavelength of a transition of a Krypton-86 atom.

In 1983 the metre was redefined in terms of the second and the speed of light in a vacuum.

*The metre, symbol \(m\),
is the SI unit of length. It is defined by taking the fixed
numerical value of the speed of light in vacuum \(c\) to be \(299 792 458\) when expressed in the
unit \(m\cdot s^{-1}\), where
the second is defined in terms of the caesium frequency \(\Delta\nu Cs\).*

The kilogram, symbol \(kg\)
is the SI unit of mass. The kilogram is curious because it has
the *kilo* multiplier. It was also the remaining SI base
unit to be based on an artifact.

The kilogram was originally defined to be the mass of \(1l\) of pure water at its freezing
point. In 1889 a 90% Platinum 10% Iridium artifact was created
which was called the International Prototype Kilogram (IPK). It
was also known as *Le Grand K*. It was stored near Paris.
There were six official copies stored around the world for
calibration. The problem was that the prototpe mass changed over
time which meant that the kilogram also changed!

In May 2019, the definition of the kilogram changed to be defined in terms of the Planck constant \(h\). The Planck constant has been fixed, rather like the speed of light was fixed to define the metre. The Kibble balance is a device used to measure the Planck constant. Now that the Planck constant is fixed, a Kibble balance can accurately measure mass.

*The kilogram, symbol \(kg\), is the SI unit of mass. It is
defined by taking the fixed numerical value of the Planck
constant \(h\) to be \(6.626 070 15 \cdot 10^{-34}\) when
expressed in the unit \(J\cdot
s\), which is equal to \(kg\cdot m^2\cdot s^{-1}\) , where
the meter and the second are defined in terms of \(c\) and \(\Delta\nu Cs\).*

*The ampere, symbol \(A\), is the SI unit of electric
current. It is defined by taking the fixed numerical value of
the elementary charge \(e\) to
be \(1.602176634\cdot
10^{-19}\) when expressed in the unit \(C\), which is equal to \(A\cdot s\), where the second is
defined in terms of \(\Delta\nu
Cs\).*

*The kelvin, symbol \(K\), is the SI unit of
thermodynamic temperature. It is defined by taking the fixed
numerical value of the Boltzmann constant \(k\) to be \(1.380649\cdot 10^{-23}\) when
expressed in the unit \(J\cdot
K^{-1}\), which is equal to \(kg\cdot m^2 \cdot s^{-2}\cdot
K^{-1}\), where the kilogram, metre and second are
defined in terms of \(h\),
\(c\) and \(\Delta\nu Cs\).*

*The mole, symbol \(mol\), is the SI unit of amount of
substance. One mole contains exactly \(6.02214076\cdot 10^23\) elementary
entities. This number is the fixed numerical value of the
Avogadro constant, \(N_A\),
when expressed in the unit \(mol^{-1}\) and is called the
Avogadro number. The amount of substance, symbol \(n\), of a system is a measure of
the number of specified elementary entities. An elementary
entity may be an atom, a molecule, an ion, an electron, any
other particle or specified group of particles.*

*The candela, symbol \(cd\), is the SI unit of luminous
intensity in a given direction. It is defined by taking the
fixed numerical value of the luminous efficacy of monochromatic
radiation of frequency \(540\cdot
10^{12} Hz\), \(K_{cd}\), to be \(683\) when expressed in the unit
\(lm\cdot W^{-1}\), which is
equal to \(cd\cdot sr\cdot
W^{-1}\), or \(cd\cdot sr\cdot
kg^{-1}\cdot m^{-2}\cdot s^3\), where the kilogram, metre
and second are defined in terms of \(h\), \(c\) and \(\Delta\nu Cs\).*

Where \(sr\) is a steradian.

Derived units are defined in terms of combinations of one or more base units or derived units.

The SI unit of frequency is the Hertz. The symbol is \(Hz\). It is named after Heinrich Rudolf Hertz. It is defined in terms of the number of cycles per second of sound waves, light waves or other vibrations \(1Hz=1s^{-1}\).

The litre, symbol \(l\), is a unit of volume \(1l=1dl^3=1,000cm^3\)

The Newton, named after Sir Isaac Newton, symbol \(N\) is the unit of force \(1N=1kg\cdot m\cdot s^{-2}\).

The Joule, named after James Prescott Joule, symbol \(J\) is the unit of energy \(1J = 1 kg\cdot m^2 s^{-2}\).