Physical Units
Historical Background
Units of all kind, generally anthropocentric, have been used for millenia,
to measure physical quantities. Length might be measured in feet or miles
or meters or kilometers, weights in stones or kilograms and so on.
A first breakthrough towards a rational system of units came with the
French revolution in 1789, when a metric system of units was introduced.
Length was measured in meters, based on the length of a metal bar, the
kilogram by the weight of a metal cylinder. They are subject to erosion,
so a more reliable system was needed. This ushered in the era of the now
almost universally accepted internatiional system of units, in shorthand
referred to as the SI system of units.
SI units
SI units, in the most cuurent incarnation, are defined by declaring the
values of certain universal physical constants to be exact. The system
defines 7 base units and has numerous derived units that can be expressed
in terms of the base units. The table below shows the base units,
$$
\large{
\begin{array}{|l|c|c|c|}
\hline
quantity & symbol & unit & defined\ by\\
\hline \hline
time & T & s & \Delta\nu_{Cs}\\
length & L & m & c\\
mass & M & kg & \hbar ,\,c\\
current & I & A & q\,(\,e^\pm\,)\\
temperature & \Theta & K & k_B\\
amount & n & mol & N_A\\
intensity & - & cd & 683\,lm/W\\
\hline
\end{array}}
$$
One starts with time and works the way down the list, if need be
using several constants, as shown below. Physical quantities X consist
of a numerical value |X| and an associated unit [X], X = |X| [X].
For dimensionless quantities one often indicates this fact by
writing [X] = 1.
$$
\large{
\begin{align}
1\,s &= \Delta\nu_{Cs}^{-1}\\
1\,m &= c\times (\, |c|^{-1}\,s \,)\\
H &= \hbar\times |\hbar|^{-1}\\
V &= c\times |c|^{-1}\\
1\,kg &= H\times\Delta\nu_{Cs}\times V^{-2}\\
\cdots\ &=\ \cdots
\end{align}}
$$