Physical quantities, units of measurement and vectors

1. Physical quantity

A physical quantity is a property of a body or of a phenomenon that can be measured, that is, compared with a standard of the same kind chosen as the unit of measurement. The measure is the number that expresses how many times the unit is contained in the quantity, and the value of the quantity is written as the product of that number and the unit:

The number depends on the chosen unit, because the same length is $2.5$ in meters and $250$ in centimeters. For this reason a numerical result in physics is complete only when it is accompanied by its unit of measurement.

2. Base and derived quantities

The base quantities (or fundamental quantities) are a small set of quantities chosen as independent, each with its own unit defined autonomously. The derived quantities are defined from the base quantities through a mathematical relation, and their unit is obtained from the same relation. The average velocity is the ratio of a displacement to a time interval, $v=\frac{\Delta x}{\Delta t}$, so it is measured in meters per second, $m/s$; the area of a rectangle is the product of two lengths, so it is measured in ${\mathrm{m}}^2$.

1. International System of Units

The International System (SI) fixes seven base quantities and the corresponding units.

Since 2019 every base unit has been defined by fixing the exact value of a constant of nature: the second by the frequency of a transition of the cesium-133 atom, the meter by the speed of light in vacuum, $c=299792458m/s$, the kilogram by the Planck constant. In mechanics the meter, the kilogram and the second are sufficient; the kelvin and the mole appear in thermodynamics.

Some derived units have a name of their own: the newton for force, $1\mathrm{N}=1\mathrm{kg}\cdot m/{\mathrm{s}}^2$, the joule for work and energy, $1\mathrm{J}=1\mathrm{N}\cdot \mathrm{m}$, the watt for power, $1\mathrm{W}=1J/s$, and the pascal for pressure, $1\mathrm{Pa}=1N/{\mathrm{m}}^2$. Each of them can always be rewritten in terms of base units; for example $1\mathrm{J}=1\mathrm{kg}\cdot {\mathrm{m}}^2/{\mathrm{s}}^2$.

2. Multiples and submultiples

To express very large or very small values, a prefix is placed in front of the unit symbol; the prefix multiplies the unit by a power of ten.

The prefix forms a single unit with the symbol, so an exponent also applies to the prefix: $1\mathrm{c}{\mathrm{m}}^2=(10^{-2}\mathrm{m}{)}^2=10^{-4}{\mathrm{m}}^2$ and $1\mathrm{c}{\mathrm{m}}^3=10^{-6}{\mathrm{m}}^3$. For mass the prefixes are applied to the gram, since the kilogram already contains the prefix kilo: $1\mathrm{mg}=10^{-3}\mathrm{g}=10^{-6}\mathrm{kg}$.