Dimensional quantity 101

"Dark and difficult times lie ahead. Soon we must all face the choice between what is right and what is easy." ― Albus Dumbledore, Harry Potter and the Goblet of Fire

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Unit systems

There are three basic unit systems in use today:

• the International System of Units (SI units, from Système international d'unités, more commonly simply called metric units)
• the English Engineering System of Units (commonly called English units)
• the British Gravitational System of Units (BG)

This library only supports SI units by library default.

Base units and dimensions

Base units have the important property that all other units derive from them. In the SI system, there are seven such base units and corresponding physical quantities:

• meter (m) for length,
• kilogram (kg) for mass,
• second (s) for time,
• kelvin (K) for temperature,
• ampere (A) for electric current,
• candela (cd) for luminous intensity, and
• mole (mol) for the amount of substance.

Mathematical Notation

We need some suitable mathematical notation to calculate with dimensions like length, mass, time, and so forth. The dimension of length is written as [ $L$ ], the dimension of mass as [ $M$ ] and the dimension of time as [ $T$ ]. The dimension of a derived unit like velocity, which is distance (length) divided by time, then becomes [ $LT^{-1}$ ] in this notation. The dimension of force, another derived unit, is the same as the dimension of mass times acceleration, and hence the dimension of force is [ $MLT^{-2}$ ].

Now let's think about multiplication and division of dimensional quantities. For example,

$$a [m] \times b [s] = c [m \cdot s]$$

$m \cdot s$ is a new unit different from both of metre and second, and can give a clear meaning that $m \cdot s$ is proportional to both of metre and second.

And $m / s$ is proportional to metre and inversely proportional to second. Also, $m /s \times s$ derives $m$.

Dimensions of common physical quantities

Many derived quantities are measured in derived units that have their own name. Force is one example: Newton (N) is a derived unit for force, equal to $kg \cdot m \cdot s^{-2}$. Another derived unit is Pascal (Pa) for pressure, i.e., force per area. The unit of Pa then equals $N/m^2$ or $kg \cdot m^{-1} \cdot s^{-2}$.

Below are more names for derived quantities, listed with their units.

Name	Symbol	Physical quantity	Unit
radian	rad	angle	$1$
hertz	Hz	frequency	$s^{-1}$
newton	N	force, weight	$kg \cdot m \cdot s^{-2}$
pascal	Pa	pressure, stress	$N/m^2$
joule	J	energy, work, heat	$N \times m$
watt	W	power	$J/s$

Some common physical quantities and their dimensions are listed next.

Quantity	Relation	Unit	Dimension
pressure	force/area	$N \cdot m^{-2}=Pa$	[ $MT^{-2}L^{-1}$ ]
density	mass/volume	$kg \cdot m^{-3}$	[ $ML^{-3}$ ]
strain	displacement/length	$1$	[ $1$ ]
Young's modulus	stress/strain	$N \cdot m^{-2}=Pa$	[ $MT^{-2}L^{-1}$ ]
Poisson's ratio	transverse strain/axial strain	$1$	[ $1$ ]
moment (of force)	distance $\times$ force	$N \cdot m$	[ $ML^2T^{-2}$ ]
impulse	force $\times$ time	$N \cdot s$	[ $MLT^{-1}$ ]
linear momentum	mass $\times$ velocity	$kg m \cdot s^{-1}$	[ $MLT^{-1}$ ]
angular momentum	distance $\times$ mass $\times$ velocity	$kg \cdot m^2 \cdot s^{-1}$	[ $ML^2T^{-1}$ ]
work	force $\times$ distance	$N \cdot m=J$	[ $ML^2T^{-2}$ ]
energy	work	$N \cdot m=J$	[ $ML^2T^{-2}$ ]
power	work/time	$N \cdot m \cdot s^{-1}=W$	[ $ML^2T^{-3}$ ]
heat	work	$J$	[ $ML^2T^{-2}$ ]
heat flux	heat rate/area	$W \cdot m^{-2}$	[ $MT^{-3}$ ]
temperature	base unit	$K$	[ $\Theta$ ]
heat capacity	heat change	$J/K$	[ $ML^2T^{-2}\Theta^{-1}$ ]
specific heat capacity	heat capacity/unit mass	$J \cdot K^{-1} \cdot kg^{-1}$	[ $L^2T^{-2}\Theta^{-1}$ ]
thermal conductivity	heat flux/temperature gradient	$W \cdot m^{-1} \cdot K^{-1}$	[ $MLT^{-3}\Theta^{-1}$ ]
dynamic viscosity	shear stress/velocity gradient	$kg \cdot m^{-1} \cdot s^{-1}$	[ $ML^{-1}T^{-1}$ ]
kinematic viscosity	dynamic viscosity/density	$m^2/s$	[ $L^2T^{-1}$ ]
surface tension	energy/area	$J \cdot s^{-2}$	[ $MT^{-2}$ ]

Metric Prefix

A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand; one kilogram is equal to one thousand grams. The prefix milli-, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousand of a metre.

The SI prefixes are standardized for use in the International System of Units (SI units).

Text	Symbol	Factor	Power
yotta	Y	1000000000000000000000000	$10^{24}$
zetta	Z	1000000000000000000000	$10^{21}$
exa	E	1000000000000000000	$10^{18}$
peta	P	1000000000000000	$10^{15}$
tera	T	1000000000000	$10^{12}$
giga	G	1000000000	$10^9$
mega	M	1000000	$10^6$
kilo	k	1000	$10^3$
hecto	h	100	$10^2$
deca	da	10	$10^1$

Text	Symbol	Factor	Power
deci	d	0.1	$10^{-1}$
centi	c	0.01	$10^{-2}$
milli	m	0.001	$10^{-3}$
micro	u	0.000001	$10^{-6}$
nano	n	0.000000001	$10^{-9}$
pico	p	0.000000000001	$10^{-12}$
femto	f	0.000000000000001	$10^{-15}$
atto	a	0.000000000000000001	$10^{-18}$
zepto	z	0.000000000000000000001	$10^{-21}$
yocto	y	0.000000000000000000000001	$10^{-24}$