Dimensional Analysis

Definition: The Dimension is the qualitative nature of a physical quantity (length, mass, time).
Square brackets denote the dimension or units of a physical quantity:

Table 2.2: Dimensions

quantity	dimension	SI units
area	[A] = L 2	m 2
volume	[V]=L 3	m 3
velocity	[v] = L/t	m/s
acceleration	[a] = L/t 2	m/s 2
mass	[m] = M	kg

Idea: Dimensional analysis can be used to derive or check formulas by treating dimensions as algebraic quantities. Quantities can be added or subtracted only if they have the same dimensions, and quantities on two sides of an equation must have the same dimensions.
Note: Dimensional analysis can't give numerical factors. For Example: The distance (x) travelled by a car in a given time (t) , starting from rest and moving with constant acceleration (a) is given by, x = ${frac{1}{2}}$ at 2. We can check this equation with dimensional analysis:

l.h.s. [x]	=	L
r.h.s. $displaystyleleft[ frac{1}{2}at^{2} right]$	=	$displaystyle{textstylefrac{1}{2}}$ [a][t 2] = $displaystyle{frac{:rm L}{{:rm t}^{2}}}$ t 2 = L.

Since the dimension of the left hand side (l.h.s.) of the equation is the same as that on the right hand side (r.h.s.), the equation is said to be dimensionally consistent.

When equating equations , one should keep this in mind that two terms can only be added only if they are dimensionally similar

Dimension may be regarded as a base to which quantities are raised just as in number system or expressing numbers as a logarithmic function .

Dimension gives us an idea of the relation a physical quantity has with mass , length and time and other basic Physical units .

Its significance lies in the fact that we can easily and quite intuitively grasp what something in the physics jargon actually has in relation to the world in terms of mass ,length , time and similar quantities

Original link