When calculating dimensions, you look at the measurement units
and ignore the numbers associated with them.
Dimensions are represented in square brackets.
[L] is a single dimension in length, [T] is time and [M] is
mass.
[L2] represents length in 2 dimensions - or an area.
Slightly more complex are density = [M][L-3] or [M]/[L3]
Addition of subtraction can only be carried out on identical
dimensions and the result is the same "term".
Thus [L2] + [L2] = [L2]. The rules for combination may look
strange mathematically, but if you describe the equation in words
they may be clear. Add an area to an area and you get and area.
Multiplication and division are "normal" and, if you are
familiar with indices, they are straightforward.
Here is an interesting example of where dimensional analysis can
take you.
Velocity = [L][T-1]
Acceleration = (Change in velocity)/Time = ([L][T-1] - [L][T-1])
/ [T]
but [X] - [X] = [X] (see rules for addition and subtraction
above)
= ([L][T-1]) /[T] = [L][T-2]
Force = Mass*Acceleration = [M]*[L][T-2] = [M][L][T-2]
Energy or mechanical Work = Force*Distance Moved =
[M][L][T-2]*[L]
= [M][L2][T-2]
= [M]*([L][T-1])2
But [L][T-1] is the dimensional representation of velocity
So, we have
Energy = [M]*velocity2
Or e =mc2 which you may have come across before!