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# Complex dimensions and dimensional analysis

last edited by 7 years, 10 months ago

## Measurements can be combined in a variety of ways: they can be added (or subtracted) or multiplied (or divided). But there are restrictions.

• We can only add, subtract, or equate quantities that have the same dimensionality.

It is important to understand why this is so. The key is change. When we change our choice of measurement scale, how does the quantity we are talking about change? If they don't change in the same way, it doesn't make sense to add or subtract them.

So, if [x] = [y] = L and [τ] = T, then it's OK to write x + y  (L + L) but not x + τ  (L + T).

Notice that because dimensions only tell you about the type of a quantity and not its magnitude, the algebraic handling of dimensions may sometimes look peculiar until you get used to it. Thus, if [x1] = [x2] = L (that is, they are both quantified by making a length measurement) then we can add them. The statement about the dimensionality of their sum, however, becomes:

[x1 + x2]= [x1] = [x2] = L

All those equalities are correct even though the values of the distances might be different.  The bracket means you are only talking about the kind of measurement -- and the dimension of a sum is equal to the dimension of each element (as we said elements can only be added up if they have the same dimension).

On the other hand multiplication (and division) is more straightforward.

• We can multiply (or divide) quantities that have the same or different dimensionalities.

This allows us to build lots of interesting and useful combination quantities:

• velocity: [v] = [Δxt] = L/T
• energy: [1/2 mv2] = [mv2] = M(L/T)2 = ML2/T2.

Yes, the first equality for the energy is correct since the brackets [..] tell us we are only concerned with dimensionalities. The 1/2 doesn't change when any of our measurements change, so it has unit dimensionality.

This is about all we can do to combine dimensioned quantities. Why can't we raise a quantity with one dimensionality, to the power of a quantity with another dimensionality e.g., xt ?