Complex Dimensions
We have defined a dimension as an arbitrary choice of scale that we make in assigning a number to a physical quantity in making a measurement. The term refers to the general fact of the choice and not to the specific unit. In our discussion of simple dimensions, we only talked about using a single measurement. Often, we have to combine multiple measurements to create a single physical quantity: like dividing a distance measurement by a time measurement to get an average velocity. To understand how this more complex situation works, we need to develop a notation for representing and analyzing dimensions.
Notating Dimension
In order to specify the kind of measurement that has to be made to determine the numerical value assigned to a physical quantity, we will introduce the bracket notation.
The dimension of a measurement is specified by the letter corresponding to the type of measurement: L, T, M, Q, ....
We write that x is a length by using the notation:
[x] = L.
NOTA BENE: I would prefer to do something different, such as Dim(x) = [L]. This would indicate that we are taking some information from x and setting it equal to a new and peculiar kind of quantity -- a dimension. This would prevent lots of confusion. With the other notation, we write "L" to stand for a dimension, but in the same problem we might be using as a variable, "L", for a particular length. Unfortunately, the notation above is quite standard so we will stick with it. You will just have to use your sensitivity for contexts to decide whether "L" stands for a dimension or a variable.
Thus, we could legitimately write
[x] = [Δx] = [y] = L.
This would not say that the three quantitites x, Δx, and y all had the same value; rather, it says they all have the same dimension (are obtained by the same kind of measurement) and therefore could in principle be compared with each other.
Combining Measurements
Measurements can be combined in a variety of ways: they can be added (or subtracted) or multipled (or divided). But there are restrictions.
- We can only add (or subtract) quantities that have the same dimension.
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 dimension of their sum, however, becomes:
[x1 + x2]= [x1] + [x2] = L + L = L
That last is correct: L + L = L, not 2L. It might look better if you read it: the sum of two length measurements is a length measurent.
On the other hand multiplication (and division) is more straightforward.
- We can multiply (or divide) quantities that have the same or different dimensions.
This allows us to build lots of interesting and useful combination quantities:
- velocity: [v] = [Δx/Δt] = 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 dimensions. The 1/2 doesn't matter. It doesn't change when any of our measurements change, so it has unit dimensions.
This is about all we can do to combine dimensioned quantities. Why can't we raise a quantity with one dimension, to the power of a quantity with another dimension e.g., xt?
