Complex dimensions and dimensional analysis

Prerequisites

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 two measured quantities don't change in the same way when we change our measurement scale, it doesn't make sense to add or subtract them.

For example, suppose that I drove for an hour ($t_1 = 1$ hour) and went a distance of 60 miles ($x_1 = 60$ miles). I then drove for a half hour ($t_2 = 0.5$ hour) but only got to go a distance of 15 miles ($x_2 = 15$ miles). It would make sense to add $x_1$ and $x_2$ to get the total distance I traveled, or to add $t_1$ and $t_2$ to get the total time I traveled, but the combination $x_1 + t_2$ wouldn't make any physical sense.

So, using the bracket notation, if [$x_1$] = [$x_2$] = L and [$t_1$] =[$t_2$] = T, then it's OK to write $x_1 + x_2$  (since this is adding L + L) but not $x_1 + t_2$ (since this is adding 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 [$x_1$] = [$x_2$] = 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:

[$x_1 + x_2$]= [$x_1$] = [$x_2$] = 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).

We say "length is a dimension" but "the quantity $x$ measured by a ruler has the dimensionality L."

Multiplying and dividing dimensions

On the other hand multiplication and division are more straightforward.

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

When we combine measurements, we express it by showing how these measurements are combined. First, using the double bracket notation, to remind you what we are doing when we are combining dimensionalities:

When we have correct equations for symbols that we know, those equations can tell us how measurements were combined to create that symbol. For example:

Rewriting this in single bracket form, these say

[$v$] = L/T

[$a$] = (L/T)/T = L/T2

and in the $F=ma$ equation

[$F$] = [$ma$] = ML/T2

Here's an interesting combination quantity: kinetic energy, $KE = \frac{1}{2}mv^2$:

[$\frac{1}{2}mv^2$] = [$mv^2$] = 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. What about the ½?

Dimensions of pure numbers

Our equations don't only contain measurements. Sometime, as in the equation above for energy, they have a pure number — like ½, 7, or $\pi$. These don't change when any of our measurements change. We might say "it has no dimension", but it's more convenient to say that a pure number "has dimensionality = 1".

It's not immediately obvious why we want to do that. It's because if you, for example, divide something with the dimensionality of a length by another quantity with the dimensionality of a length (for example, $[x/y]$ = L/L then you get the same result whatever units you use (as long as you use the same ones for $x$ and $y$). It looks like "it has no dimensionality" but since "L/L = 1" in algebra, it's convenient to call this 1. It works since when you divide or multiply by 1 it doesn't change anything.

So if we have an equation with a pure number in it, if we take its dimensionality as 1, it won't change the dimensions of what it's multiplying (or dividing).

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 ? (This is an illegal move!) Think about this and then see the page on Powers and exponentials.

Joe Redish 9/2/13

Article 251