Dimensionalities

An essential idea in the use of math in science is that when we model something in the real world with math, we do so by making a measurement. As a result, the quantities we typically assign symbols to when modeling with math are not just numbers. They're more complex quantities.

When we figure out how to assign a number to a quantity, we have to make an arbitrary choice of a standard unit. Since there are many kinds of measurements we have to make, we need a way to track the various kinds of measurement choices we have made.

When we report a measurement or write an equation containing measurements, it is, therefore, essential that we specify

• what kind of measurement we made (the dimensionality) and
• what specific standard we chose (the unit). 

We stress "arbitrary" for an important reason. Measurements in science are not just for ourselves; they are about communicating something to someone else. The arbitrariness of a measurement standard means that different people may choose to use different standards.*

We refer to "length" as a dimensionality of the quantity. (Sometimes this is called dimension, but we will often use the longer term so as not to get confused with the number of space dimensions we are considering in a particular problem; for example: motion in a plane is motion in 2 dimensions.)  The specific choice we make to measure length with — feet, furlongs, meters, or light years — is referred to as a unit.  

Dimensionalities

Since we are mapping physical measurements into math, most of the quantities we use in physics are NOT NUMBERS. They are MEASUREMENTS.

In our class we will typically use five different kinds of measurement which we will indicate using 5 icons:

A measurement made with a ruler (a length)
A measurement made with a clock (a time)
A measurement made with a scale (a mass)
A measurement made with a thermometer (a temperature)
A measurement made with an ammeter (a charge)

Measurements always depend on an arbitrary scale we have chosen.

In order that the equations we write keep their validity (the equation still holds) when we change our arbitrary scale, dimensions must match on both sides of the equation.

This means that equations in physics (and science in general) tend to be different from what you have seen in math. Figuring out what kinds of measurement were used in getting a value is called dimensional analysis.

Notating dimensionality

Equation in physics typically contain a lot of symbols. To make sense of the equation we have to ask each symbol: "What measurements are you made of and how?" For the moment, we will indicate the question by using double square brackets and the answer by using the icons given above. In dimensional analysis, we express the statement in words to the left by the (visual) equation on the right.

1. A displacement (change in position) is found using a ruler
(making a length measurement - L)

2. A time interval is found using a clock
(making a time measurement - T)

3. A mass is found using a scale
(making a mass measurement - M)

4. A temperature is found using a thermometer
(making a temperature measurement - Θ)

5. A charge is found using an ammeter
(making a current-time measurement - Q) 

(Note that the variable for temperature is typically, T, but the symbol for the dimension of temperature is a capital theta (Θ). There is always the possibility of confusion when we use the same symbol for different things. Keep track by context.)

We've been very careful in this discussion to use markers that are very different from what we do in standard algebra and the way we usually write equations. Double brackets are rarely used in doing algebra and our funny little icons, while being a very good reminder of what we are doing (specifying the kind of measurement, NOT a value), are not commonly used (and are not available on typical keyboards).

As a result, instead of using a little ruler, we will write "L", instead of using a little clock, we will write "T", instead of using a little scale, we will write "M", instead of using a little thermometer, we will write "Θ", and instead of using a little ammeter, we will write "Q". 

L =

T =

M =

Θ =

Q =

Worse than that, instead of writing double brackets, from laziness, we will use single brackets! So our "what measurements were used to get your value?" question will look like this for velocity and force:

[$v$] = L/T

[$F$] = ML/T²

This can be a real pain! In particular, we might sometimes see the equation $v = L/T$, where we mean $L$ and $T$ to be values rather than icons. You just have to be careful and pay attention to the context. What question are you asking? Are you doing a dimensional analysis (asking what measurements are made to get the value) or doing a calculation (using the values of the measurements)? The symbols mean very different things in the two cases. It might help you when doing dimensional analysis to think of the symbols M, L, T, and Q as our little icons.

We'll try to be consistent in using dimensions as non-italic letters and symbols that take values as italic letters. This won't work in handwriting, alas, so you'll have to be careful in watching out for the square brackets that tell you we are NOT writing an equation with symbols but rather asking about dimensionalities.

Think of the bracket as a kind of function: one that asks the thing inside, "What kind of measurement was made to get your value?"**

 Thus, we could legitimately write

This is really tricky. In a very real sense the "=" in these equations don't mean what they usually mean. It does not say that the three quantities $x$, $Δx$, and $y$ all have the same value. Rather, it only says they all have the same dimensionality — they are obtained by the same kind of measurement — and therefore could in principle be compared with each other.

* It is amusing to realize is that the choice of dimension is arbitrary and it depends on our current state of knowledge.  For example, we typically measure distance and time, but we now know that there is a fundamental speed associated with the universe: the speed of light, c. This speed is an invariant (in empty space) -- the same for all observers. We could therefore choose c = 1 (with no units) and take our measure of distance to be the amount of time it takes light to travel that distance. We could thus measure all of our distances using time units. (1 meter corresponds to about 3 nanoseconds.)

**  We might well 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 dimensionality. This would prevent lots of confusion. With the other notation, we write "L" to stand for dimensionality, but in the same problem we might be using as a variable, "L", for a particular length. Unfortunately, the notation above is standard so we will stick with it. You will just have to use your sensitivity for contexts to decide whether "L" stands for a dimensionality or a variable.

Joe Redish and Wolfgang Losert 8/29/12