Class Content I >Modeling with mathematics > Using math in science
2.1.3
Pre-requisites:
When we figure out how to assign a number to a quantity (make a measurement), 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.
We're stressing "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. 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 dimension) and what specific standard we chose (the unit). Thus, a "length" is a dimension. The specific choice we make to measure length with — feet, furlongs, meters, or light years — is referred to as a unit.
Dimensions
A dimension is what we call an arbitrary* choice of scale that we make in assigning a number to a physical quantity in making a measurement. Typically in working with mechanics, we choose three distinct and arbitrary dimensions to describe our systems:
- length, or distance (L)
- time (T)
- mass (M)
As we go to more complex physics, we can add additional dimensions.
- charge (Q)
- temperature (Θ {theta})
Units
When we are creating a physical equation, typically we are writing equations that say two physical quantities are equal. Thus, we can say the length of two objects is the same. We would then write an equation
L1 = L2.
Note that this equation will be true no matter what unit we choose — as long as we have chosen the same standard — the same unit —for each. But if we tried to equate a length and a time, say, to consider 1 meter = 1 second, then we would need something else - meters are missing on the right hand side of the equation plus something to equate with seconds.
The crucial result is
We can only equate quantities that have the same dimension and we can only expect the numbers of both sides to be the same if we use the same units.
The last part of that statement can be shown with the example
1 inch = 2.54 cm
The numbers are different; 1 is not equal to 2.54. This equation makes no sense if you think of it as a statement about numbers. But it makes perfect sense physically: the length 1 inch and the length 2.54 cm represent the same physical distance. (Actually, this is the definition of the inch.)
Notating Dimension
We refer to the properties of a variable in terms of dimensions as its dimensionality. In order to specify in symbols the kind of measurement that has to be made to assign a numerical value to something, we will introduce the bracket notation. The dimensionality 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.
Thus, we could legitimately write
[x] = [Δx] = [y] = L.
This would not say that the three quantities x, Δx, and y all had the same value; rather, it says they all have the same dimensionality (are obtained by the same kind of measurement) and therefore could in principle be compared with each other.
Dimensional analysis is one of our most powerful and useful tools in our toolbelt of methods for using math in science. For more details and some examples, see the discussion of dimensional analysis as a tool.
* 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.)
** 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 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.
Follow Ons:
Joe Redish 7/7/11
Wolfgang Losert 8/29/12
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