*Class content* >*Modeling with mathematics* > *Using math in science*

## Dimensions

When we figured out how to assign a number to a quantity via a measurement, we had 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.

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 (T)

(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.)

## Units

Note that the term "dimension" refers to the general fact that we measure a physical quantity. and so have an arbitrary choice to make when we define a particular kind of measurement. 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*.

The arbitrariness of the standard unit is the key for the implications of dimensions. 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

*L*_{1} = *L*_{2}.

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.**

### 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.

*NOTE: 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.

* *

Thus, we could legitimately write

[*x*] = [Δ*x*] = [*y*] = L.

This would not say that the three quantities *x*, Δ*x*, 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.

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Joe Redish 7/7/11

Wolfgang Losert 8/29/12

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