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

When we model something in the physical world with mathematics, we assign a number to a physical quantity. One common way to do this is an **operational definition**. This describes a procedure for assigning a number to a physical quantity. An example is *length*. We have an intuitive sense of what we mean by length. But how do we assign a number to it?

To assign a number to a length, we have to:

- Choose a standard unit of length (e.g, an inch or a centimeter)
- Compare our standard to the length we are trying to measure by counting how many of the standard units can be removed from our initial unit. (Another way of saying this is to consider how many times our standard length can fit into our length to be measured.)
- When we are left with something smaller than our standard unit, we break the standard unit up into equal fractions and start again.

Notice that there are a number of assumptions in making this definition.

- We can move our standard from one place to another without it changing.
- We can say when two lengths are equal.
- The physical object we are measuring actually has a well-defined length.
- The length we are measuring can be fit with a reasonable number of pieces of our measuring stick. (If we have to divide it, it divides without being destroyed.)

Point 1 is usually OK. We only have to worry about it when the properties of space change (as they do in general relativity near black holes). Point 2 only works if the quantity we are trying to represent with a number from our standard is "of the same type" (whatever that means). If you think about trying to measure an area by fitting a standard length against it and counting the number of times it fits in you will have a problem, since our standard length has a length but no width. You could fit an infinite number of them in an area.

Points 3 and 4 limit what we can do. If we are measuring the height of a door and the door has been cut by a power saw and not sanded, there may be grooves on the edges of the door of a few millimeters or more. We could not define "the height of the door" to better than that accuracy. Even if it were sanded very smooth, the door is made up of atoms -- as is our standard measuring stick. We could not break our standard measuring stick into pieces less than a nanometer in size in order to count how many fit against the door. Nor could we measure the distance to the moon with a measuring stick. We need to find other operational definitions to extend our measurement to these regimes.

Joe Redish 7/7/11

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