*Class Content I* > *Modeling with mathematics* > *Using math in science*

2.1.2

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 means that we have 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: Simply put, we need to count how many times our standard length can fit into the length to be measured.
- If what we want to measure is smaller than our standard unit, we break the standard unit into equal fractions and then compare the length we are trying to measure to those standard fractions.

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.
- 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.

Point 1 is usually OK (as long as we are not near black holes...).

Points 2 and 3 limit what we can do. For living systems such as cells or animals, the length of an object can expand and contract so living systems generally do not have an exact length. We can still measure a length at each time, but it would not make sense to determine it down to an atomic scale.

Even if the object is inanimate, the length of an object is usually not exactly identifiable. 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 three foot measuring stick. We need to find other operational definitions to extend our measurement to these regimes. And we need to be aware that at whatever scale we are measuring, a measured concept such as "length" is not an exact number — not even in principle.

**Follow ons:**

Joe Redish 7/7/11

Wolfgang Losert 8/31/2013

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