# Measurement

#### Prerequisite

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 through an **operational definition**. This means that we have a procedure for assigning a number to a physical quantity.

## An example of an operational definition: *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 (for example, an inch or a centimeter).
- Count how many times our standard length can fit into the length to be measured (for example, 3 inches or 12 centimeters). If there's some length left over that is less than our standard unit,
- 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 this definition.

- Our standard doesn't change its size over time or when we move it from one place to another.
- 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.

Why would we bother about saying point 1. Isn't that obvious? Well, its usually OK. But we might imagine a ruler that has a large coefficient of thermal expansion not behaving well if we did one measurement in a hot room and one in a cold room. Our ruler would be of (slightly) different sizes in the two rooms. (This actually becomes interesting when we are in curved spacetime 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.

## Measurements are not perfect numbers

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 nanometer or extra-terrestrial 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.

This doesn't mean that you can be sloppy with your measurements!

*In science we need to be as accurate as is needed for what we are trying to do — but we need to understand how certain we are of our measurements. *

One thing it means is that you need to pay careful attention to significant (and insignificant) figures!

Joe Redish and Wolfgang Losert 8/31/13

#### Follow-ons

Last Modified: February 19, 2021