131 C1.02: Reporting
Section 1. Reporting Measurements
The following examples use the metric system for measuring lengths. The ideas are applicable to measurement in other contexts, such as volume or area, and using the English system as well as the metric system. These examples were chosen because it is easier to convey precision with decimals than fractions, and the metric system makes it easy to use decimals. Look at your ruler and notice that the centimeter is divided into tenths (each of those is called a millimeter) and the inch is divided into either eighths or sixteenths. So when we want to make measurements with a ruler smaller than one centimeter, we can easily use decimals, but when we want to make measurements smaller than one inch with a ruler, it is natural to use fractions rather than decimals.
Suppose we are using a ruler to measure the lengths of some pieces of cardboard and we give each measurement, along with a phrase describing how precisely we measured it.
| Measurement | ||
| 12 cm, correct to the nearest cm. | ||
| 18.2 cm, correct to the nearest tenth of a cm. | ||
| 33 cm, correct to the nearest tenth of a cm. | ||
| 30 cm, correct to the nearest cm. | ||
| Estimate of 20 cm, correct to the nearest ten cm. | ||
| Estimate of 180 cm, correct to the nearest ten cm. | ||
| Estimate of 300 cm, correct to the nearest ten cm. | ||
| Estimate of 300 cm, correct to the nearest hundred cm. |
It would be more convenient to give each number in a way that conveys the precision instead of having to write the phrase afterwards. The generally accepted method for doing that is to report exactly the same number of digits as were observed. That is straightforward in many situations. It is somewhat less straightforward when the most precise digit observed is a “trailing zero.” But that is also easily taken care of for the third number in our list, 33 cm, rounded to the nearest cm. Here we use a trailing zero after the decimal place. In arithmetic courses, we learned that when we compute with the numbers 33 and 33.0, we will obtain the same results. So if someone writes 33.0 instead of 33, they must be intending to convey something more than just the exact value of 33. They are conveying that the number is approximately 33 and also conveying the precision of that approximation.
The fourth number in our list, 30 cm, correct to the nearest cm, can be written as 30. cm. When we explicitly include a decimal we indicate that each of the digits before the decimal was measured precisely. I think of this as only a fairly clear report rather than a clear report, because a decimal point at the end of the number is so easily overlooked or forgotten in copying the number.
| Measurement | Completely clear correct report | Fairly clear correct report |
| 12 cm, correct to the nearest cm. | 12 cm | |
| 18.2 cm, correct to the nearest tenth of a cm. | 18.2 cm | |
| 33 cm, correct to the nearest tenth of a cm. | 33.0 cm | |
| 30 cm, correct to the nearest cm