Uncertainty

Problem 2

When measuring the rectangle's length and width and when calculating the rectangle's area you had to make several decisions:

- How do you define the rectangle? Should you measure its length and width from the outside edges of the border, from the inside edges or from the middle of the border?
- Should you estimate the rectangle's length and width to the nearest millimeter, or should you estimate between the millimeter markings and report the length and width to the nearest tenth of a millimeter?
- To how many decimal places should you report the area?

The first of these decisions is important because the border has a measurable thickness of approximately 1 mm. How you define the rectangle, therefore, affects your measurement of its length and width.

Task 1. Suppose a rectangle's "official" definition includes the entire border. If an analyst measures the rectangle's length using the border's inside edge, has he or she made a determinate or indeterminate error? Explain.

Task 2. Your second decision is important because estimating introduces uncertainty. If you report an object's length as 154 mm, you indicate absolute certainty for the digits in the hundred's and ten's place and uncertainty about the digit in the one's place. If the uncertainty in the last digit is ±1 mm, for example, then the length might be as small as 153 mm and as large as 155 mm. On the other hand, if you reported an object's length as 154.3 mm and the uncertainty is ±0.2 mm, then the length is between 154.1 mm and 154.5 mm.

You may recall that the we refer to the digits in our measurements as significant figures. Basically, a significant figure is any number in which we can express confidence, including those digits known exactly and the one digit whose value is an estimate. The lengths 154 mm and 154.3 mm have three and four significant digits, respectively. The number of significant figures in a measurement is important because it affects the number of significant figures in a result based on that measurement.

Suppose the length and width of a rectangle are 154 mm and 32 mm, what is the rectangle's area to the correct number of significant figures? What if the length is 154.3 mm and the width is 31.8 mm? Click here for a review of significant figures.

When you are finished with these tasks, proceed to Problem 3.