It is necessary to say a few words about a perennial source of difficulty: significant figures. Mathematics is a subject where numbers and relationships can be as precise as desired, but physics deals with a real world of ambiguity. It is important in science and engineering to state clearly what you know about a situation—no less and, especially, no more. Numbers provide one way to specify your knowledge.

If you report that a length has a value of $6.2\text{m,}$ the implication is that the actual value falls between $6.15\text{m}$ and $6.25\text{m}$ and thus rounds to $6.2\text{m}$. If that is the case, then reporting a value of simply 6 m is saying less than you know; you are withholding information. On the other hand, to report the number as $6.213\text{m}$ is wrong. Any person reviewing your work—perhaps a client who hired you—would interpret the number $6.213\text{m}$ as meaning that the actual length falls between $6.2125\text{m}$ and $6.2135\text{m,}$ thus rounding to $6.213\text{m}$. In this case, you are claiming to have knowledge and information that you do not really possess.

The way to state your knowledge precisely is through the proper use of significant figures. You can think of a significant figure as being a digit that is reliably known. A number such as $6.2\text{m}$ has two significant figures because the next decimal place—the one-hundredths—is not reliably known. As FIGURE 1.22 shows, the best way to determine how many significant figures a number has is to write it in scientific notation.

FIGURE 1.22 Determining significant figures.

What about numbers like 320 m and 20 kg? Whole numbers with trailing zeros are ambiguous unless written in scientific notation. Even so, writing $2.0\times {10}^1\text{kg}$ is tedious, and few practicing scientists or engineers would do so. In this textbook, we’ll adopt the rule that whole numbers always have at least two significant figures, even if one of those is a trailing zero. By this rule, 320 m, 20 kg, and 8000 s each have two significant figures, but 8050 s would have three.

Calculations with numbers follow the “weakest link” rule. The saying, which you probably know, is that “a chain is only as strong as its weakest link.” If nine out of ten links in a chain can support a 1000 pound weight, that strength is meaningless if the tenth link can support only 200 pounds. Nine out of the ten numbers used in a calculation might be known with a precision of 0.01%; but if the tenth number is poorly known, with a precision of only 10%, then the result of the calculation cannot possibly be more precise than 10%.

Proper use of significant figures is part of the “culture” of science and engineering. We will frequently emphasize these “cultural issues” because you must learn to speak the same language as the natives if you wish to communicate effectively. Most students know the rules of significant figures, having learned them in high school, but many fail to apply them. It is important to understand the reasons for significant figures and to get in the habit of using them properly.