3.2 Significant figures
Measurements are not infinitely precise. For example, if you measure something with a ruler that has markings every eighth of an inch, and the measurement lies between two markings, you need to estimate the length. If you perform a calculation involving this measurement, your final answer is only going to be as precise as the ruler you used to make the measurement.
Now, remember that every number that goes into a physics calculation represents a real physical quantity that was measured with some device or another. So, the result of any calculation is inherently limited by the precision of the tools used to get the measurements. In practical terms, this is to say that the string of decimals your calculator gives you as a final result is mostly meaningless.
The term significant figures refers to the rules we use to determine how to handle this limitation. A figure (or digit) is significant if it has real meaning in regards to measurement. Different scientific disciplines—and even different groups of researchers within the same discipline—have different ways of defining what “significant” means in practice. A general rule of thumb is the result of a calculation is rounded to the number of significant figures of the least precise measurement that went into the calculation. You will certainly come across situations where there are more strict rules than this. The important thing is that everyone involved uses the same set of rules for determining significant figures. (In fact, you will probably also run across situations where the rules are less strict that this; it is not uncommon in introductory physics to just round all answers to three significant figures, no matter what went into the calculation.)
If you have never encountered significant figures previously, it can be difficult to wrap your head around—numbers are numbers, right? The thing to keep in mind is that everything we do in physics needs to relate back to some physical quantity that must be measured. Here are some examples to help you figure it out:
Example 3.2
Add two following lengths together: 2.5 cm and 4.509 cm.
You would get 2.5 + 4.509 = 7.009, but result is not correct. The first number has two significant figures, and the second has four. Since the least precise measurement only had 2 significant figures, we need to round our result to two significant figures as well, giving us 7.0 cm as the correct result.
In general, a zero after a decimal place is significant. So, the zero in the number 4.509 cm and the zero in 7.0 cm would both be significant. However, leading zeros are not significant. For example, if we write 4.509 cm as 0.04509 m, the two zeros at the very beginning are not significant, but the zero between the 5 and 9 is significant.
Example 3.3
Add the two lengths: 0.0025 cm and 0.045 09 cm.
Your calculator would give you 0.0025 + 0.04509 = 0.04759, but your calculator knows nothing about significant figures. The first value has two significant figures, and the second has four significant figures. Again, the least precise value is still only known to two significant figures. So our result needs to be rounded to just two significant figures, and we have 0.048 cm.
In this example, the zero between the 5 and 9 in 0.04509cm was significant, but none of the other zeros were.
Example 3.4
Multiply 12.4571 m/s and 13.28 s.
The calculation is (12.4571)(13.28) = 165.430288
The first value has six significant figures and the second one has four. Our answer is 164.4 m, which has four significant figures.
Example 3.5
Divide 195.0 m by 42 s.
The calculation is \(\frac{195.0}{42} = 4.642857143\)
The fist number has four significant figures, and the second only has two; we need to round our result to two significant figures, giving us 4.6 m/s.