Significant figures what are they for and what do they have to do with uncertainty?

Suppose I am working on a problem and I wish to calculate the density of something. I measure the mass to be *m* = 24.5 grams and the volume is *V* = 10 cm³. In this case the density would be:

ALERT! ALERT! ALERT! This is not a test!!!! Something is drastically wrong! Clearly I messed up. How can I have the mass measured to **3** significant figures, the volume measured to **1** significant figure, but the density calculated to **3** significant figures? Isn't this a violation of some fundamental principle that could be worse than the Large Hardron Collider coming online and destroying the world?

No, we can all calm down. This really isn't a big deal. Unfortunately many people (*hint* like chemists) do get all freaky about significant figures. Now calm down chemists. I am not saying significant figures are entirely stupid. They do have a purpose. What I AM saying is that they are not some fundamental rule that can not be broken.

So what is the deal with significant figures?

Suppose I want to measure the time it takes for an apple to fall off a tree and hit the ground. I pay two 8^th graders to sit in front of the tree with stop watches. The first 8^th grader times how long until the apple falls using his watch and the second 8^th grader uses the stopwatch to measure the time the apple falls from the tree to the ground. 8th grader 1 measures a time of 4 hours, 23 minutes for the apple to leave the tree. The second 8th grader uses the stopwatch to measure the falling time and reads 0.9345 seconds. So, what is the total time. These 8th graders are pretty smart. They know that both the times have to be in seconds to add them. 8th grader 1 converts his time to seconds and gets:


And of course, the second 8th grader gets:


These students then decide to report a time of:


Hopefully, we can all see how silly this is. The point is: why report the time to such a large number of significant figures if some of the data is insignificant. Thus, the goal of significant figures is to somehow give an estimate of the uncertainty in a calculation. Really, there is a better way to do this but all ways are just an approximation.

And here is my problem with significant figures. I believe that many people (you know who you are) either treat significant figures as some fundamental truth, or they haven't thought about them one way or the other. This is a huge problem when you are sig-fig-stickler.

There is a better way of dealing with the idea that measurements are not exact. One way is to include an uncertainty value with any measurement (or calculation). In this way, the time for the apple to fall from the tree to the ground could be reported as:


This says that the value of time is somewhere between:


(well, at least it is MOST likely to be in this range)