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:
![Sigfig 1](http://blog.dotphys.net/wp-content/uploads/2008/09/sigfig-1.jpg)

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?

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