Why physicists care about significant figures? I came across the concept of significant figures through physics books not math books, surprisingly. I know the rules of them and they are trivial but why physicists care much about them? Why the topic is introduced in physics textbooks at the beginning while in a typical Calculus or Linear Algebra textbooks, there is nothing regarding significant figures even though numbers are critical in determining for example a solution for a linear system or a domain of a function in calculus. Can any one show a case where ignoring significant figures can lead to catastrophic consequences?    
 A: What physicists (and indeed all quantitative scientists) care about is understanding and communicating the precision (or lack there of) of measurements, and of values calculated on the basis of measurements.
It's one thing to say that is is "about a mile" from one end of campus to the other, and a very different thing to suggest that you could set a world record by running the distance 3 minutes 43.12 second (as of mid 2016). For the latter to be true we would have to know that the distance run was a mile to within less than an inch (and that the path was level and the wind was within parameters, but that's another story).
Significant figures are an attempt to offer a low precision (but also low overhead) start on the job of communicating the precision of measurements. This is the job of the rules about which figures are to be considered significant and how you write values down.
The second task of a error-system is to deal with the results of computations. How fast were you going on average during that hypothetical record setting mile run? This is the task of the rules about how many (or which) digits are still significant after a computation.
Significant figure are a very rough but relatively low-overhead means of doing something that approximates both tasks of a error-asignment-and-tracking system. There are much better—more consistent and precise—means of tracking uncertainties, but they all impose more overhead.
A: The story is told of a guard at the Egyptology exhibit of the British National Museum, who was asked how old a particular mummy was. 
"3011 years, sir."
"Really? How can you tell?"
"I started work here 11 years ago, and on my first day on the job I heard the Director say that that mummy was 3000 years old."
As has been mentioned, significant figures are a crude tool for error estimation, but they have the virtue of being a quick and easy first approximation, and prevent the drawing of silly conclusions.
Is a measurement reported as 1/10 larger, equal to, or smaller than 0.1000001? If you answered anything other than "I haven't the faintest idea", you ignored the fact that using 1/10 is a pretty sure tipoff that the measurement was probably not very precise, and so no conclusions can be reliably drawn. 
