Significant Figures in Equations: Applying the Rules of Precision in Physics

[jk4]

I'm sorry if this doesn't fit the proper format for this section of the forum but it seems like the best place to ask this to me.

I know all the basic rules of significant figures, however I'm just curious about how it should be applied in equations. For example say I was trying to solve:
$$\sqrt{1-\frac{(0.99c)^{2}}{c^{2}}}$$

I get $$0.99^{2} = 0.9801$$ which I would then round to 0.98. Then:
1 - 0.98 = 0.02
So now, the addition/subtraction significant figure rules say that I only take 2 digits past the decimal place which will give me 0.02 of course, which is very different from 0.020
so am I now supposed to use only 1 significant figure? Or do I solve the entire equation before applying significant figures?
If the later is the case then wouldn't this present big problems because there are usually several different ways to find an answer in physics so different people with different methods would get different answers...

 

[Hurkyl]

Keep in mind that significant figures are just a weak approximation to error analysis -- their goal is not to do things right, but to do things in an easy way that's usually good enough.

Anyways, 1 - 0.98 = 0.02 is what the method of significant figures tells you to do. You've obliterated most of your precision by subtracting two numbers that are nearly equal, so you should expect the difference to have poor precision.


Incidentally, your example is not solving an equation -- you're merely computing the value of an arithmetic expression.

 

[jk4]

because I actually got that example out of a textbook and when I compute the entire problem similar to the way I was doing above my final answer is 300m due to significant figures right?
Well the book reports 210m and if I perform the calculations in a calculator without using any sig figs whatsoever I get 210.4m so that's where they get their answer from, so what am I doing wrong? I need to know what is the acceptable way to use significant figures when solving problems like this. Or is the book wrong?

 

[kamerling]

I would compute the entire problem and only round at the end. Significant figures are a rough approximation of the real precision, especially if the error is large, and rounding before the end only makes it worse.
You can always compute the expression for both the minimum and maximum values of your independent variable(s). In this case v=0.985 and v=0.995 this would tell you that your expression is between 0.099 and 0.173 Using significant figures you'd compute sqrt(0.02) = 0.1414 and if you then round it to 1 significant figure you get 0.1 which should mean that the answer is between 0.05 and 0.15.