Rounding up to $3$ significant figures when adding If $a,\ b$ and $c$ are real numbers and you are required to find $a + b + c$ to $3$ significant figures, to how many significant figures could $a,\ b$ and $c$ be rounded up to to give the result?

In general, if you want $\mathrm{round}_3(a+b+c) = \mathrm{round}_{3+k}(a)+\mathrm{round}_{3+k}(b)+\mathrm{round}_{3+k}(c)$ you should make $k$ as large as you possibly can. 
A simple example is $a=499001,b=499.001,c=0.499001$.


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*If you don't do any rounding before addition you get $\mathrm{round}_3(a+b+c) = \mathrm{round}_3(499001 + 499.001 + 0.499001) = \mathrm{round}_3(499500.500001) = 500000$.

*If you round $a,b,c$ before addition with $k<3$ you get $\mathrm{round}_3(a+b+c) = \mathrm{round}_{3+k}(499000)+\mathrm{round}_{3+k}(499)+\mathrm{round}_{3+k}(0.499) = \mathrm{round}_3(499499.499)=499000$
This non-intuitive behaviour results from the fact that rounding functions "jump", i.e. they are unstable at the point where they round up or down.
