Significant figures for special functions (square roots)

[yucheng]

TL;DR

I am trying to decide which scheme to follow....

I am using square roots, however, I am confused over how many significant figures (s.f.) to keep.
Suppose I have $\sqrt{3.0}$, which has 2 s.f.

From three different sources, I'll put a summary in brackets:
https://www.kpu.ca/sites/default/files/downloads/signfig.pdf
(if 2 s.f. in the data, keep 3 s.f. in the result; this also appeared in the answers for Kleppner's Introduction to Mechanics)
http://cda.morris.umn.edu/~mcintogc/classes/modern/sigfig.htm
(if 2 s.f., keep 2s.f.)
https://math.stackexchange.com/ques...cant-figures-involving-radicals-and-exponents
(if 2 s.f., keep 2s.f.)

So, which is correct (not for high school, but for Physics in general)? Thanks in advance.

 

[Dale]

For physics in general none of the approaches is correct. Significant figures is a rule of thumb for students.

In a scientific paper you would report your uncertainty explicitly. So if you measured something to be 3.0 with a standard uncertainty of 0.2 then you would report it as $3.0 \pm 0.2$ or more concisely $3.0(2)$.

You would use the propagation of errors formula to report the error after the square root.

 

[jbriggs444]

I am using square roots, however, I am confused over how many significant figures (s.f.) to keep.
Suppose I have $\sqrt{3.0}$, which has 2 s.f.

If you use the propagation of errors formula, you will find that taking the square root cuts the relative error in half. This buys you 3 dB -- 30% of one significant digit. [If you know x to plus or minus 1 percent, you know $\sqrt{x}$ to plus or minus half a percent].

Conversely, squaring a number worsens the relative error by a factor of two. This costs you 30% of a significant digit. [If you know know x to plus or minus 1 percent, you only know $x^2$ to plus or minus 2 percent].