I am reading the book "Experimentation : an introduction to measurement theory and experiment design" by D. C. Baird and I have come across a paragraph where it says that the absolute uncertainty in an indirect measurement cannot have more significant figures than that of the original direct measurements. Since there are different ways to achieve an estimation of the final uncertainty (and the author has previously explained a calculus-based approach to do so) I wonder why this additional rule regarding significant figures is needed.
I leave the aforementioned paragraph down below. For further clarification, I feel that there is nothing wrong with the expression $R = 9.06 \, \pm \, 0.59 \, \Omega$ because the original uncertainties in $V$ and $I$ have already been taken into consideration when computing the $0.59 \, \Omega$ uncertainty.
Because computations tend to produce answers consisting of long strings of numbers, we must be careful to quote the final answer sensibly. If, for example, we are given the voltage across a resistor as $15.4 ± 0.1 $ V and the current as $1.7 ± 0.1 $ A, we can calculate a value for the resistance. The ratio $V/I$ comes out on my calculator as $9.0588235 \, \Omega$. Is this the answer? Clearly not. A brief calculation shows that the absolute uncertainty in the resistance is close to $0.59 \, \Omega$. So, if the first two places of decimals in the value for the resistance are uncertain, the rest are clearly meaningless. A statement like $R = 9.0588235 ± 0.59 \, \Omega$ is, therefore, nonsense. We should quote our results in such a way that the answer and its uncertainty are consistent, perhaps something like $R = 9.06 ± 0.59 \, \Omega$. But is even this statement really valid? Remember that the originally quoted uncertainties for $V$ and $I$ had the value $±0.1$, containing one significant figure. If we do not know these uncertainties any more precisely than that, we have no right to claim two significant figures for the uncertainty in $R$. Our final, valid, and self-consistent statement is, therefore, $R = 9.1 ± 0.6 \, \Omega$. Only if we had a good reason to believe that our original uncertainty was accurate to two significant figures, could we lay claim to two significant figures in the final uncer tainty and a correspondingly more precisely quoted value for $R$.