# Number of significant figures relative to true value of x

I've been stuck on this relatively simple matter for a while and I'd really appreciate some insight into what the actual answer should be. Say I'm given an approximate x value $x_A = 28.271$, and a true x value $x_T = 28.254$.

Finding the error of $x_A$ relative to $x_T$ would simply be $x_T - x_A = -0.017$

My textbook says that to find the number of significant figures would be to compare this error with $x_T$ and find the $mth$ value where the zeros end.

i.e $\space$$(x_T)= 28.254$

$E(x_A) = -00.017$

Counting where the zeroes end is at the $3rd$ value, so $3$ significant figures. Simple.

Yet my prof says to use the concept where the error should be less than half the $mth$ digit of $x_T$ to find the number of significant figures.

i.e |$E(x_A)$| $\leq$ ${1\over2} 10^{-m}$

So $0.017 \leq {1\over2} 10^{-1} = 0.05$.

Yet how does $m = -1$ relate to $x_T$?

I think what your professor meant was to take the $m$th digit of $x_T$ and turn all other digits into 0, and then cut this value in half, and then as soon as your error is greater than this amount then you say you have $m-1$ significant figures.