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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 $x_T$?

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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.

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I think that you are confusing two different things.

First you describe (accurately) a way to determine the number of significant figures that the approximation is correct to.

Then you describe a way to determine the number of significant figures that the "true value" is correct to (independent of the approximate value).

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m=1 means that your approximate x value 28.271 is accurate upto 1st decimal place hence having 3 significant digit i.e the 1st three digits of approx. value are same as of the True value 28.254.

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