Suppose that $\tilde x$ is an approximation to x. Given that a necessary and sufficient condition for $n$ correct decimal places in $\tilde x$ is $|x − \tilde x| < 0.5 \times 10^{−n}$, so, how to derive a sufficient condition for $n$ correct significant figures in $\tilde x$, in which the upper bound on the relative error does not depend on $x$.
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1I don't really get what you are aiming for here. In particular what is the shape of the bound you would like to obtain? – user70925 May 18 '20 at 13:05
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You can't. Relative error is defined as error divided by the actual value. If $x \approx 10^9$ you can get five significant figures with an absolute error of $5\cdot 10^4$. If $x \approx 10^{-9}$ to get five significant figures you need an absolute error of $5 \cdot 10^{-14}$
Ross Millikan
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