# relative error relation

Let $x$ be a non-null quantity. Let $\hat{x}$ be its approximation. I am trying to find the relation between: $\frac{\left | x-\hat{x} \right |}{\left | x \right |}$ and $\frac{\left | x-\hat{x} \right |}{\left | \hat{x} \right |}$?

According to what I understood, I did the following:

$\frac{\left | x-\hat{x} \right |}{\left | x \right |}=\frac{\left | x-\hat{x} \right |}{\left | \hat{x} \right |}\frac{\left | \hat{x} \right |}{\left | x \right |}$

but, I am not sure if this is how the problem is supposed to be solved. Any ideas? Thanks

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If it is a good approximation, they may be equal to the required precision. You would like $\left | x-\hat{x} \right | \ll |x|$ –  Ross Millikan Sep 19 '12 at 4:28
@RossMillikan: I didn't understand what you mean by your comment. Can you elaborate? –  C. Lambda Sep 19 '12 at 4:31
Say $x$ is $1001$ and $\hat x$ is $1000$. $\frac 1{1001}$ and $\frac 1{1000}$ are very close to each other. If you are using an approximation, you probably don't care about the difference. –  Ross Millikan Sep 19 '12 at 4:38