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The question is as under:

Find the relative error in Z if

  1. Z= A/B
  2. Z= A(B^-1)

    Relative error in A is ΔA/A and in B is ΔB/B

My solutions are :

  1. As A and B are being divided the relative error in Z is the sum of the relative errors in A and B.

ΔZ/Z = ΔA/A + ΔB/B

  1. B is raised to power -1. Then the relative error in Z will be ΔZ/Z = ΔA/A - ΔB/B

I have written - ΔB/B because when a physical quantity is raised to a power then the relative error will be the product of power and the relative error in the original quantity.

Now the cases 1 and 2 give the same meaning but the relative errors are different. Why is this so.

Please give me the explanation.

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  • $\begingroup$ Your expression for compounding relative error is incorrect. $\endgroup$ – garyp Apr 30 at 14:52
  • $\begingroup$ @garyp Please tell me the correct expression $\endgroup$ – Ashok Sharma Apr 30 at 15:27
  • $\begingroup$ The correct expression is in the answer by probably_someone $\endgroup$ – garyp Apr 30 at 22:00
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This:

the relative error in Z is the sum of the relative errors in A and B.

is incorrect.

This:

when a physical quantity is raised to a power then the relative error will be the product of power and the relative error in the original quantity.

is also incorrect.

When you have a function $f(A,B)$ of two variables, then the uncertainty in $f$ (labeled $\sigma_f$) is related to the uncertainties $\sigma_A$ and $\sigma_B$ in $A$ and $B$ in the following way:

$$\sigma_f^2=\left(\frac{\partial f}{\partial A}\right)^2\sigma_A^2+\left(\frac{\partial f}{\partial B}\right)^2\sigma_B^2$$

Inserting $f(A,B)=A/B$ and $f(A,B)=AB^{-1}$ into the above formula will give you the same results for $\sigma_f$, and therefore will give you the same results for the relative error $\sigma_f/f$.

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  • $\begingroup$ Thank you Sir. But I have read in many books the rules which I stated. Is the rule you have stated a general rule ? Please tell me the name of this rule or something related so that I can Google it to learn more. $\endgroup$ – Ashok Sharma Apr 30 at 15:27
  • $\begingroup$ @AshokSharma This is the most general rule that I know of. You might want to re-check your books - for example, the books I've read say that when two quantities are multiplied or divided, the relative errors are added in quadrature. Adding $A$ and $B$ in quadrature gives you $\sqrt{A^2+B^2}$ (this is just the definition of the term). In general, the rules for propagation of uncertainty can be found all over the internet, for example: lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm $\endgroup$ – probably_someone Apr 30 at 15:38

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