Question on finding the relative error

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.

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

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 $$\sigma_f/f$$.

• 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. – Ashok Sharma Apr 30 at 15:27
• @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 – probably_someone Apr 30 at 15:38