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I am trying to find the length of a hypotenuse with error, when the measurements of the two legs have an error.

So for this equation $$C = \sqrt{A^2+B^2}$$

when $A$ and $B$ each have an error of $\sigma_A$ and $\sigma_B$ respectively.

Sangchul Lee
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unknownnine9
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  • Please add some context. I would like to see this problem stated better so that it doesn't get closed and deleted, because I would like to undelete my answer. – robjohn Jun 30 '20 at 18:34
  • there is no need of context, as the tag is error-propagation the term error is clear. – L F Jun 30 '20 at 21:34
  • @LuisFelipe: Please follow the link I provided in my last comment and read what "context" I am talking about. This question already has 3 close votes against it. I am simply trying to help keep this question from being closed and possibly deleted. The reason for all 3 close votes is "lack of context". – robjohn Jun 30 '20 at 21:41
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    The reality of MSE is that if the questions is from "low level maths" and seems not effort to resolve question, then is flagget as "low quality" and then closed, But if the question is of high level maths, like galois theory, quotient space etc, adn there is just question but not effort to answer, then that questions will have a lot of upvotes, anwers and comments like "improve your question/ missing context, etc". This is because users in this side believees that every new user only wants to do their homework and leave. – L F Jun 30 '20 at 21:50
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    @LuisFelipe: that is not completely true. I have seen many high level PSQs (problem statement questions), closed and deleted as well. I do not know about the galois theory questions, as I don't answer those questions, but it could be that the people who patrol for PSQs do not watch all topics equally. – robjohn Jun 30 '20 at 21:54

1 Answers1

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Using multivariable Taylor series we have $$\sqrt{(A\pm\sigma_A)^2+(B\pm\sigma_B)^2}=\sqrt{A^2+B^2}\pm\left(\sigma_A\frac{A}{\sqrt{A^2+B^2}}+\sigma_B\frac{B}{\sqrt{A^2+B^2}}\right)+O(\sigma_A^2+\sigma_B^2)$$ So one could say for small $\sigma_A$ and $\sigma_B$ that the error in $C$ is $$\sigma_C\approx\sigma_A\frac{A}{\sqrt{A^2+B^2}}+\sigma_B\frac{B}{\sqrt{A^2+B^2}}$$

Peter Foreman
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