How Is Uncertainty Calculated for (a+b)/(c+d) When a=b=c=d?

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Homework Help Overview

The discussion centers on calculating the uncertainty of the expression \(\frac{a+b}{c+d}\) under the condition that \(a = b = c = d\) and that their uncertainties are equal, denoted as \(\sigma_a = \sigma_b = \sigma_c = \sigma_d\). Participants are exploring the implications of this setup on the resulting uncertainty.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to derive the uncertainty using known formulas for the propagation of uncertainty, leading to a specific expression for \(\sigma_{\frac{a+b}{c+d}}\). Some participants question the validity of the result and its implications regarding the relationship between the uncertainty of the ratio and that of a single measurement.

Discussion Status

There is a mix of agreement and questioning among participants regarding the calculations presented. While one participant expresses confidence in the result, another raises concerns about its interpretation, suggesting that the uncertainties in ratios should be considered differently than in single measurements. This indicates an ongoing exploration of the topic without a clear consensus.

Contextual Notes

Participants are discussing the assumptions related to equal values and uncertainties for \(a\), \(b\), \(c\), and \(d\), which may influence the interpretation of the results. The mention of evaluating a different expression, \(\frac{a+b+c}{e+f+g}\), suggests a consideration of how changes in the variables might affect the uncertainty calculations.

lavster
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Homework Statement


what is the uncertainty of \frac{a+b}{c+d} if a=b=c=d and σ_a = σ_b =σ_c=_d

Homework Equations



σ_{a+b}=√(σ_a^2+σ_b^2)

σ_{\frac{a}{b}}=√((\frac{σ_a}{a})^2+(\frac{σ_b}{b}^2))

The Attempt at a Solution


since a=b=c=d

<br /> σ_{a+b}=√2 σ_a

σ_{\frac{a}{b}}=√2 \frac{σ_a}{a}

so σ_{\frac{a+b}{c+d}} = \frac{√2 √2 σ _a}{2a} = \frac{σ_a}{a}

is this correct?!

Thanks
 
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The result looks good to me!
 
Great thanks :) any idea why it is the same as the uncertainty in a single measurement... Just doesn't seem right to me! Xxx
 
It is NOT the same as the uncertainty in a single measurement! That would be ##\sigma_a##. Since you are evaluating a ratio, only relative errors matter. The factors ##\sqrt 2## and 2 just happen to cancel.

You can repeat the exercise with ##{a+b+c}\over e+f+g## and see what happens...
 

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