How Should Uncertainties Be Calculated When Multiplying Values Near Zero?

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SUMMARY

The discussion centers on calculating uncertainties when multiplying values A (±a) and B (±b) that approach zero. The correct formula for determining the uncertainty in the product C (±c) is c = A * B * sqrt((a/A)² + (b/B)²), which addresses the issue of infinite uncertainty as values approach zero. Participants suggest adding a small constant to A or B to mitigate divide-by-zero errors. Simulations indicate that while the formula works for one value nearing zero, it fails when both A and B are close to zero.

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Homework Statement


Dear All,

Suppose I had to values: A (±a) and B (±b) where A not equal to a and B not equal to b. I have to find the product of A and B taking into consideration uncertainties.


Homework Equations




A (±a) * B (±b) = C (±c) where c = (a/A+b/B)

The Attempt at a Solution



My question would be: Suppose A and B represented change in mass of a sample. As A and/or B tended to zero the uncertainty would tend to infinite. Am I doing anything wrong?

max
 
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I think your formula for c is suspect. I have seen and used c= A*B*sqrt((a/A)2 + (b/B)2).

To avoid the divide by zero issue, add a small number to A or B (much smaller than a or b).

Your question is intriguing. This formula seems to work if either A or B is (almost) zero, but if A and B are both (almost) zero, the results do not compare very well, based on simulations.
 

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