Error propagation when you take the inverse?

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Discussion Overview

The discussion centers on the topic of error propagation when taking the inverse of a value, specifically in the context of a voltage measurement with an associated uncertainty. Participants explore mathematical approaches and concepts related to how errors are affected by inversion.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant poses a question about the error associated with the inverse of a voltage measurement, specifically asking for clarification on how to calculate it.
  • Another participant provides a mathematical derivation related to the inverse function, assuming that the error is small compared to the value.
  • Some participants suggest using percentage error as a method to determine the error in the inverse, indicating that it remains consistent with the original measurement.
  • A participant expresses confusion and requests further elaboration with examples to clarify the mathematical concepts being discussed.
  • One participant calculates the error in the original voltage measurement and applies it to the inverse, suggesting that the percentage error remains the same, leading to a specific numerical error for the inverse value.
  • Another participant introduces a general formula for uncertainty in functions of one variable, stating that the error remains unchanged when taking the inverse.

Areas of Agreement / Disagreement

Participants express differing levels of understanding regarding the mathematical principles involved, with some agreeing on the use of percentage error while others seek clarification. The discussion does not reach a consensus on the best approach or interpretation of the error propagation in this context.

Contextual Notes

Some assumptions are made regarding the size of the error relative to the value, and there is a reliance on specific mathematical formulations that may not be universally accepted or understood by all participants.

Who May Find This Useful

This discussion may be useful for individuals interested in error analysis, particularly in the context of measurements in physics or engineering, as well as those looking to understand mathematical approaches to error propagation.

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Say something is a value +/- .05. What happens when you take the inverse of the value? For example, 30 V +/- .05 V. 1/V...what would the error be?
 
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This is a math question. 1/(x+y)=1/(x(1+y/x)).=.(1/x)(1-y/x)=1/x-y/x2.

The assumption is|y|<<|x|, .=. means approx =

I'll let you do the arithmetic.
 
When you take the inverse, use % error. That is the same for the inverse as for the original.
 
Sorry, I have the same qns but i don't get what both of you are saying, elaborate with example? thanks
 
In the original question, the error in V is 0.05 V or (0.05/30)*100% = 0.1667%.

1/V = 0.0333 V^{-1}. The error in this is also 0.1667%, or about 0.0000556 V^{-1}.
 
The uncertainty in any function of one variable is \delta y = \left|\frac{dy}{dx}\right| \delta x. If y = x^n (in your case n = -1), then \frac{\delta y}{|y|} = |n| \frac{\delta x}{|x|}. For your case, the error is unchanged.

Taylor's book "An introduction to error analysis" is well worth reading.
 

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