# Error propagation when you take the inverse?

• homestar
In summary, when taking the inverse of a value, the error remains the same as the original value. This can be calculated using the percentage error formula and the uncertainty in any function of one variable can be determined using the formula from Taylor's book "An introduction to error analysis".
homestar
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?

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.

## 1. What is error propagation when taking the inverse?

Error propagation when taking the inverse is the process of determining how errors in the input variables of a function will affect the output when the function involves taking the inverse of one or more of the input variables. This is important in scientific research, as it allows for a better understanding of the accuracy and reliability of experimental data.

## 2. How is error propagation calculated when taking the inverse?

Error propagation when taking the inverse is typically calculated using the formula: Δf/f = |df/dx| * Δx/x, where Δf is the error in the output, f is the function, |df/dx| is the absolute value of the derivative of f with respect to x, Δx is the error in the input, and x is the input variable. This formula is based on the assumption that the errors in the input variables are independent and follow a normal distribution.

## 3. Why is error propagation important in scientific research?

Error propagation is important in scientific research because it allows for a more accurate and thorough analysis of experimental data. By understanding how errors in the input variables affect the output of a function, researchers can determine the precision and reliability of their results. This information is crucial in making conclusions and drawing insights from scientific data.

## 4. What are some common sources of error in error propagation when taking the inverse?

Some common sources of error in error propagation when taking the inverse include measurement errors, rounding errors, and systematic errors. Measurement errors can occur due to limitations in the equipment used to collect data, while rounding errors can arise from using a finite number of digits in calculations. Systematic errors, on the other hand, are consistent errors that occur due to faulty equipment or flawed experimental design.

## 5. How can errors in error propagation when taking the inverse be minimized?

To minimize errors in error propagation when taking the inverse, it is important to use precise and accurate measurement tools and to follow careful experimental procedures. It is also essential to understand the assumptions and limitations of the error propagation formula used and to properly calculate and report the errors. In some cases, repeating experiments and averaging results can also help reduce errors in error propagation.

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