If i take the inverse of a value, what happens to its uncertainty?
For example, if I have C +/- deltC and then i do 1/C. What happens to delta C?
Thank you
Well, if you take the inverse of a value, it's like flipping it upside down. So, if the original value had an uncertainty, it's going to feel a little uncertain about its new direction too! The uncertainty, delta C, gets flipped as well when you take the inverse. So, be prepared for a less certain outcome when dealing with the inverse. It's like trying to dance the tango with two left feet!
When you take the inverse of a value, the uncertainty or error associated with that value will also change. To determine the uncertainty of the inverse, you can use the formula for the propagation of uncertainty.
Let's say you have a value C with an uncertainty ΔC, and you want to calculate the inverse 1/C.
The formula for the uncertainty of the inverse is given by:
Δ(1/C) = |(d(1/C)/dC)| * ΔC,
where | | denotes the absolute value and d(1/C)/dC represents the derivative of the inverse with respect to C.
Taking the derivative, we get:
d(1/C)/dC = -1/C².
Now, substitute this value into the uncertainty formula:
Δ(1/C) = |-1/C²| * ΔC.
Therefore, the uncertainty of the inverse 1/C is given by:
Δ(1/C) = |1/C²| * ΔC,
which can also be written as:
Δ(1/C) = ΔC / |C|².
In conclusion, to find the uncertainty of the inverse, you divide the original uncertainty by the square of the value.
When you take the inverse of a value, the uncertainty associated with that value (represented by delta C) changes in the following manner:
1. If C is a fixed value with no uncertainty, taking the inverse of C (1/C) will also be a fixed value with no uncertainty. In this case, delta C remains zero.
2. If C has uncertainty (C +/- delta C), taking the inverse of C (1/C) can result in a change in delta C. The change depends on the magnitude of the uncertainty and the value of C. To analyze this situation, consider the relative uncertainty, which is the ratio of delta C to C. Let's denote the relative uncertainty as delta C/C.
When taking the inverse of C, the relative uncertainty of 1/C can be estimated using the formula:
(delta (1/C))/(1/C) = (delta C)/C
This equation states that the relative uncertainty of 1/C is equal to the relative uncertainty of C. So, the absolute uncertainty of 1/C can be calculated as:
delta (1/C) = (delta C)/C^2
In simple terms, the uncertainty of the inverse value is equal to the uncertainty of the original value divided by the square of the original value.
To summarize, when you take the inverse of a value, the uncertainty associated with that value changes by becoming smaller in magnitude relative to the original value. The larger the value of C, the smaller the uncertainty of 1/C will be. Conversely, the smaller the value of C, the larger the uncertainty of 1/C will be.
It is a little more complicated.
First calculate relative uncertainity.
If A=N+-a
then relative uncertainity = a/N
and your report this as relative uncertainity.
Now if you are looking to avoid propagating error,
If Result= A/B or AxB
relativeuncertainity/result=sqrt(a^2/A^2 + b^2/B^2 )