Uncertainty propagation for sum divided by product

[lapo3399]

For the following calculation, would the uncertainty propagate as I have estimated?


$$\frac{\left(a \pm \Delta a \right) + \left(b \pm \Delta b \right)}{\left(c \pm \Delta c \right) \cdot \left(d \pm \Delta d \right)} = \frac{a+b}{cd} \pm \frac{a+b}{cd} \left( \frac{\Delta a + \Delta b}{a + b} + \frac{\Delta c}{c} + \frac{\Delta d}{d} \right)$$

Thanks.

 

[learningphysics]

Yes, looks good to me.

 

[lapo3399]

Thanks!

One more quick question - is the following uncertainty propagation correct also?

$$\frac{1}{ \left( r \pm \Delta r \right)} = \frac{1}{r} \pm \frac{1}{ \Delta r}$$

Thanks again.

 

[learningphysics]

Thanks!

One more quick question - is the following uncertainty propagation correct also?

$$\frac{1}{ \left( r \pm \Delta r \right)} = \frac{1}{r} \pm \frac{1}{ \Delta r}$$

Thanks again.

That doesn't look right to me. To get the relative uncertainty of the fraction... add the relative uncertainty of the top (0)... to the relative uncertainty of the bottom...

So the relative uncertainty of the fraction seems to be $$\frac{\Delta r}{r}$$ so the absolute uncertainty would be $$\frac{1}{r}\times \frac{\Delta r}{r}$$