Wednesday, 29 January 2020

error analysis - Has my textbook given the incorrect equation for calculating uncertainty in multiplication?



I am 99% sure that the calculation to obtain the uncertainty of two multiplied values given by my textbook and this university (http://web.uvic.ca/~jalexndr/192UncertRules.pdf) is incorrect.


They both say this:


$$(A \pm a)(B \pm b) = AB \pm (\varepsilon_A + \varepsilon_B)$$


Which should be equivalent to (correct me if I'm wrong):


$$(A \pm a)(B \pm b) = AB \pm (AB (\frac{a}{A} + \frac{b}{B})) = AB \pm (Ba + Ab))$$


Lets consider: $(4 \pm 1)(2 \pm 1)$ This should equal (8 \pm 6) right?


meaning the smallest value possible is 2 while the largest is 14. However we can see that the largest possible value is actually 15 (5 * 3) and the smallest is 3 (3 * 1). If this is plot on a 3D graph you will see there is no way to obtain lower or larger values than 3 and 15. Therefore there must be a problem with the formula. After some math I came up with the following equation which I believe is the correct equation:


$$(A \pm a)(B \pm b) = (AB + ab) ± (Ab + Ba))$$


As long as $A ≥ a$ and $B ≥ b$ and $a, b ≥ 0$.


Is the equation in the textbook (and given by this university) incorrect or have I just missed something?





No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...