Correction for uncertainty of multiplications and divisions

Question

The conventional means of obtaining uncertainty of $c$ where $c = a \cdot b$ is adding percentage uncertainty of $a$ and $b$. This method seems to have a flaw as shown below (please excuse me if I am yet to get to the correction in my studies).

You have a square and a very bad ruler and want to find the area of the square. You find that the dimensions of the square are $L = 4\;\mathrm{cm} \pm 2\;\mathrm{cm}$ and $W = 4\;\mathrm{cm} \pm 2\;\mathrm{cm}$. The percentage uncertainty for each of these measurements is $50\%$, you add the two $50\%$ uncertainties to get $100\%$ percentage uncertainty. $$ A = 16\;\mathrm{cm}^2 \pm 16\;\mathrm{cm}^2 $$ I was wondering if there was a method of correction to avoid problems like these arising.

Answer 1

The problem you encounter is that you are applying the rule that is applicable to "linear addition" of uncertainty, to non-linear applications (multiplication).

Valid application .. $(4 \frac{+}{-}2)$ + $(8 \frac{+}{-}3) = 12 \frac{+}{-}5 $ (7 -> 17 range)
Invalid application $(4 \frac{+}{-}2)$ x $(8 \frac{+}{-}3) = 32 \frac{+}{-}? $

To obtain the correct answer, when multiplying, you have to use the rules of multiplication, which will give you: $(4 \frac{+}{-}2)$ x $(8 \frac{+}{-}3) = 32 \frac{+34}{-22} $ (10 -> 66 range).