# Correction for uncertainty of multiplications and divisions

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.

• The possible results range from 4 to 36, not 0 to 32. Commented Apr 19, 2016 at 4:09

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).