Monday 11 May 2020

Why do you add percentage uncertainties?

This is a very regularly asked question. You are multiplying two readings together and yet you ADD the percentage uncertainties. Why?
Here's how it goes. Suppose that we are calculating w by multiplying two readings called x and y. So w=xy. The absolute uncertainties are shown below.
 Since the absolute uncertainties are small, we say that delta(x).delta(y) is negligibly small and ignore it. Also, since w=xy, that cancels on both sides. Now divide what remains through by w but remember that w=xy.
 Look! The fractional uncertainty in w = fractional uncertainty in x ADD fractional uncertainty in y.
Finally multiply through by 100 and you'll see that when you multiply x and y, you ADD their percentage uncertainties.
THE SAME APPLIES FOR DIVIDING so eg if w=x/y, we still ADD the %Us.