Question

if W=1/X is the inverse of X. show that W=w±Δw with w=1/x and Δw/w= Δx/x. Thus, the percentage error of the inverse of a quantity is the same as that of the quantity.

Answer 1

\[\ln W=-\ln x\]
So,
\[\Delta\ln W=-\Delta\ln x\]
i.e.
\[\frac{\Delta W}{W}=-\frac{\Delta x}{x}\]

Answer 2

Another way is Wx=1,
Thus,
\[\Delta (Wx)=0\]
but then,
\[\Delta (Wx)=W\Delta x+x\Delta W=0\]
which again means that,
\[\frac{\Delta W}{W}=-\frac{\Delta x}{x}\]